3.4.98 \(\int \frac {\sec ^3(c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [398]

Optimal. Leaf size=747 \[ -\frac {b^{5/3} \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )} \]

[Out]

-1/4*(a+7*b)*ln(1-sin(d*x+c))/(a+b)^3/d+1/4*(a-7*b)*ln(1+sin(d*x+c))/(a-b)^3/d+1/9*b^(5/3)*(4*a^2+3*a^(4/3)*b^
(2/3)+2*b^2)*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(5/3)/(a^2-b^2)^2/d+1/3*b^(5/3)*(3*b^(2/3)*(3*a^2+b^2)+4*a^(2/3)
*(a^2+2*b^2))*ln(a^(1/3)+b^(1/3)*sin(d*x+c))/a^(1/3)/(a^2-b^2)^3/d-1/18*b^(5/3)*(4*a^2+3*a^(4/3)*b^(2/3)+2*b^2
)*ln(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(5/3)/(a^2-b^2)^2/d-1/6*b^(5/3)*(3*b^(2/3)*(3*
a^2+b^2)+4*a^(2/3)*(a^2+2*b^2))*ln(a^(2/3)-a^(1/3)*b^(1/3)*sin(d*x+c)+b^(2/3)*sin(d*x+c)^2)/a^(1/3)/(a^2-b^2)^
3/d+2/3*a*b*(a^2+5*b^2)*ln(a+b*sin(d*x+c)^3)/(a^2-b^2)^3/d+1/4/(a+b)^2/d/(1-sin(d*x+c))-1/4/(a-b)^2/d/(1+sin(d
*x+c))-1/3*b*(a*(a^2+2*b^2)-b*sin(d*x+c)*(2*a^2+b^2-3*a*b*sin(d*x+c)))/a/(a^2-b^2)^2/d/(a+b*sin(d*x+c)^3)-1/9*
b^(5/3)*(4*a^2-3*a^(4/3)*b^(2/3)+2*b^2)*arctan(1/3*(a^(1/3)-2*b^(1/3)*sin(d*x+c))/a^(1/3)*3^(1/2))/a^(5/3)/(a^
2-b^2)^2/d*3^(1/2)-1/3*b^(5/3)*(4*a^(8/3)-9*a^2*b^(2/3)+8*a^(2/3)*b^2-3*b^(8/3))*arctan(1/3*(a^(1/3)-2*b^(1/3)
*sin(d*x+c))/a^(1/3)*3^(1/2))/a^(1/3)/(a^2-b^2)^3/d*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.68, antiderivative size = 747, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3302, 2099, 1868, 1874, 31, 648, 631, 210, 642, 1885, 266} \begin {gather*} -\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )\right )}{3 a d \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^3}-\frac {b^{5/3} \left (-3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (8 a^{2/3} b^2+4 a^{8/3}-9 a^2 b^{2/3}-3 b^{8/3}\right ) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} d \left (a^2-b^2\right )^3}-\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2-b^2\right )^3}+\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} d \left (a^2-b^2\right )^2}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2-b^2\right )^3}+\frac {1}{4 d (a+b)^2 (1-\sin (c+d x))}-\frac {1}{4 d (a-b)^2 (\sin (c+d x)+1)}-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac {(a-7 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

-1/3*(b^(5/3)*(4*a^2 - 3*a^(4/3)*b^(2/3) + 2*b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]
)/(Sqrt[3]*a^(5/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(4*a^(8/3) - 9*a^2*b^(2/3) + 8*a^(2/3)*b^2 - 3*b^(8/3))*ArcTan[
(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 - b^2)^3*d) - ((a + 7*b)*Log[1 -
Sin[c + d*x]])/(4*(a + b)^3*d) + ((a - 7*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (b^(5/3)*(4*a^2 + 3*a^(4/
3)*b^(2/3) + 2*b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2)^2*d) + (b^(5/3)*(3*b^(2/3)*(3*
a^2 + b^2) + 4*a^(2/3)*(a^2 + 2*b^2))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(1/3)*(a^2 - b^2)^3*d) - (b^(5
/3)*(4*a^2 + 3*a^(4/3)*b^(2/3) + 2*b^2)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/
(18*a^(5/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(3*b^(2/3)*(3*a^2 + b^2) + 4*a^(2/3)*(a^2 + 2*b^2))*Log[a^(2/3) - a^(1
/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(1/3)*(a^2 - b^2)^3*d) + (2*a*b*(a^2 + 5*b^2)*Log[a +
b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^3*d) + 1/(4*(a + b)^2*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)^2*d*(1 + Sin[c +
d*x])) - (b*(a*(a^2 + 2*b^2) - b*Sin[c + d*x]*(2*a^2 + b^2 - 3*a*b*Sin[c + d*x])))/(3*a*(a^2 - b^2)^2*d*(a +
b*Sin[c + d*x]^3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1868

