\(\int \frac {\sec (c+d x)}{(a+b \sin ^3(c+d x))^2} \, dx\) [397]

Optimal result

Integrand size = 21, antiderivative size = 587 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=-\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} \left (a^2-b^2\right ) d}+\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3302, 2099, 1868, 1874, 31, 648, 631, 210, 642, 1885, 266} \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a d \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (-2 a^{2/3} b^{4/3}+a^2+b^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} d \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} d \left (a^2-b^2\right )}+\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2-b^2\right )^2}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2-b^2\right )}-\frac {\sqrt [3]{b} \left (2 a^{2/3} b^{4/3}+a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2-b^2\right )^2}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]

[In]

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

[Out]

-1/3*(b^(1/3)*(a^(4/3) - 2*b^(4/3))*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)*d) - (b^(1/3)*(a^2 - 2*a^(2/3)*b^(4/3) + b^2)*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)^2*d) - Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sin[c + d*x
]]/(2*(a - b)^2*d) - (b^(1/3)*(a^(4/3) + 2*b^(4/3))*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(9*a^(5/3)*(a^2 - b^2
)*d) - (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3*a^(1/3)*(a^2 - b^2)^2*
d) + (b^(1/3)*(a^(4/3) + 2*b^(4/3))*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)*d) + (b^(1/3)*(a^2 + 2*a^(2/3)*b^(4/3) + 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)^2*d) - (2*a*b*Log[a + b*Sin[c + d*x]^3])/(3*(a^2 - b^2)^2*d)
+ (b*(a - Sin[c + d*x]*(b - a*Sin[c + d*x])))/(3*a*(a^2 - b^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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{2 (a+b)^2 (-1+x)}+\frac {1}{2 (a-b)^2 (1+x)}+\frac {b \left (b-a x+b x^2\right )}{\left (-a^2+b^2\right ) \left (a+b x^3\right )^2}+\frac {b \left (-2 a b+\left (a^2+b^2\right ) x-2 a b x^2\right )}{\left (a^2-b^2\right )^2 \left (a+b x^3\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}+\frac {b \text {Subst}\left (\int \frac {-2 a b+\left (a^2+b^2\right ) x-2 a b x^2}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}-\frac {b \text {Subst}\left (\int \frac {b-a x+b x^2}{\left (a+b x^3\right )^2} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right ) d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )}+\frac {b \text {Subst}\left (\int \frac {-2 a b+\left (a^2+b^2\right ) x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{\left (a^2-b^2\right )^2 d}-\frac {\left (2 a b^2\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 )^2 d}+\frac {b \text {Subst}\left (\int \frac {-2 b+a x}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{3 a \left (a^2-b^2\right ) d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )}+\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (-4 a b^{4/3}+\sqrt [3]{a} \left (a^2+b^2\right )\right )+\sqrt [3]{b} \left (2 a b^{4/3}+\sqrt [3]{a} \left (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 )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac {b^{2/3} \text {Subst}\left (\int \frac {\sqrt [3]{a} \left (a^{4/3}-4 b^{4/3}\right )+\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\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 ) d}-\frac {\left (b \left (\frac {a^{2/3}}{\sqrt [3]{b}}+\frac {2 b}{a^{2/3}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{9 a \left (a^2-b^2\right ) d}-\frac {\left (b^{2/3} \left (a^2+2 a^{2/3} b^{4/3}+b^2\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 )^2 d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (b^{2/3} \left (\frac {a^{4/3}}{\sqrt [3]{b}}+2 b\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 ) d}+\frac {\left (b^{2/3} \left (1-\frac {2 b^{4/3}}{a^{4/3}}\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 )}{6 \left (a^2-b^2\right ) d}+\frac {\left (b^{2/3} \left (a^2-2 a^{2/3} b^{4/3}+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 )}{2 \left (a^2-b^2\right )^2 d}+\frac {\left (\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+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 )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2 d} \\ & = -\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\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 ) d}+\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+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 )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )}+\frac {\left (\sqrt [3]{b} \left (1-\frac {2 b^{4/3}}{a^{4/3}}\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 )}{3 \sqrt [3]{a} \left (a^2-b^2\right ) d}+\frac {\left (\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}+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 )}{\sqrt [3]{a} \left (a^2-b^2\right )^2 d} \\ & = -\frac {\sqrt [3]{b} \left (a^{4/3}-2 b^{4/3}\right ) \arctan \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 ) d}-\frac {\sqrt [3]{b} \left (a^2-2 a^{2/3} b^{4/3}+b^2\right ) \arctan \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 )^2 d}-\frac {\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} \left (a^2-b^2\right ) d}-\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}+\frac {\sqrt [3]{b} \left (a^{4/3}+2 b^{4/3}\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 ) d}+\frac {\sqrt [3]{b} \left (a^2+2 a^{2/3} b^{4/3}+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 )}{6 \sqrt [3]{a} \left (a^2-b^2\right )^2 d}-\frac {2 a b \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {b (a-\sin (c+d x) (b-a \sin (c+d x)))}{3 a \left (a^2-b^2\right ) d \left (a+b \sin ^3(c+d x)\right )} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 4.38 (sec) , antiderivative size = 564, normalized size of antiderivative = 0.96 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {12 \sqrt {3} \sqrt [3]{a} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\left (a^2-b^2\right )^2}+\frac {4 \sqrt {3} b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac {9 \log (1-\sin (c+d x))}{(a+b)^2}+\frac {9 \log (1+\sin (c+d x))}{(a-b)^2}-\frac {12 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{\left (a^2-b^2\right )^2}-\frac {4 b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{5/3} \left (a^2-b^2\right )}+\frac {6 \sqrt [3]{a} b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {2 b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{a^{5/3} \left (a^2-b^2\right )}-\frac {12 a b \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {9 b \left (a^2+b^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a \left (a^2-b^2\right )^2}+\frac {9 b \operatorname {Hypergeometric2F1}\left (\frac {2}{3},2,\frac {5}{3},-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{a^3-a b^2}+\frac {6 b}{\left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}-\frac {6 b^2 \sin (c+d x)}{a \left (a^2-b^2\right ) \left (a+b \sin ^3(c+d x)\right )}}{18 d} \]

