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

Optimal. Leaf size=51 \[ \frac{2 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} \left (c+d x^3\right )\right )+b}{\sqrt{a^2-b^2}}\right )}{3 d \sqrt{a^2-b^2}} \]

[Out]

(2*ArcTan[(b + a*Tan[(c + d*x^3)/2])/Sqrt[a^2 - b^2]])/(3*Sqrt[a^2 - b^2]*d)

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Rubi [A]  time = 0.0778821, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3379, 2660, 618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} \left (c+d x^3\right )\right )+b}{\sqrt{a^2-b^2}}\right )}{3 d \sqrt{a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Sin[c + d*x^3]),x]

[Out]

(2*ArcTan[(b + a*Tan[(c + d*x^3)/2])/Sqrt[a^2 - b^2]])/(3*Sqrt[a^2 - b^2]*d)

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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])

Rubi steps

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

Mathematica [A]  time = 0.0946136, size = 51, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} \left (c+d x^3\right )\right )+b}{\sqrt{a^2-b^2}}\right )}{3 d \sqrt{a^2-b^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[(b + a*Tan[(c + d*x^3)/2])/Sqrt[a^2 - b^2]])/(3*Sqrt[a^2 - b^2]*d)

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Maple [A]  time = 0.016, size = 49, normalized size = 1. \begin{align*}{\frac{2}{3\,d}\arctan \left ({\frac{1}{2} \left ( 2\,a\tan \left ( 1/2\,d{x}^{3}+c/2 \right ) +2\,b \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 154.506, size = 10905, normalized size = 213.82 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*arctan2(-2*(4*(a^2*b^4 - b^6)*cos(d*x^3 + 2*c)^4*cos(c)*sin(c) - 4*(a^2*b^4 - b^6)*cos(c)*sin(d*x^3 + 2*c)
^4*sin(c) - 4*((a^3*b^3 - a*b^5)*cos(c)^3 + 3*(a^3*b^3 - a*b^5)*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c)^3 - 4*(3*(a^
3*b^3 - a*b^5)*cos(c)^2*sin(c) + (a^3*b^3 - a*b^5)*sin(c)^3 + ((a^2*b^4 - b^6)*cos(c)^2 - (a^2*b^4 - b^6)*sin(
c)^2)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^3 + 4*((4*a^4*b^2 - 5*a^2*b^4 + b^6)*cos(c)^3*sin(c) + (4*a^4*b^2 - 5
*a^2*b^4 + b^6)*cos(c)*sin(c)^3)*cos(d*x^3 + 2*c)^2 - 4*((4*a^4*b^2 - 5*a^2*b^4 + b^6)*cos(c)^3*sin(c) + (4*a^
4*b^2 - 5*a^2*b^4 + b^6)*cos(c)*sin(c)^3 + 3*((a^3*b^3 - a*b^5)*cos(c)^3 - (a^3*b^3 - a*b^5)*cos(c)*sin(c)^2)*
cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^2 - 4*((2*a^5*b - 3*a^3*b^3 + a*b^5)*cos(c)^5 + 2*(2*a^5*b - 3*a^3*b^3 + a*
b^5)*cos(c)^3*sin(c)^2 + (2*a^5*b - 3*a^3*b^3 + a*b^5)*cos(c)*sin(c)^4)*cos(d*x^3 + 2*c) - 4*((2*a^5*b - 3*a^3
*b^3 + a*b^5)*cos(c)^4*sin(c) + 2*(2*a^5*b - 3*a^3*b^3 + a*b^5)*cos(c)^2*sin(c)^3 + (2*a^5*b - 3*a^3*b^3 + a*b
^5)*sin(c)^5 + ((a^2*b^4 - b^6)*cos(c)^2 - (a^2*b^4 - b^6)*sin(c)^2)*cos(d*x^3 + 2*c)^3 - 3*((a^3*b^3 - a*b^5)
*cos(c)^2*sin(c) - (a^3*b^3 - a*b^5)*sin(c)^3)*cos(d*x^3 + 2*c)^2 + ((4*a^4*b^2 - 5*a^2*b^4 + b^6)*cos(c)^4 -
(4*a^4*b^2 - 5*a^2*b^4 + b^6)*sin(c)^4)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c) + (b^5*cos(d*x^3 + 2*c)^5*cos(c) -
4*a*b^4*cos(d*x^3 + 2*c)^4*cos(c)*sin(c) + b^5*sin(d*x^3 + 2*c)^5*sin(c) + (b^5*cos(d*x^3 + 2*c)*cos(c) + 4*a*
b^4*cos(c)*sin(c))*sin(d*x^3 + 2*c)^4 + 2*((2*a^2*b^3 - b^5)*cos(c)^3 + 3*(2*a^2*b^3 - b^5)*cos(c)*sin(c)^2)*c
os(d*x^3 + 2*c)^3 + 2*(b^5*cos(d*x^3 + 2*c)^2*sin(c) + 3*(2*a^2*b^3 - b^5)*cos(c)^2*sin(c) + (2*a^2*b^3 - b^5)
*sin(c)^3 + 2*(a*b^4*cos(c)^2 - a*b^4*sin(c)^2)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^3 - 4*((4*a^3*b^2 - 3*a*b^4
)*cos(c)^3*sin(c) + (4*a^3*b^2 - 3*a*b^4)*cos(c)*sin(c)^3)*cos(d*x^3 + 2*c)^2 + 2*(b^5*cos(d*x^3 + 2*c)^3*cos(
c) + 2*(4*a^3*b^2 - 3*a*b^4)*cos(c)^3*sin(c) + 2*(4*a^3*b^2 - 3*a*b^4)*cos(c)*sin(c)^3 + 3*((2*a^2*b^3 - b^5)*
cos(c)^3 - (2*a^2*b^3 - b^5)*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^2 + ((8*a^4*b - 8*a^2*b^3 + b
^5)*cos(c)^5 + 2*(8*a^4*b - 8*a^2*b^3 + b^5)*cos(c)^3*sin(c)^2 + (8*a^4*b - 8*a^2*b^3 + b^5)*cos(c)*sin(c)^4)*
cos(d*x^3 + 2*c) + (b^5*cos(d*x^3 + 2*c)^4*sin(c) + (8*a^4*b - 8*a^2*b^3 + b^5)*cos(c)^4*sin(c) + 2*(8*a^4*b -
8*a^2*b^3 + b^5)*cos(c)^2*sin(c)^3 + (8*a^4*b - 8*a^2*b^3 + b^5)*sin(c)^5 + 4*(a*b^4*cos(c)^2 - a*b^4*sin(c)^
2)*cos(d*x^3 + 2*c)^3 - 6*((2*a^2*b^3 - b^5)*cos(c)^2*sin(c) - (2*a^2*b^3 - b^5)*sin(c)^3)*cos(d*x^3 + 2*c)^2
+ 4*((4*a^3*b^2 - 3*a*b^4)*cos(c)^4 - (4*a^3*b^2 - 3*a*b^4)*sin(c)^4)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c))*sqrt
(a^2 - b^2))/(b^6*cos(d*x^3 + 2*c)^6 + 6*a*b^5*cos(c)*sin(d*x^3 + 2*c)^5 + b^6*sin(d*x^3 + 2*c)^6 - 6*a*b^5*co
s(d*x^3 + 2*c)^5*sin(c) + (32*a^6 - 48*a^4*b^2 + 18*a^2*b^4 - b^6)*cos(c)^6 + 3*(32*a^6 - 48*a^4*b^2 + 18*a^2*
b^4 - b^6)*cos(c)^4*sin(c)^2 + 3*(32*a^6 - 48*a^4*b^2 + 18*a^2*b^4 - b^6)*cos(c)^2*sin(c)^4 + (32*a^6 - 48*a^4
