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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.02152, antiderivative size = 747, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {3223, 2074, 1854, 1860, 31, 634, 617, 204, 628, 1871, 260} \[ -\frac{b \left (a \left (a^2+2 b^2\right )-b \sin (c+d x) \left (2 a^2-3 a b \sin (c+d x)+b^2\right )\right )}{3 a d \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}-\frac{b^{5/3} \left (3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{18 a^{5/3} d \left (a^2-b^2\right )^2}-\frac{b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 \sqrt [3]{a} d \left (a^2-b^2\right )^3}+\frac{2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac{b^{5/3} \left (3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{9 a^{5/3} d \left (a^2-b^2\right )^2}+\frac{b^{5/3} \left (3 b^{2/3} \left (3 a^2+b^2\right )+4 a^{2/3} \left (a^2+2 b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \sqrt [3]{a} d \left (a^2-b^2\right )^3}-\frac{b^{5/3} \left (-3 a^{4/3} b^{2/3}+4 a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{5/3} d \left (a^2-b^2\right )^2}-\frac{b^{5/3} \left (-9 a^2 b^{2/3}+8 a^{2/3} b^2+4 a^{8/3}-3 b^{8/3}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} d \left (a^2-b^2\right )^3}+\frac{1}{4 d (a+b)^2 (1-\sin (c+d x))}-\frac{1}{4 d (a-b)^2 (\sin (c+d x)+1)}-\frac{(a+7 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a-7 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 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 1854

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

Mathematica [C]  time = 6.37817, size = 657, normalized size = 0.88 \[ \frac{-\frac{3 b^3 \left (3 a^2+b^2\right ) \sin ^2(c+d x) \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right )}{2 a \left (a^2-b^2\right )^3}-\frac{3 b^3 \sin ^2(c+d x) \, _2F_1\left (\frac{2}{3},2;\frac{5}{3};-\frac{b \sin ^3(c+d x)}{a}\right )}{2 a \left (a^2-b^2\right )^2}+\frac{a b^2 \left (\frac{b^2}{a^2}+2\right ) \sin (c+d x)}{3 \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}-\frac{b \left (a^2+2 b^2\right )}{3 \left (a^2-b^2\right )^2 \left (a+b \sin ^3(c+d x)\right )}+\frac{2 a b \left (a^2+5 b^2\right ) \log \left (a+b \sin ^3(c+d x)\right )}{3 \left (a^2-b^2\right )^3}+\frac{4 \sqrt [3]{a} b^{5/3} \left (a^2+2 b^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3}-\frac{2 \sqrt [3]{a} \left (a^2+2 b^2\right ) \left (b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )+2 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{3 \left (a^2-b^2\right )^3}+\frac{\left (\frac{b^2}{a^2}+2\right ) \left (2 \sqrt [3]{a} b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )-\sqrt [3]{a} \left (b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )+2 \sqrt{3} b^{5/3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )\right )\right )}{9 \left (a^2-b^2\right )^2}+\frac{1}{4 (a+b)^2 (1-\sin (c+d x))}-\frac{1}{4 (a-b)^2 (\sin (c+d x)+1)}-\frac{(a+7 b) \log (1-\sin (c+d x))}{4 (a+b)^3}+\frac{(a-7 b) \log (\sin (c+d x)+1)}{4 (a-b)^3}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.208, size = 1309, normalized size = 1.8 \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)^2,x)

[Out]

-5/3/d*b^2/(a-b)^3/(a+b)^3*a^2/(a/b)^(1/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))+2/3/d*b^2/(a-b)
^3/(a+b)^3/(a+b*sin(d*x+c)^3)*a^3*sin(d*x+c)-2/3/d*b^4/(a-b)^3/(a+b)^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*
(2/(a/b)^(1/3)*sin(d*x+c)-1))-1/4/d/(a+b)^2/(sin(d*x+c)-1)-1/4/(a-b)^2/d/(1+sin(d*x+c))-2/9/d*b^5/(a-b)^3/(a+b
)^3/a/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1))+16/9/d*b/(a-b)^3/(a+b)^3*a^3/(a/b)^
(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1))-10/3/d*b^2/(a-b)^3/(a+b)^3*a^2*3^(1/2)/(a/b)^(1
/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1))+22/9/d*b^3/(a-b)^3/(a+b)^3*a/(a/b)^(2/3)*3^(1/2)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1))-1/4/d/(a+b)^3*ln(sin(d*x+c)-1)*a-7/4/d/(a+b)^3*ln(sin(d*x+c)-1)*b+1/4
/d/(a-b)^3*ln(1+sin(d*x+c))*a-7/4/d/(a-b)^3*ln(1+sin(d*x+c))*b-1/3/d*b^3/(a-b)^3/(a+b)^3/(a+b*sin(d*x+c)^3)*a^
2+2/3/d*b^4/(a-b)^3/(a+b)^3/(a/b)^(1/3)*ln(sin(d*x+c)+(a/b)^(1/3))-1/3/d*b^4/(a-b)^3/(a+b)^3/(a/b)^(1/3)*ln(si
n(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))+2/3/d*b/(a-b)^3/(a+b)^3*a^3*ln(a+b*sin(d*x+c)^3)+10/3/d*b^3/(a-
b)^3/(a+b)^3*a*ln(a+b*sin(d*x+c)^3)+1/d*b^5/(a-b)^3/(a+b)^3/(a+b*sin(d*x+c)^3)*sin(d*x+c)^2-1/3/d*b/(a-b)^3/(a
+b)^3/(a+b*sin(d*x+c)^3)*a^4+16/9/d*b/(a-b)^3/(a+b)^3*a^3/(a/b)^(2/3)*ln(sin(d*x+c)+(a/b)^(1/3))+22/9/d*b^3/(a
-b)^3/(a+b)^3*a/(a/b)^(2/3)*ln(sin(d*x+c)+(a/b)^(1/3))-2/9/d*b^5/(a-b)^3/(a+b)^3/a/(a/b)^(2/3)*ln(sin(d*x+c)+(
a/b)^(1/3))-8/9/d*b/(a-b)^3/(a+b)^3*a^3/(a/b)^(2/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))-11/9/d
*b^3/(a-b)^3/(a+b)^3*a/(a/b)^(2/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))+1/9/d*b^5/(a-b)^3/(a+b)
^3/a/(a/b)^(2/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b)^(2/3))-1/d*b^3/(a-b)^3/(a+b)^3/(a+b*sin(d*x+c)^3
)*sin(d*x+c)^2*a^2-1/3/d*b^4/(a-b)^3/(a+b)^3/(a+b*sin(d*x+c)^3)*a*sin(d*x+c)-1/3/d*b^6/(a-b)^3/(a+b)^3/(a+b*si
n(d*x+c)^3)/a*sin(d*x+c)+10/3/d*b^2/(a-b)^3/(a+b)^3*a^2/(a/b)^(1/3)*ln(sin(d*x+c)+(a/b)^(1/3))+2/3/d*b^5/(a-b)
^3/(a+b)^3/(a+b*sin(d*x+c)^3)

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [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)^2,x, algorithm="fricas")

[Out]

Timed out

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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)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.27934, size = 1064, normalized size = 1.42 \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)^2,x, algorithm="giac")

[Out]

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