[In]

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

[Out]

((12*Sqrt[3]*a^(1/3)*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^2 - b^2)^2 + (4*
Sqrt[3]*b^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(a^(5/3)*(a^2 - b^2)) - (9*Log[1
- Sin[c + d*x]])/(a + b)^2 + (9*Log[1 + Sin[c + d*x]])/(a - b)^2 - (12*a^(1/3)*b^(5/3)*Log[a^(1/3) + b^(1/3)*
Sin[c + d*x]])/(a^2 - b^2)^2 - (4*b^(5/3)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(a^(5/3)*(a^2 - b^2)) + (6*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])/(a^2 - b^2)^2 + (2*b^(5/3)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(a^(5/3)*(a^2 - b^2)) - (12*a*b*Log[a + b
*Sin[c + d*x]^3])/(a^2 - b^2)^2 + (9*b*(a^2 + b^2)*Hypergeometric2F1[2/3, 1, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin
[c + d*x]^2)/(a*(a^2 - b^2)^2) + (9*b*Hypergeometric2F1[2/3, 2, 5/3, -((b*Sin[c + d*x]^3)/a)]*Sin[c + d*x]^2)/
(a^3 - a*b^2) + (6*b)/((a^2 - b^2)*(a + b*Sin[c + d*x]^3)) - (6*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*(a + b*Sin[c
+ d*x]^3)))/(18*d)