*b^2 + 18*a^2*b^4 - b^6)*sin(c)^6 + 3*((2*a^2*b^4 - b^6)*cos(c)^2 + 5*(2*a^2*b^4 - b^6)*sin(c)^2)*cos(d*x^3 +
2*c)^4 + 3*(b^6*cos(d*x^3 + 2*c)^2 - 2*a*b^5*cos(d*x^3 + 2*c)*sin(c) + 5*(2*a^2*b^4 - b^6)*cos(c)^2 + (2*a^2*b
^4 - b^6)*sin(c)^2)*sin(d*x^3 + 2*c)^4 - 4*(3*(4*a^3*b^3 - 3*a*b^5)*cos(c)^2*sin(c) + 5*(4*a^3*b^3 - 3*a*b^5)*
sin(c)^3)*cos(d*x^3 + 2*c)^3 + 4*(3*a*b^5*cos(d*x^3 + 2*c)^2*cos(c) + 5*(4*a^3*b^3 - 3*a*b^5)*cos(c)^3 - 6*(2*
a^2*b^4 - b^6)*cos(d*x^3 + 2*c)*cos(c)*sin(c) + 3*(4*a^3*b^3 - 3*a*b^5)*cos(c)*sin(c)^2)*sin(d*x^3 + 2*c)^3 +
3*((8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^4 + 6*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^2*sin(c)^2 + 5*(8*a^4*b^2 -
8*a^2*b^4 + b^6)*sin(c)^4)*cos(d*x^3 + 2*c)^2 + 3*(b^6*cos(d*x^3 + 2*c)^4 - 4*a*b^5*cos(d*x^3 + 2*c)^3*sin(c)
+ 5*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^4 + 6*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^2*sin(c)^2 + (8*a^4*b^2 -
8*a^2*b^4 + b^6)*sin(c)^4 + 6*((2*a^2*b^4 - b^6)*cos(c)^2 + (2*a^2*b^4 - b^6)*sin(c)^2)*cos(d*x^3 + 2*c)^2 -
4*(3*(4*a^3*b^3 - 3*a*b^5)*cos(c)^2*sin(c) + (4*a^3*b^3 - 3*a*b^5)*sin(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c
)^2 - 6*((16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)^4*sin(c) + 2*(16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)^2*sin(
c)^3 + (16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*sin(c)^5)*cos(d*x^3 + 2*c) + 6*(a*b^5*cos(d*x^3 + 2*c)^4*cos(c) + (16
*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)^5 - 4*(2*a^2*b^4 - b^6)*cos(d*x^3 + 2*c)^3*cos(c)*sin(c) + 2*(16*a^5*b -
20*a^3*b^3 + 5*a*b^5)*cos(c)^3*sin(c)^2 + (16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)*sin(c)^4 + 2*((4*a^3*b^3 -
3*a*b^5)*cos(c)^3 + 3*(4*a^3*b^3 - 3*a*b^5)*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 4*((8*a^4*b^2 - 8*a^2*b^4 +
b^6)*cos(c)^3*sin(c) + (8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)*sin(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c) - 2*
(3*b^5*cos(c)*sin(d*x^3 + 2*c)^5 - 3*b^5*cos(d*x^3 + 2*c)^5*sin(c) + (16*a^5 - 16*a^3*b^2 + 3*a*b^4)*cos(c)^6
+ 3*(16*a^5 - 16*a^3*b^2 + 3*a*b^4)*cos(c)^4*sin(c)^2 + 3*(16*a^5 - 16*a^3*b^2 + 3*a*b^4)*cos(c)^2*sin(c)^4 +
(16*a^5 - 16*a^3*b^2 + 3*a*b^4)*sin(c)^6 + 3*(a*b^4*cos(c)^2 + 5*a*b^4*sin(c)^2)*cos(d*x^3 + 2*c)^4 + 3*(5*a*b
