3.387 \(\int \frac{\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx\)

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

[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]))

________________________________________________________________________________________

Rubi [A]  time = 0.504331, antiderivative size = 385, normalized size of antiderivative = 1., 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 = {3223, 2074, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} d \left (a^2-b^2\right )^2}+\frac{b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^2}+\frac{b^{5/3} \left (3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} d \left (a^2-b^2\right )^2}-\frac{b^{5/3} \left (-3 a^{4/3} b^{2/3}+2 a^2+b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}-\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]

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 3223

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])

Rule 2074

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 1871

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 1860

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 31

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

Rule 634

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 617

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 204

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

Rule 628

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 260

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]

Rubi steps

\begin{align*} \int \frac{\sec ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\frac{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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 ) \operatorname{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} \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 2.32315, size = 333, normalized size = 0.86 \[ -\frac{\frac{18 b^3 \sin ^2(c+d x) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right )}{\left (a^2-b^2\right )^2}-\frac{4 b \left (a^2+2 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{\left (a^2-b^2\right )^2}-\frac{4 b^{5/3} \left (2 a^2+b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac{2 b^{5/3} \left (2 a^2+b^2\right ) \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{a^{2/3} \left (a^2-b^2\right )^2}+\frac{3}{(a+b) (\sin (c+d x)-1)}+\frac{3}{(a-b) (\sin (c+d x)+1)}+\frac{3 (a+4 b) \log (1-\sin (c+d x))}{(a+b)^2}-\frac{3 (a-4 b) \log (\sin (c+d x)+1)}{(a-b)^2}}{12 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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*Sqrt[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])))/(12*d)

________________________________________________________________________________________

Maple [B]  time = 0.146, size = 668, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 25.194, size = 20282, normalized size = 52.68 \begin{align*} \text{result too large to display} \end{align*}

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(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 + 3/4*sqrt(1/3)*(3*(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))*d^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*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)) + 2*(8*a^2*b^3 + b^5)*sin(d*x + c)) -
((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(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*sqr
t(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*sq
rt(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 + 3/4*sqrt(1/3)*(3*(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))*d^2
+ 2*(a^5*b - 2*a^3*b^3 + a*b^5)*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)) - 2*(8*a^2*b^3 + b^5)*sin(d*x + c)) + 6*(a^3 - a
*b^2)*sin(d*x + c))/((a^4 - 2*a^2*b^2 + b^4)*d*cos(d*x + c)^2)

________________________________________________________________________________________

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

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]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.22952, size = 689, normalized size = 1.79 \begin{align*} \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{12 \,{\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 )} \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 )}{\sqrt{3} a^{5} - 2 \, \sqrt{3} a^{3} b^{2} + \sqrt{3} 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{align*}

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) + 12*
(3*(-a*b^2)^(2/3)*a*b + (2*a^2*b + 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))/(sqrt(3)*a^5 - 2*sqrt(3)*a^3*b^2 + sqrt(3)*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