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2}}+\frac {b \left (\frac {\left (\frac {a^{2}}{3}-\frac {b^{2}}{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )-\frac {b \left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (-4 a^{2} b +b^{3}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}+a \,b^{2}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 a^{2} \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}}{a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\)	\(389\)
default	\(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{2}}+\frac {b \left (\frac {\left (\frac {a^{2}}{3}-\frac {b^{2}}{3}\right ) \left (\sin ^{2}\left (d x +c \right )\right )-\frac {b \left (a^{2}-b^{2}\right ) \sin \left (d x +c \right )}{3 a}+\frac {a^{2}}{3}-\frac {b^{2}}{3}}{a +b \left (\sin ^{3}\left (d x +c \right )\right )}+\frac {\frac {2 \left (-4 a^{2} b +b^{3}\right ) \left (\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {2 \left (2 a^{3}+a \,b^{2}\right ) \left (-\frac {\ln \left (\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\sin ^{2}\left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{3}} \sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {2 a^{2} \ln \left (a +b \left (\sin ^{3}\left (d x +c \right )\right )\right )}{3}}{a}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}}{d}\)	\(389\)
risch	\(-\frac {i x}{a^{2}-2 a b +b^{2}}-\frac {i c}{d \left (a^{2}-2 a b +b^{2}\right )}+\frac {i x}{a^{2}+2 a b +b^{2}}+\frac {i c}{d \left (a^{2}+2 a b +b^{2}\right )}+\frac {4 i a^{6} b \,d^{3} x}{a^{9} d^{3}-2 a^{7} b^{2} d^{3}+a^{5} b^{4} d^{3}}+\frac {4 i a^{6} b \,d^{2} c}{a^{9} d^{3}-2 a^{7} b^{2} d^{3}+a^{5} b^{4} d^{3}}-\frac {2 b \left (i a \,{\mathrm e}^{5 i \left (d x +c \right )}-6 i a \,{\mathrm e}^{3 i \left (d x +c \right )}+2 b \,{\mathrm e}^{4 i \left (d x +c \right )}+i a \,{\mathrm e}^{i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 d \left (-a^{2}+b^{2}\right ) a \left (b \,{\mathrm e}^{6 i \left (d x +c \right )}-3 b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a^{2}-2 a b +b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a^{2}+2 a b +b^{2}\right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (729 a^{9} d^{3}-1458 a^{7} b^{2} d^{3}+729 a^{5} b^{4} d^{3}\right ) \textit {\_Z}^{3}+729 a^{6} b \,d^{2} \textit {\_Z}^{2}+27 a^{3} b^{2} d \textit {\_Z} +8 a^{2} b -b^{3}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (\frac {324 i a^{11} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {486 i a^{9} b^{2} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {162 i a^{5} b^{6} d^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) \textit {\_R}^{2}+\left (\frac {216 i a^{8} b d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {396 i a^{6} b^{3} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {144 i a^{4} b^{5} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {18 i a^{2} b^{7} d}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) \textit {\_R} -\frac {28 i a^{5} b^{2}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {10 i a^{3} b^{4}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {8 a^{6} b}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {28 a^{4} b^{3}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}+\frac {10 a^{2} b^{5}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}-\frac {b^{7}}{8 a^{6} b +28 a^{4} b^{3}-10 a^{2} b^{5}+b^{7}}\right )\right )\)	\(946\)

[In]

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

[Out]

1/d*(-1/2/(a+b)^2*ln(sin(d*x+c)-1)+1/2/(a-b)^2*ln(1+sin(d*x+c))+b/(a-b)^2/(a+b)^2*(((1/3*a^2-1/3*b^2)*sin(d*x+
c)^2-1/3*b*(a^2-b^2)/a*sin(d*x+c)+1/3*a^2-1/3*b^2)/(a+b*sin(d*x+c)^3)+2/3/a*((-4*a^2*b+b^3)*(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)))+(2*a^3+a*b^2)*(-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)*arctan(1/3*3^(1/2)*(2/(1/b*a)^(1/3)*sin(d*x+c)-1)))-a^2*ln(a+b*sin(d*x+c)^3))))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.60 (sec) , antiderivative size = 10855, normalized size of antiderivative = 18.49 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.82 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\frac {4 \, \sqrt {3} {\left (2 \, a^{3} {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} - 2 \, a^{2} b {\left (2 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {a}{b}\right )} + a b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{3} \left (\frac {a}{b}\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 )}{{\left (a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (2 \, a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2\right )} - 2 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + b^{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^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {4 \, {\left (a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4\right )} + 2 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - b^{3}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {6 \, {\left (a b \sin \left (d x + c\right )^{2} - b^{2} \sin \left (d x + c\right ) + a b\right )}}{a^{4} - a^{2} b^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )^{3}} + \frac {9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {9 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{18 \, d} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

sage0*x

Mupad [B] (verification not implemented)

