3.4.87 \(\int \frac {\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [387]

Optimal. Leaf size=385 \[ -\frac {b^{5/3} \left (2 a^2-3 a^{4/3} b^{2/3}+b^2\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \left (a^2-b^2\right )^2 d}-\frac {b^{5/3} \left (2 a^2+3 a^{4/3} b^{2/3}+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \left (a^2-b^2\right )^2 d}+\frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^2 d}+\frac {1}{4 (a+b) d (1-\sin (c+d x))}-\frac {1}{4 (a-b) d (1+\sin (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3302, 2099, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} \frac {b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (-3 a^{4/3} b^{2/3}+2 a^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 )}{\sqrt {3} a^{2/3} d \left (a^2-b^2\right )^2}-\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+2 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 a^{2/3} d \left (a^2-b^2\right )^2}+\frac {b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} d \left (a^2-b^2\right )^2}+\frac {1}{4 d (a+b) (1-\sin (c+d x))}-\frac {1}{4 d (a-b) (\sin (c+d x)+1)}-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac {(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^(5/3)*(2*a^2 - 3*a^(4/3)*b^(2/3) + b^2)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))])/(Sq
rt[3]*a^(2/3)*(a^2 - b^2)^2*d)) - ((a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a - 4*b)*Log[1 + Sin[c
+ d*x]])/(4*(a - b)^2*d) + (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + b^2)*Log[a^(1/3) + b^(1/3)*Sin[c + d*x]])/(3
*a^(2/3)*(a^2 - b^2)^2*d) - (b^(5/3)*(2*a^2 + 3*a^(4/3)*b^(2/3) + 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^(2/3)*(a^2 - b^2)^2*d) + (b*(a^2 + 2*b^2)*Log[a + b*Sin[c + d*x]^3])/(3*(a
^2 - b^2)^2*d) + 1/(4*(a + b)*d*(1 - Sin[c + d*x])) - 1/(4*(a - b)*d*(1 + Sin[c + d*x]))

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.58, size = 333, normalized size = 0.86 \begin {gather*} -\frac {\frac {3 (a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac {3 (a-4 b) \log (1+\sin (c+d x))}{(a-b)^2}-\frac {4 b^{5/3} \left (2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac {2 b^{5/3} \left (2 a^2+b^2\right ) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt {3} \sqrt [3]{a}}\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 )\right )}{a^{2/3} \left (a^2-b^2\right )^2}-\frac {4 b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}+\frac {3}{(a+b) (-1+\sin (c+d x))}+\frac {18 b^3 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {b \sin ^3(c+d x)}{a}\right ) \sin ^2(c+d x)}{\left (a^2-b^2\right )^2}+\frac {3}{(a-b) (1+\sin (c+d x))}}{12 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 1.38, size = 372, normalized size = 0.97 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 470, normalized size = 1.22 \begin {gather*} -\frac {\frac {4 \, \sqrt {3} {\left (a b^{2} {\left (9 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4\right )} - b^{3} {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {4 \, a}{b}\right )} - 2 \, a^{2} b {\left (3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} + \frac {a}{b}\right )} + 2 \, a^{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^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {6 \, {\left (b^{3} {\left (4 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} + 2 \, a^{2} b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} - 1\right )} - 3 \, a b^{2} \left (\frac {a}{b}\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^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {12 \, {\left (b^{3} {\left (2 \, \left (\frac {a}{b}\right )^{\frac {2}{3}} + 1\right )} + a^{2} b {\left (\left (\frac {a}{b}\right )^{\frac {2}{3}} + 2\right )} + 3 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \log \left (\left (\frac {a}{b}\right )^{\frac {1}{3}} + \sin \left (d x + c\right )\right )}{a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {9 \, {\left (a - 4 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {9 \, {\left (a + 4 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {18 \, {\left (a \sin \left (d x + c\right ) - b\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2} + b^{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),x, algorithm="maxima")

[Out]