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 3302

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps

\begin {align*} \int \frac {\sec ^3(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{4 (a+b)^2 (-1+x)^2}+\frac {-a-7 b}{4 (a+b)^3 (-1+x)}+\frac {1}{4 (a-b)^2 (1+x)^2}+\frac {a-7 b}{4 (a-b)^3 (1+x)}+\frac {b^2 \left (2 a^2+b^2-3 a b x+\left (a^2+2 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^2 \left (a+b x^3\right )^2}+\frac {b^2 \left (4 a \left (a^2+2 b^2\right )-3 b \left (3 a^2+b^2\right ) x+2 a \left (a^2+5 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^3 \left (a+b x^3\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}+\frac {b^2 \text {Subst}\left (\int \frac {4 a \left (a^2+2 b^2\right )-3 b \left (3 a^2+b^2\right ) x+2 a \left (a^2+5 b^2\right ) x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac {b^2 \text {Subst}\left (\int \frac {2 a^2+b^2-3 a b x+\left (a^2+2 b^2\right ) x^2}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {b^2 \text {Subst}\left (\int \frac {4 a \left (a^2+2 b^2\right )-3 b \left (3 a^2+b^2\right ) x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}-\frac {b^2 \text {Subst}\left (\int \frac {-2 \left (2 a^2+b^2\right )+3 a b x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a \left (a^2-b^2\right )^2 d}+\frac {\left (2 a b^2 \left (a^2+5 b^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {b^{5/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-3 \sqrt [3]{a} b \left (3 a^2+b^2\right )+8 a \sqrt [3]{b} \left (a^2+2 b^2\right )\right )+\sqrt [3]{b} \left (-3 \sqrt [3]{a} b \left (3 a^2+b^2\right )-4 a \sqrt [3]{b} \left (a^2+2 b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (3 a^{4/3} b-4 \sqrt [3]{b} \left (2 a^2+b^2\right )\right )+\sqrt [3]{b} \left (3 a^{4/3} b+2 \sqrt [3]{b} \left (2 a^2+b^2\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (b^2 \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 a^{4/3} \left (a^2-b^2\right )^2 d}-\frac {\left (b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^2 \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right )\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{2 \left (a^2-b^2\right )^3 d}-\frac {\left (b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right )\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}\\ &=-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (b^{5/3} \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {\left (b^{5/3} \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} \left (a^2-b^2\right )^3 d}\\ &=-\frac {b^{5/3} \left (4 a^2-3 a^{4/3} b^{2/3}+2 b^2\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (4 a^{8/3}-9 a^2 b^{2/3}+8 a^{2/3} b^2-3 b^{8/3}\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3 d}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3 d}+\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right )^2 d}+\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}-\frac {b^{5/3} \left (4 a^2+3 a^{4/3} b^{2/3}+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^3 d}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {1}{4 (a+b)^2 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^2 d (1+\sin (c+d x))}-\frac {b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2+b^2-3 a b \sin (c+d x)\right )\right )}{3 a \left (a^2-b^2\right )^2 d \left (a+b \sin ^3(c+d x)\right )}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 6.30, size = 657, normalized size = 0.88 \begin {gather*} \frac {-\frac {(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3}+\frac {(a-7 b) \log (1+\sin (c+d x))}{4 (a-b)^3}+\frac {4 \sqrt [3]{a} b^{5/3} \left (a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3}-\frac {2 \sqrt [3]{a} \left (a^2+2 b^2\right ) \left (2 \sqrt {3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )\right )}{3 \left (a^2-b^2\right )^3}+\frac {\left (2+\frac {b^2}{a^2}\right ) \left (2 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-\sqrt [3]{a} \left (2 \sqrt {3} b^{5/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )\right )\right )}{9 \left (a^2-b^2\right )^2}+\frac {2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3}+\frac {1}{4 (a+b)^2 (1-\sin (c+d x))}-\frac {3 b^3 \left (3 a^2+b^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{2 a \left (a^2-b^2\right )^3}-\frac {3 b^3 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{2 a \left (a^2-b^2\right )^2}-\frac {1}{4 (a-b)^2 (1+\sin (c+d x))}-\frac {b \left (a^2+2 b^2\right )}{3 \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}+\frac {a b^2 \left (2+\frac {b^2}{a^2}\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x]^3)^2,x]