Time = 14.82 (sec) , antiderivative size = 980, normalized size of antiderivative = 1.67 \[ \int \frac {\sec (c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {\sum _{k=1}^3\ln \left (\frac {\frac {8\,b^6}{27}-\frac {16\,a^2\,b^4}{27}}{a^7-2\,a^5\,b^2+a^3\,b^4}+\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\frac {\frac {128\,a^3\,b^5}{27}+\frac {32\,a\,b^7}{27}}{a^7-2\,a^5\,b^2+a^3\,b^4}-\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\frac {27\,a^9\,b^3+34\,a^7\,b^5-77\,a^5\,b^7+16\,a^3\,b^9}{a^7-2\,a^5\,b^2+a^3\,b^4}+\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )\,\left (\frac {180\,a^{10}\,b^4-324\,a^8\,b^6+108\,a^6\,b^8+36\,a^4\,b^{10}}{a^7-2\,a^5\,b^2+a^3\,b^4}+\frac {\sin \left (c+d\,x\right )\,\left (1458\,a^{11}\,b^3+1458\,a^9\,b^5-7290\,a^7\,b^7+4374\,a^5\,b^9\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (2484\,a^8\,b^4-1836\,a^6\,b^6-864\,a^4\,b^8+216\,a^2\,b^{10}\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\frac {388\,a^6\,b^4}{9}-\frac {353\,a^4\,b^6}{9}+\frac {64\,a^2\,b^8}{9}}{a^7-2\,a^5\,b^2+a^3\,b^4}+\frac {\sin \left (c+d\,x\right )\,\left (447\,a^5\,b^5-408\,a^3\,b^7+96\,a\,b^9\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-236\,a^4\,b^4+134\,a^2\,b^6+16\,b^8\right )}{27\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )+\frac {8\,a\,b^5\,\sin \left (c+d\,x\right )}{9\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}\right )\,\mathrm {root}\left (1458\,a^7\,b^2\,z^3-729\,a^5\,b^4\,z^3-729\,a^9\,z^3-1458\,a^6\,b\,z^2-108\,a^3\,b^2\,z-64\,a^2\,b+8\,b^3,z,k\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{d\,\left (2\,a^2+4\,a\,b+2\,b^2\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{d\,\left (2\,a^2-4\,a\,b+2\,b^2\right )}+\frac {\frac {b}{3\,\left (a^2-b^2\right )}+\frac {b\,{\sin \left (c+d\,x\right )}^2}{3\,\left (a^2-b^2\right )}-\frac {b^2\,\sin \left (c+d\,x\right )}{3\,a\,\left (a^2-b^2\right )}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^3+a\right )} \]

[In]

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

[Out]

symsum(log(((8*b^6)/27 - (16*a^2*b^4)/27)/(a^7 + a^3*b^4 - 2*a^5*b^2) + root(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^
3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3, z, k)*(((32*a*b^7)/27 + (128*a^3*b^5)/27)
/(a^7 + a^3*b^4 - 2*a^5*b^2) - root(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^
3*b^2*z - 64*a^2*b + 8*b^3, z, k)*(root(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 10
8*a^3*b^2*z - 64*a^2*b + 8*b^3, z, k)*((16*a^3*b^9 - 77*a^5*b^7 + 34*a^7*b^5 + 27*a^9*b^3)/(a^7 + a^3*b^4 - 2*
a^5*b^2) + root(1458*a^7*b^2*z^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b +
8*b^3, z, k)*((36*a^4*b^10 + 108*a^6*b^8 - 324*a^8*b^6 + 180*a^10*b^4)/(a^7 + a^3*b^4 - 2*a^5*b^2) + (sin(c +
d*x)*(4374*a^5*b^9 - 7290*a^7*b^7 + 1458*a^9*b^5 + 1458*a^11*b^3))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (sin(c
+ d*x)*(216*a^2*b^10 - 864*a^4*b^8 - 1836*a^6*b^6 + 2484*a^8*b^4))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + ((64*a
^2*b^8)/9 - (353*a^4*b^6)/9 + (388*a^6*b^4)/9)/(a^7 + a^3*b^4 - 2*a^5*b^2) + (sin(c + d*x)*(96*a*b^9 - 408*a^3
*b^7 + 447*a^5*b^5))/(27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (sin(c + d*x)*(16*b^8 + 134*a^2*b^6 - 236*a^4*b^4))/(
27*(a^7 + a^3*b^4 - 2*a^5*b^2))) + (8*a*b^5*sin(c + d*x))/(9*(a^7 + a^3*b^4 - 2*a^5*b^2)))*root(1458*a^7*b^2*z
^3 - 729*a^5*b^4*z^3 - 729*a^9*z^3 - 1458*a^6*b*z^2 - 108*a^3*b^2*z - 64*a^2*b + 8*b^3, z, k), k, 1, 3)/d - lo
g(sin(c + d*x) - 1)/(d*(4*a*b + 2*a^2 + 2*b^2)) + log(sin(c + d*x) + 1)/(d*(2*a^2 - 4*a*b + 2*b^2)) + (b/(3*(a
^2 - b^2)) + (b*sin(c + d*x)^2)/(3*(a^2 - b^2)) - (b^2*sin(c + d*x))/(3*a*(a^2 - b^2)))/(d*(a + b*sin(c + d*x)
^3))