-1/36*(4*sqrt(3)*(a*b^2*(9*(a/b)^(2/3) + 4) - b^3*(3*(a/b)^(1/3) + 4*a/b) - 2*a^2*b*(3*(a/b)^(1/3) + a/b) + 2*
a^3)*arctan(-1/3*sqrt(3)*((a/b)^(1/3) - 2*sin(d*x + c))/(a/b)^(1/3))/((a^4*(a/b)^(2/3) - 2*a^2*b^2*(a/b)^(2/3)
+ b^4*(a/b)^(2/3))*(a/b)^(1/3)) - 6*(b^3*(4*(a/b)^(2/3) - 1) + 2*a^2*b*((a/b)^(2/3) - 1) - 3*a*b^2*(a/b)^(1/3
))*log(sin(d*x + c)^2 - (a/b)^(1/3)*sin(d*x + c) + (a/b)^(2/3))/(a^4*(a/b)^(2/3) - 2*a^2*b^2*(a/b)^(2/3) + b^4
*(a/b)^(2/3)) - 12*(b^3*(2*(a/b)^(2/3) + 1) + a^2*b*((a/b)^(2/3) + 2) + 3*a*b^2*(a/b)^(1/3))*log((a/b)^(1/3) +
sin(d*x + c))/(a^4*(a/b)^(2/3) - 2*a^2*b^2*(a/b)^(2/3) + b^4*(a/b)^(2/3)) - 9*(a - 4*b)*log(sin(d*x + c) + 1)
/(a^2 - 2*a*b + b^2) + 9*(a + 4*b)*log(sin(d*x + c) - 1)/(a^2 + 2*a*b + b^2) + 18*(a*sin(d*x + c) - b)/((a^2 -
b^2)*sin(d*x + c)^2 - a^2 + b^2))/d

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Fricas [C] Result contains complex when optimal does not.
time = 2.17, size = 10135, normalized size = 26.32 \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),x, algorithm="fricas")

[Out]

1/12*(2*(a^4 - 2*a^2*b^2 + b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a
^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^
3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^
2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*
d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d))
+ 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*s
qrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d*cos(d*x + c)^2*log(7*a^3*b^2 + 2*a*b^4 + 3/4*
(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d
- 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b
^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^
2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3
+ a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2
*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(
3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))^2*d^2 - 1/2*(10*a^5*b + 16*a^3*b^3 + a*b^5)*(2*(1/2
)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt
(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*
d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/
((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*
b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b
^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d
- 2*a^2*b^2*d + b^4*d))*d - (8*a^2*b^3 + b^5)*sin(d*x + c)) + 3*(a^3 - 2*a^2*b - 7*a*b^2 - 4*b^3)*cos(d*x + c)
^2*log(sin(d*x + c) + 1) - 3*(a^3 + 2*a^2*b - 7*a*b^2 + 4*b^3)*cos(d*x + c)^2*log(-sin(d*x + c) + 1) - 6*a^2*b
+ 6*b^3 - ((a^4 - 2*a^2*b^2 + b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^
2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b +
2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d -
2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*
b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4
*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*
(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d*cos(d*x + c)^2 - 3*sqrt(1/3)*(a^4 - 2*a^2
*b^2 + b^4)*d*sqrt(-(4*a^4*b^2 - 80*a^2*b^4 - 32*b^6 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(2*(1/2
)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt
(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*
d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/
((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*
b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b
^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d
- 2*a^2*b^2*d + b^4*d))^2*d^2 - 4*(a^6*b - 3*a^2*b^5 + 2*b^7)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b
^4*d^2) - (a^2*b + 2*b^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt(3) + 1)/(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 +
a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(
a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2 - b^2)^4*a^2*d^3))^(1/3) - (1/2)^(1
/3)*(b^3/(a^6*d^3 - 2*a^4*b^2*d^3 + a^2*b^4*d^3) - 3*(a^2*b + 2*b^3)*b^2/((a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2)*
(a^4*d - 2*a^2*b^2*d + b^4*d)) + 2*(a^2*b + 2*b^3)^3/(a^4*d - 2*a^2*b^2*d + b^4*d)^3 + (8*a^2 + b^2)*b^5/((a^2
- b^2)^4*a^2*d^3))^(1/3)*(I*sqrt(3) + 1) + 2*(a^2*b + 2*b^3)/(a^4*d - 2*a^2*b^2*d + b^4*d))*d)/((a^8 - 4*a^6*
b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*d^2))*cos(d*x + c)^2 - 6*(a^2*b + 2*b^3)*cos(d*x + c)^2)*log(7*a^3*b^2 + 2*
a*b^4 + 3/4*(a^7 - 2*a^5*b^2 + a^3*b^4)*(2*(1/2)^(2/3)*(b^2/(a^4*d^2 - 2*a^2*b^2*d^2 + b^4*d^2) - (a^2*b + 2*b
^3)^2/(a^4*d - 2*a^2*b^2*d + b^4*d)^2)*(-I*sqrt...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{3}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \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),x)