[Out]

(-1/4*((a + 7*b)*Log[1 - Sin[c + d*x]])/(a + b)^3 + ((a - 7*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3) + (4*a^(1/
3)*b^(5/3)*(a^2 + 2*b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*(a^2 - b^2)^3) - (2*a^(1/3)*(a^2 + 2*b^2)*(2*
Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b
^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2]))/(3*(a^2 - b^2)^3) + ((2 + b^2/a^2)*(2*a^(1/3)*b^(5/3)*Log[a^(1
/3) + b^(1/3)*Sin[c + d*x]] - a^(1/3)*(2*Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^
(1/3))] + b^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])))/(9*(a^2 - b^2)^2) +
(2*a*b*(a^2 + 5*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^3) + 1/(4*(a + b)^2*(1 - Sin[c + d*x])) - (3*b^
3*(3*a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(2*a*(a^2 - b^2)^3) -
(3*b^3*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/(2*a*(a^2 - b^2)^2) - 1/(4*(a -
b)^2*(1 + Sin[c + d*x])) - (b*(a^2 + 2*b^2))/(3*(a^2 - b^2)^2*(a + b*Sin[c + d*x]^3)) + (a*b^2*(2 + b^2/a^2)*
Sin[c + d*x])/(3*(a^2 - b^2)^2*(a + b*Sin[c + d*x]^3)))/d

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Maple [A]
time = 2.84, size = 483, normalized size = 0.65

method	result	size
derivativedivides	\(\frac {\frac {b^{2} \left (\frac {\left (-a^{2} b +b^{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )+\frac {\left (2 a^{4}-a^{2} b^{2}-b^{4}\right ) \sin \left (d x +c \right )}{3 a}-\frac {a^{4}+a^{2} b^{2}-2 b^{4}}{3 b}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (8 a^{4}+11 a^{2} b^{2}-b^{4}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (-15 a^{3} b -3 a \,b^{3}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {2 \left (3 a^{4}+15 a^{2} b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{9 b}}{a}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{4 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -7 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{3}}}{d}\)	\(483\)
default	\(\frac {\frac {b^{2} \left (\frac {\left (-a^{2} b +b^{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )+\frac {\left (2 a^{4}-a^{2} b^{2}-b^{4}\right ) \sin \left (d x +c \right )}{3 a}-\frac {a^{4}+a^{2} b^{2}-2 b^{4}}{3 b}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (8 a^{4}+11 a^{2} b^{2}-b^{4}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (-15 a^{3} b -3 a \,b^{3}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {2 \left (3 a^{4}+15 a^{2} b^{2}\right ) \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{9 b}}{a}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {1}{4 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -7 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -7 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{3}}}{d}\)	\(483\)
risch	\(\text {Expression too large to display}\)	\(1831\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(b^2/(a-b)^3/(a+b)^3*(((-a^2*b+b^3)*sin(d*x+c)^2+1/3*(2*a^4-a^2*b^2-b^4)/a*sin(d*x+c)-1/3*(a^4+a^2*b^2-2*b
^4)/b)/(a+b*sin(d*x+c)^3)+2/3/a*((8*a^4+11*a^2*b^2-b^4)*(1/3/b/(1/b*a)^(2/3)*ln(sin(d*x+c)+(1/b*a)^(1/3))-1/6/
b/(1/b*a)^(2/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3/b/(1/b*a)^(2/3)*3^(1/2)*arctan(1/3
*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))+(-15*a^3*b-3*a*b^3)*(-1/3/b/(1/b*a)^(1/3)*ln(sin(d*x+c)+(1/b*a)^(1/3
))+1/6/b/(1/b*a)^(1/3)*ln(sin(d*x+c)^2-(1/b*a)^(1/3)*sin(d*x+c)+(1/b*a)^(2/3))+1/3*3^(1/2)/b/(1/b*a)^(1/3)*arc
tan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))+1/3*(3*a^4+15*a^2*b^2)/b*ln(a+b*sin(d*x+c)^3)))-1/4/(a-b)^2/(
1+sin(d*x+c))+1/4*(a-7*b)/(a-b)^3*ln(1+sin(d*x+c))-1/4/(a+b)^2/(sin(d*x+c)-1)+1/4/(a+b)^3*(-a-7*b)*ln(sin(d*x+
c)-1))