^4*cos(c)^2 - b^5*cos(d*x^3 + 2*c)*sin(c) + a*b^4*sin(c)^2)*sin(d*x^3 + 2*c)^4 - 2*(3*(4*a^2*b^3 - b^5)*cos(c)
^2*sin(c) + 5*(4*a^2*b^3 - b^5)*sin(c)^3)*cos(d*x^3 + 2*c)^3 + 2*(3*b^5*cos(d*x^3 + 2*c)^2*cos(c) - 12*a*b^4*c
os(d*x^3 + 2*c)*cos(c)*sin(c) + 5*(4*a^2*b^3 - b^5)*cos(c)^3 + 3*(4*a^2*b^3 - b^5)*cos(c)*sin(c)^2)*sin(d*x^3
+ 2*c)^3 + 6*((2*a^3*b^2 - a*b^4)*cos(c)^4 + 6*(2*a^3*b^2 - a*b^4)*cos(c)^2*sin(c)^2 + 5*(2*a^3*b^2 - a*b^4)*s
in(c)^4)*cos(d*x^3 + 2*c)^2 - 6*(b^5*cos(d*x^3 + 2*c)^3*sin(c) - 5*(2*a^3*b^2 - a*b^4)*cos(c)^4 - 6*(2*a^3*b^2
- a*b^4)*cos(c)^2*sin(c)^2 - (2*a^3*b^2 - a*b^4)*sin(c)^4 - 3*(a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*cos(d*x^3 + 2
*c)^2 + (3*(4*a^2*b^3 - b^5)*cos(c)^2*sin(c) + (4*a^2*b^3 - b^5)*sin(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^
2 - 3*((16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)^4*sin(c) + 2*(16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)^2*sin(c)^3 + (16
*a^4*b - 12*a^2*b^3 + b^5)*sin(c)^5)*cos(d*x^3 + 2*c) + 3*(b^5*cos(d*x^3 + 2*c)^4*cos(c) - 8*a*b^4*cos(d*x^3 +
2*c)^3*cos(c)*sin(c) + (16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)^5 + 2*(16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)^3*sin(
c)^2 + (16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)*sin(c)^4 + 2*((4*a^2*b^3 - b^5)*cos(c)^3 + 3*(4*a^2*b^3 - b^5)*cos
(c)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 16*((2*a^3*b^2 - a*b^4)*cos(c)^3*sin(c) + (2*a^3*b^2 - a*b^4)*cos(c)*sin(c)
^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c))*sqrt(a^2 - b^2)), (b^6*cos(d*x^3 + 2*c)^6 + 6*a*b^5*cos(c)*sin(d*x^3 +
2*c)^5 + b^6*sin(d*x^3 + 2*c)^6 - 6*a*b^5*cos(d*x^3 + 2*c)^5*sin(c) + (8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^6
+ 3*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^4*sin(c)^2 + 3*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^2*sin(c)^4 + (8*a
^4*b^2 - 8*a^2*b^4 + b^6)*sin(c)^6 + ((4*a^2*b^4 - b^6)*cos(c)^2 + 5*(4*a^2*b^4 - b^6)*sin(c)^2)*cos(d*x^3 + 2
*c)^4 + (3*b^6*cos(d*x^3 + 2*c)^2 - 6*a*b^5*cos(d*x^3 + 2*c)*sin(c) + 5*(4*a^2*b^4 - b^6)*cos(c)^2 + (4*a^2*b^
4 - b^6)*sin(c)^2)*sin(d*x^3 + 2*c)^4 - 4*(3*(2*a^3*b^3 - a*b^5)*cos(c)^2*sin(c) + 5*(2*a^3*b^3 - a*b^5)*sin(c
)^3)*cos(d*x^3 + 2*c)^3 + 4*(3*a*b^5*cos(d*x^3 + 2*c)^2*cos(c) + 5*(2*a^3*b^3 - a*b^5)*cos(c)^3 - 2*(4*a^2*b^4