[Out]

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

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Giac [A]
time = 0.50, size = 510, normalized size = 1.32 \begin {gather*} \frac {\frac {4 \, {\left (3 \, a^{5} b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 6 \, a^{3} b^{6} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 3 \, a b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - b^{9}\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^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}} + \frac {4 \, {\left (3 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} a b + {\left (2 \, \sqrt {3} a^{2} b + \sqrt {3} b^{3}\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^{5} - 2 \, a^{3} b^{2} + a b^{4}} - \frac {2 \, {\left (3 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b - {\left (2 \, a^{2} b + b^{3}\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^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac {4 \, {\left (a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right )^{3} + a \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac {3 \, {\left (a - 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac {3 \, {\left (a + 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {6 \, {\left (a^{2} b \sin \left (d x + c\right )^{2} + 2 \, b^{3} \sin \left (d x + c\right )^{2} - a^{3} \sin \left (d x + c\right ) + a b^{2} \sin \left (d x + c\right ) - 3 \, b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}}}{12 \, 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),x, algorithm="giac")

[Out]

1/12*(4*(3*a^5*b^4*(-a/b)^(1/3) - 6*a^3*b^6*(-a/b)^(1/3) + 3*a*b^8*(-a/b)^(1/3) - 2*a^6*b^3 + 3*a^4*b^5 - b^9)
*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9) + 4*(
3*sqrt(3)*(-a*b^2)^(2/3)*a*b + (2*sqrt(3)*a^2*b + sqrt(3)*b^3)*(-a*b^2)^(1/3))*arctan(1/3*sqrt(3)*((-a/b)^(1/3
) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a^5 - 2*a^3*b^2 + a*b^4) - 2*(3*(-a*b^2)^(2/3)*a*b - (2*a^2*b + b^3)*(-a*b^
2)^(1/3))*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x + c) + (-a/b)^(2/3))/(a^5 - 2*a^3*b^2 + a*b^4) + 4*(a^2*b
+ 2*b^3)*log(abs(b*sin(d*x + c)^3 + a))/(a^4 - 2*a^2*b^2 + b^4) + 3*(a - 4*b)*log(abs(sin(d*x + c) + 1))/(a^2
- 2*a*b + b^2) - 3*(a + 4*b)*log(abs(sin(d*x + c) - 1))/(a^2 + 2*a*b + b^2) + 6*(a^2*b*sin(d*x + c)^2 + 2*b^3*
sin(d*x + c)^2 - a^3*sin(d*x + c) + a*b^2*sin(d*x + c) - 3*b^3)/((a^4 - 2*a^2*b^2 + b^4)*(sin(d*x + c)^2 - 1))
)/d