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Maxima [A]
time = 0.51, size = 788, normalized size = 1.05 \begin {gather*} -\frac {\frac {8 \, \sqrt {3} {\left (5 \, a^{3} b^{2} {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} - a^{2} b^{3} {\left (11 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {10 \, a}{b}\right )} - 2 \, a^{4} b {\left (4 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {a}{b}\right )} + 3 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, a^{5}\right )} \arctan \left (-\frac {\sqrt {3} {\left (\left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{{\left (a^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{5} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {4 \, {\left (a^{2} b^{3} {\left (30 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 11\right )} + 2 \, a^{4} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4\right )} - 15 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}} + b^{5}\right )} \log \left (\sin \left (d x + c\right )^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{5} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {8 \, {\left (a^{2} b^{3} {\left (15 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 11\right )} + a^{4} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 8\right )} + 15 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}} - b^{5}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 3 \, a^{5} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + 3 \, a^{3} b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {9 \, {\left (a - 7 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {9 \, {\left (a + 7 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {6 \, {\left (3 \, {\left (a^{3} b + 3 \, a b^{3}\right )} \sin \left (d x + c\right )^{4} - 8 \, a^{3} b - 4 \, a b^{3} - 2 \, {\left (5 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{3} + 2 \, {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{2} + {\left (3 \, a^{4} + 7 \, a^{2} b^{2} + 2 \, b^{4}\right )} \sin \left (d x + c\right )\right )}}{a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} - {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )^{2}}}{36 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

-1/36*(8*sqrt(3)*(5*a^3*b^2*(3*(a/b)^(2/3) + 2) - a^2*b^3*(11*(a/b)^(1/3) + 10*a/b) - 2*a^4*b*(4*(a/b)^(1/3) +
a/b) + 3*a*b^4*(a/b)^(2/3) + b^5*(a/b)^(1/3) + 2*a^5)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b
)^(1/3))/((a^7*(a/b)^(2/3) - 3*a^5*b^2*(a/b)^(2/3) + 3*a^3*b^4*(a/b)^(2/3) - a*b^6*(a/b)^(2/3))*(a/b)^(1/3)) -
4*(a^2*b^3*(30*(a/b)^(2/3) - 11) + 2*a^4*b*(3*(a/b)^(2/3) - 4) - 15*a^3*b^2*(a/b)^(1/3) - 3*a*b^4*(a/b)^(1/3)
+ b^5)*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(a^7*(a/b)^(2/3) - 3*a^5*b^2*(a/b)^(2/3)
+ 3*a^3*b^4*(a/b)^(2/3) - a*b^6*(a/b)^(2/3)) - 8*(a^2*b^3*(15*(a/b)^(2/3) + 11) + a^4*b*(3*(a/b)^(2/3) + 8) +
15*a^3*b^2*(a/b)^(1/3) + 3*a*b^4*(a/b)^(1/3) - b^5)*log((a/b)^(1/3) + sin(d*x + c))/(a^7*(a/b)^(2/3) - 3*a^5*b
^2*(a/b)^(2/3) + 3*a^3*b^4*(a/b)^(2/3) - a*b^6*(a/b)^(2/3)) - 9*(a - 7*b)*log(sin(d*x + c) + 1)/(a^3 - 3*a^2*b
+ 3*a*b^2 - b^3) + 9*(a + 7*b)*log(sin(d*x + c) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 6*(3*(a^3*b + 3*a*b^3)
*sin(d*x + c)^4 - 8*a^3*b - 4*a*b^3 - 2*(5*a^2*b^2 + b^4)*sin(d*x + c)^3 + 2*(a^3*b - a*b^3)*sin(d*x + c)^2 +
(3*a^4 + 7*a^2*b^2 + 2*b^4)*sin(d*x + c))/(a^6 - 2*a^4*b^2 + a^2*b^4 - (a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x + c
)^5 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x + c)^3 - (a^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c)^2))/d

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Fricas [C] Result contains complex when optimal does not.
time = 6.24, size = 15989, normalized size = 21.40 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm="fricas")

[Out]

-1/36*(18*a^5*b - 36*a^3*b^3 + 18*a*b^5 - 18*(a^5*b + 2*a^3*b^3 - 3*a*b^5)*cos(d*x + c)^4 + 48*(a^5*b + a^3*b^
3 - 2*a*b^5)*cos(d*x + c)^2 + 2*((a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*cos(d*x + c)^2 - ((a^7*b - 3*a^5*b^
3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c)^4 - (a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*cos(d*x + c)^2)*sin(d*x +
c))*(3*4^(2/3)*(-I*sqrt(3) + 1)*(3*(a^3*b + 5*a*b^3)^2/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^2 - (3*a^2*
b^2 - b^4)/(a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2))/((27*a^2*b^3 - b^5)/(a^11*d^3 - 3*a^9*b^2*
d^3 + 3*a^7*b^4*d^3 - a^5*b^6*d^3) + 54*(a^3*b + 5*a*b^3)^3/(a^6*d - 3*a^4*b^2*d + 3*a^2*b^4*d - b^6*d)^3 - 27
*(a^3*b + 5*a*b^3)*(3*a^2*b^2 - b^4)/((a^8*d^2 - 3*a^6*b^2*d^2 + 3*a^4*b^4*d^2 - a^2*b^6*d^2)*(a^6*d - 3*a^4*b
^2*d + 3*a^2*b^4*d - b^6*d)) + (512*a^6 + 273*a^4*b^2 - 30*a^2*b^4 + b^6)*b^5/((a^2 - b^2)^6*a^5*d^3))^(1/3) +
4^(1/3)*(I*sqrt(3) + 1)*((27*a^2*b^3 - b^5)/(a^11*d^3 - 3*a^9*b^2*d^3 + 3*a^7*b^4*d^3 - a^5*b^6*d^3) + 54*(a^
3*b + 5*a* ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**3/(a+b*sin(d*x+c)**3)**2,x)

[Out]

Timed out

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Giac [A]
time = 0.59, size = 790, normalized size = 1.06 \begin {gather*} \frac {\frac {8 \, {\left (15 \, a^{10} b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 42 \, a^{8} b^{6} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 36 \, a^{6} b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{4} b^{10} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 3 \, a^{2} b^{12} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 8 \, a^{11} b^{3} + 13 \, a^{9} b^{5} + 10 \, a^{7} b^{7} - 28 \, a^{5} b^{9} + 14 \, a^{3} b^{11} - a b^{13}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | -\left (-\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a^{15} b - 6 \, a^{13} b^{3} + 15 \, a^{11} b^{5} - 20 \, a^{9} b^{7} + 15 \, a^{7} b^{9} - 6 \, a^{5} b^{11} + a^{3} b^{13}} + \frac {8 \, {\left (3 \, {\left (5 \, \sqrt {3} a^{3} b + \sqrt {3} a b^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} + {\left (8 \, \sqrt {3} a^{4} b + 11 \, \sqrt {3} a^{2} b^{3} - \sqrt {3} b^{5}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (\left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} - \frac {4 \, {\left (3 \, {\left (5 \, a^{3} b + a b^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} - {\left (8 \, a^{4} b + 11 \, a^{2} b^{3} - b^{5}\right )} \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x + c\right ) + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}} + \frac {24 \, {\left (a^{3} b + 5 \, a b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {9 \, {\left (a - 7 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {9 \, {\left (a + 7 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {6 \, {\left (3 \, a^{3} b \sin \left (d x + c\right )^{4} + 9 \, a b^{3} \sin \left (d x + c\right )^{4} - 10 \, a^{2} b^{2} \sin \left (d x + c\right )^{3} - 2 \, b^{4} \sin \left (d x + c\right )^{3} + 2 \, a^{3} b \sin \left (d x + c\right )^{2} - 2 \, a b^{3} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + 7 \, a^{2} b^{2} \sin \left (d x + c\right ) + 2 \, b^{4} \sin \left (d x + c\right ) - 8 \, a^{3} b - 4 \, a b^{3}\right )}}{{\left (b \sin \left (d x + c\right )^{5} - b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - a\right )} {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}}{36 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3/(a+b*sin(d*x+c)^3)^2,x, algorithm="giac")