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Mupad [B]
time = 15.27, size = 898, normalized size = 2.33 \begin {gather*} \frac {\left (\sum _{k=1}^3\ln \left (-\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (-\frac {28\,a\,b^7-6\,a^3\,b^5}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (-\frac {18\,a^5\,b^4-\frac {219\,a^3\,b^6}{4}+12\,a\,b^8}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (\frac {\frac {27\,a^7\,b^3}{2}-87\,a^5\,b^5+\frac {51\,a^3\,b^7}{2}+48\,a\,b^9}{a^4-2\,a^2\,b^2+b^4}+\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\,\left (\frac {180\,a^7\,b^4-324\,a^5\,b^6+108\,a^3\,b^8+36\,a\,b^{10}}{a^4-2\,a^2\,b^2+b^4}+\frac {\sin \left (c+d\,x\right )\,\left (216\,a^8\,b^3+216\,a^6\,b^5-1080\,a^4\,b^7+648\,a^2\,b^9\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-600\,a^4\,b^6+552\,a^2\,b^8+48\,b^{10}\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (-171\,a^4\,b^5+120\,a^2\,b^7+96\,b^9\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {\sin \left (c+d\,x\right )\,\left (61\,a^2\,b^6+40\,b^8\right )}{4\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )+\frac {a\,b^6}{2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {2\,b^7\,\sin \left (c+d\,x\right )}{a^4-2\,a^2\,b^2+b^4}\right )\,\mathrm {root}\left (54\,a^4\,b^2\,z^3-27\,a^2\,b^4\,z^3-27\,a^6\,z^3+54\,a^2\,b^3\,z^2+27\,a^4\,b\,z^2-9\,a^2\,b^2\,z+b^3,z,k\right )\right )+\frac {\frac {b}{2\,\left (a^2-b^2\right )}-\frac {a\,\sin \left (c+d\,x\right )}{2\,\left (a^2-b^2\right )}}{{\sin \left (c+d\,x\right )}^2-1}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+4\,b\right )}{4\,a^2+8\,a\,b+4\,b^2}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-4\,b\right )}{4\,a^2-8\,a\,b+4\,b^2}}{d} \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)),x)

[Out]

(symsum(log((a*b^6)/(2*(a^4 + b^4 - 2*a^2*b^2)) - root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b
^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k)*(root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b
^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k)*(root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b
^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k)*((48*a*b^9 + (51*a^3*b^7)/2 - 87*a^5*b^5 + (27*a^7*b^3)/2)/(a
^4 + b^4 - 2*a^2*b^2) + root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54*a^2*b^3*z^2 + 27*a^4*b*z^2 - 9*
a^2*b^2*z + b^3, z, k)*((36*a*b^10 + 108*a^3*b^8 - 324*a^5*b^6 + 180*a^7*b^4)/(a^4 + b^4 - 2*a^2*b^2) + (sin(c
+ d*x)*(648*a^2*b^9 - 1080*a^4*b^7 + 216*a^6*b^5 + 216*a^8*b^3))/(4*(a^4 + b^4 - 2*a^2*b^2))) + (sin(c + d*x)
*(48*b^10 + 552*a^2*b^8 - 600*a^4*b^6))/(4*(a^4 + b^4 - 2*a^2*b^2))) - (12*a*b^8 - (219*a^3*b^6)/4 + 18*a^5*b^
4)/(a^4 + b^4 - 2*a^2*b^2) + (sin(c + d*x)*(96*b^9 + 120*a^2*b^7 - 171*a^4*b^5))/(4*(a^4 + b^4 - 2*a^2*b^2)))
- (28*a*b^7 - 6*a^3*b^5)/(a^4 + b^4 - 2*a^2*b^2) + (sin(c + d*x)*(40*b^8 + 61*a^2*b^6))/(4*(a^4 + b^4 - 2*a^2*
b^2))) + (2*b^7*sin(c + d*x))/(a^4 + b^4 - 2*a^2*b^2))*root(54*a^4*b^2*z^3 - 27*a^2*b^4*z^3 - 27*a^6*z^3 + 54*
a^2*b^3*z^2 + 27*a^4*b*z^2 - 9*a^2*b^2*z + b^3, z, k), k, 1, 3) + (b/(2*(a^2 - b^2)) - (a*sin(c + d*x))/(2*(a^
2 - b^2)))/(sin(c + d*x)^2 - 1) - (log(sin(c + d*x) - 1)*(a + 4*b))/(8*a*b + 4*a^2 + 4*b^2) + (log(sin(c + d*x
) + 1)*(a - 4*b))/(4*a^2 - 8*a*b + 4*b^2))/d

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