[Out]

1/36*(8*(15*a^10*b^4*(-a/b)^(1/3) - 42*a^8*b^6*(-a/b)^(1/3) + 36*a^6*b^8*(-a/b)^(1/3) - 6*a^4*b^10*(-a/b)^(1/3
) - 3*a^2*b^12*(-a/b)^(1/3) - 8*a^11*b^3 + 13*a^9*b^5 + 10*a^7*b^7 - 28*a^5*b^9 + 14*a^3*b^11 - a*b^13)*(-a/b)
^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^15*b - 6*a^13*b^3 + 15*a^11*b^5 - 20*a^9*b^7 + 15*a^7*b^9 - 6
*a^5*b^11 + a^3*b^13) + 8*(3*(5*sqrt(3)*a^3*b + sqrt(3)*a*b^3)*(-a*b^2)^(2/3) + (8*sqrt(3)*a^4*b + 11*sqrt(3)*
a^2*b^3 - sqrt(3)*b^5)*(-a*b^2)^(1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^8 -
3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6) - 4*(3*(5*a^3*b + a*b^3)*(-a*b^2)^(2/3) - (8*a^4*b + 11*a^2*b^3 - b^5)*(-a*b
^2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b
^6) + 24*(a^3*b + 5*a*b^3)*log(abs(b*sin(d*x + c)^3 + a))/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) + 9*(a - 7*b)*lo
g(abs(sin(d*x + c) + 1))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 9*(a + 7*b)*log(abs(sin(d*x + c) - 1))/(a^3 + 3*a^2
*b + 3*a*b^2 + b^3) - 6*(3*a^3*b*sin(d*x + c)^4 + 9*a*b^3*sin(d*x + c)^4 - 10*a^2*b^2*sin(d*x + c)^3 - 2*b^4*s
in(d*x + c)^3 + 2*a^3*b*sin(d*x + c)^2 - 2*a*b^3*sin(d*x + c)^2 + 3*a^4*sin(d*x + c) + 7*a^2*b^2*sin(d*x + c)
+ 2*b^4*sin(d*x + c) - 8*a^3*b - 4*a*b^3)/((b*sin(d*x + c)^5 - b*sin(d*x + c)^3 + a*sin(d*x + c)^2 - a)*(a^5 -
2*a^3*b^2 + a*b^4)))/d

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Mupad [B]
time = 15.75, size = 1605, normalized size = 2.15 \begin {gather*} \frac {\sum _{k=1}^3\ln \left (\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (-\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (\frac {-\frac {63\,a^{10}\,b^4}{2}+\frac {4153\,a^8\,b^6}{12}+325\,a^6\,b^8-\frac {1017\,a^4\,b^{10}}{4}+\frac {32\,a^2\,b^{12}}{3}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (\frac {\frac {27\,a^{13}\,b^3}{2}-239\,a^{11}\,b^5+188\,a^9\,b^7+303\,a^7\,b^9-\frac {563\,a^5\,b^{11}}{2}+16\,a^3\,b^{13}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )\,\left (\frac {180\,a^{14}\,b^4-684\,a^{12}\,b^6+936\,a^{10}\,b^8-504\,a^8\,b^{10}+36\,a^6\,b^{12}+36\,a^4\,b^{14}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {\sin \left (c+d\,x\right )\,\left (5832\,a^{15}\,b^3-5832\,a^{13}\,b^5-34992\,a^{11}\,b^7+81648\,a^9\,b^9-64152\,a^7\,b^{11}+17496\,a^5\,b^{13}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )-\frac {\sin \left (c+d\,x\right )\,\left (2916\,a^{12}\,b^4+50004\,a^{10}\,b^6-96660\,a^8\,b^8+30780\,a^6\,b^{10}+13824\,a^4\,b^{12}-864\,a^2\,b^{14}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )-\frac {\sin \left (c+d\,x\right )\,\left (10449\,a^9\,b^5+31542\,a^7\,b^7-68247\,a^5\,b^9+7200\,a^3\,b^{11}-384\,a\,b^{13}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )+\frac {-20\,a^7\,b^5+\frac {847\,a^5\,b^7}{3}+\frac {2173\,a^3\,b^9}{27}+\frac {32\,a\,b^{11}}{27}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {\sin \left (c+d\,x\right )\,\left (-9234\,a^6\,b^6-29860\,a^4\,b^8+4758\,a^2\,b^{10}+64\,b^{12}\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )+\frac {\frac {10\,a^4\,b^6}{3}+\frac {122\,a^2\,b^8}{27}+\frac {28\,b^{10}}{27}}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {\sin \left (c+d\,x\right )\,\left (2568\,a^3\,b^7+1080\,a\,b^9\right )}{108\,\left (a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8\right )}\right )\,\mathrm {root}\left (2187\,a^9\,b^2\,z^3-2187\,a^7\,b^4\,z^3+729\,a^5\,b^6\,z^3-729\,a^{11}\,z^3+7290\,a^6\,b^3\,z^2+1458\,a^8\,b\,z^2-972\,a^5\,b^2\,z+324\,a^3\,b^4\,z+216\,a^2\,b^3-8\,b^5,z,k\right )}{d}+\frac {\frac {b\,{\sin \left (c+d\,x\right )}^2}{3\,\left (a^2-b^2\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {a^2\,b}{2}+\frac {3\,b^3}{2}\right )}{a^4-2\,a^2\,b^2+b^4}-\frac {2\,b\,\left (2\,a^2+b^2\right )}{3\,{\left (a^2-b^2\right )}^2}+\frac {\sin \left (c+d\,x\right )\,\left (3\,a^4+7\,a^2\,b^2+2\,b^4\right )}{6\,a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {5\,a^2\,b^2}{3}+\frac {b^4}{3}\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (-b\,{\sin \left (c+d\,x\right )}^5+b\,{\sin \left (c+d\,x\right )}^3-a\,{\sin \left (c+d\,x\right )}^2+a\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+7\,b\right )}{d\,\left (4\,a^3+12\,a^2\,b+12\,a\,b^2+4\,b^3\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-7\,b\right )}{d\,\left (4\,a^3-12\,a^2\,b+12\,a\,b^2-4\,b^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^3*(a + b*sin(c + d*x)^3)^2),x)

[Out]

symsum(log(root(2187*a^9*b^2*z^3 - 2187*a^7*b^4*z^3 + 729*a^5*b^6*z^3 - 729*a^11*z^3 + 7290*a^6*b^3*z^2 + 1458
*a^8*b*z^2 - 972*a^5*b^2*z + 324*a^3*b^4*z + 216*a^2*b^3 - 8*b^5, z, k)*(((32*a*b^11)/27 + (2173*a^3*b^9)/27 +
(847*a^5*b^7)/3 - 20*a^7*b^5)/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) - root(2187*a^9*b^2*z^3 -
2187*a^7*b^4*z^3 + 729*a^5*b^6*z^3 - 729*a^11*z^3 + 7290*a^6*b^3*z^2 + 1458*a^8*b*z^2 - 972*a^5*b^2*z + 324*a^
3*b^4*z + 216*a^2*b^3 - 8*b^5, z, k)*(((32*a^2*b^12)/3 - (1017*a^4*b^10)/4 + 325*a^6*b^8 + (4153*a^8*b^6)/12 -
(63*a^10*b^4)/2)/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) + root(2187*a^9*b^2*z^3 - 2187*a^7*b^4*
z^3 + 729*a^5*b^6*z^3 - 729*a^11*z^3 + 7290*a^6*b^3*z^2 + 1458*a^8*b*z^2 - 972*a^5*b^2*z + 324*a^3*b^4*z + 216
*a^2*b^3 - 8*b^5, z, k)*((16*a^3*b^13 - (563*a^5*b^11)/2 + 303*a^7*b^9 + 188*a^9*b^7 - 239*a^11*b^5 + (27*a^13
*b^3)/2)/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) + root(2187*a^9*b^2*z^3 - 2187*a^7*b^4*z^3 + 729
*a^5*b^6*z^3 - 729*a^11*z^3 + 7290*a^6*b^3*z^2 + 1458*a^8*b*z^2 - 972*a^5*b^2*z + 324*a^3*b^4*z + 216*a^2*b^3
- 8*b^5, z, k)*((36*a^4*b^14 + 36*a^6*b^12 - 504*a^8*b^10 + 936*a^10*b^8 - 684*a^12*b^6 + 180*a^14*b^4)/(a^11
+ a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2) + (sin(c + d*x)*(17496*a^5*b^13 - 64152*a^7*b^11 + 81648*a^9*b^
9 - 34992*a^11*b^7 - 5832*a^13*b^5 + 5832*a^15*b^3))/(108*(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2)
)) - (sin(c + d*x)*(13824*a^4*b^12 - 864*a^2*b^14 + 30780*a^6*b^10 - 96660*a^8*b^8 + 50004*a^10*b^6 + 2916*a^1
2*b^4))/(108*(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))) - (sin(c + d*x)*(7200*a^3*b^11 - 384*a*b^1
3 - 68247*a^5*b^9 + 31542*a^7*b^7 + 10449*a^9*b^5))/(108*(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^4 - 4*a^9*b^2))
) + (sin(c + d*x)*(64*b^12 + 4758*a^2*b^10 - 29860*a^4*b^8 - 9234*a^6*b^6))/(108*(a^11 + a^3*b^8 - 4*a^5*b^6 +
6*a^7*b^4 - 4*a^9*b^2))) + ((28*b^10)/27 + (122*a^2*b^8)/27 + (10*a^4*b^6)/3)/(a^11 + a^3*b^8 - 4*a^5*b^6 + 6
*a^7*b^4 - 4*a^9*b^2) + (sin(c + d*x)*(1080*a*b^9 + 2568*a^3*b^7))/(108*(a^11 + a^3*b^8 - 4*a^5*b^6 + 6*a^7*b^
4 - 4*a^9*b^2)))*root(2187*a^9*b^2*z^3 - 2187*a^7*b^4*z^3 + 729*a^5*b^6*z^3 - 729*a^11*z^3 + 7290*a^6*b^3*z^2
+ 1458*a^8*b*z^2 - 972*a^5*b^2*z + 324*a^3*b^4*z + 216*a^2*b^3 - 8*b^5, z, k), k, 1, 3)/d + ((b*sin(c + d*x)^2
)/(3*(a^2 - b^2)) + (sin(c + d*x)^4*((a^2*b)/2 + (3*b^3)/2))/(a^4 + b^4 - 2*a^2*b^2) - (2*b*(2*a^2 + b^2))/(3*
(a^2 - b^2)^2) + (sin(c + d*x)*(3*a^4 + 2*b^4 + 7*a^2*b^2))/(6*a*(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^3*(b
^4/3 + (5*a^2*b^2)/3))/(a*(a^4 + b^4 - 2*a^2*b^2)))/(d*(a - a*sin(c + d*x)^2 + b*sin(c + d*x)^3 - b*sin(c + d*
x)^5)) - (log(sin(c + d*x) - 1)*(a + 7*b))/(d*(12*a*b^2 + 12*a^2*b + 4*a^3 + 4*b^3)) + (log(sin(c + d*x) + 1)*
(a - 7*b))/(d*(12*a*b^2 - 12*a^2*b + 4*a^3 - 4*b^3))

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