- b^6)*cos(d*x^3 + 2*c)*cos(c)*sin(c) + 3*(2*a^3*b^3 - a*b^5)*cos(c)*sin(c)^2)*sin(d*x^3 + 2*c)^3 + ((8*a^4*b
^2 - 4*a^2*b^4 - b^6)*cos(c)^4 + 6*(8*a^4*b^2 - 4*a^2*b^4 - b^6)*cos(c)^2*sin(c)^2 + 5*(8*a^4*b^2 - 4*a^2*b^4
- b^6)*sin(c)^4)*cos(d*x^3 + 2*c)^2 + (3*b^6*cos(d*x^3 + 2*c)^4 - 12*a*b^5*cos(d*x^3 + 2*c)^3*sin(c) + 5*(8*a^
4*b^2 - 4*a^2*b^4 - b^6)*cos(c)^4 + 6*(8*a^4*b^2 - 4*a^2*b^4 - b^6)*cos(c)^2*sin(c)^2 + (8*a^4*b^2 - 4*a^2*b^4
- b^6)*sin(c)^4 + 6*((4*a^2*b^4 - b^6)*cos(c)^2 + (4*a^2*b^4 - b^6)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 12*(3*(2*a
^3*b^3 - a*b^5)*cos(c)^2*sin(c) + (2*a^3*b^3 - a*b^5)*sin(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^2 - 2*((8*a
^5*b - 5*a*b^5)*cos(c)^4*sin(c) + 2*(8*a^5*b - 5*a*b^5)*cos(c)^2*sin(c)^3 + (8*a^5*b - 5*a*b^5)*sin(c)^5)*cos(
d*x^3 + 2*c) + 2*(3*a*b^5*cos(d*x^3 + 2*c)^4*cos(c) + (8*a^5*b - 5*a*b^5)*cos(c)^5 - 4*(4*a^2*b^4 - b^6)*cos(d
*x^3 + 2*c)^3*cos(c)*sin(c) + 2*(8*a^5*b - 5*a*b^5)*cos(c)^3*sin(c)^2 + (8*a^5*b - 5*a*b^5)*cos(c)*sin(c)^4 +
6*((2*a^3*b^3 - a*b^5)*cos(c)^3 + 3*(2*a^3*b^3 - a*b^5)*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 4*((8*a^4*b^2 -
4*a^2*b^4 - b^6)*cos(c)^3*sin(c) + (8*a^4*b^2 - 4*a^2*b^4 - b^6)*cos(c)*sin(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3
+ 2*c) - 4*(b^5*cos(c)*sin(d*x^3 + 2*c)^5 - b^5*cos(d*x^3 + 2*c)^5*sin(c) + (2*a^3*b^2 - a*b^4)*cos(c)^6 + 3*(
2*a^3*b^2 - a*b^4)*cos(c)^4*sin(c)^2 + 3*(2*a^3*b^2 - a*b^4)*cos(c)^2*sin(c)^4 + (2*a^3*b^2 - a*b^4)*sin(c)^6
+ (a*b^4*cos(c)^2 + 5*a*b^4*sin(c)^2)*cos(d*x^3 + 2*c)^4 + (5*a*b^4*cos(c)^2 - b^5*cos(d*x^3 + 2*c)*sin(c) + a
*b^4*sin(c)^2)*sin(d*x^3 + 2*c)^4 - 2*(3*a^2*b^3*cos(c)^2*sin(c) + 5*a^2*b^3*sin(c)^3)*cos(d*x^3 + 2*c)^3 + 2*
(b^5*cos(d*x^3 + 2*c)^2*cos(c) + 5*a^2*b^3*cos(c)^3 - 4*a*b^4*cos(d*x^3 + 2*c)*cos(c)*sin(c) + 3*a^2*b^3*cos(c
)*sin(c)^2)*sin(d*x^3 + 2*c)^3 + 2*(a^3*b^2*cos(c)^4 + 6*a^3*b^2*cos(c)^2*sin(c)^2 + 5*a^3*b^2*sin(c)^4)*cos(d
*x^3 + 2*c)^2 + 2*(5*a^3*b^2*cos(c)^4 - b^5*cos(d*x^3 + 2*c)^3*sin(c) + 6*a^3*b^2*cos(c)^2*sin(c)^2 + a^3*b^2*
sin(c)^4 + 3*(a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 3*(3*a^2*b^3*cos(c)^2*sin(c) + a^2*b^3*sin
(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c)^2 - ((4*a^4*b + 2*a^2*b^3 - b^5)*cos(c)^4*sin(c) + 2*(4*a^4*b + 2*a^
2*b^3 - b^5)*cos(c)^2*sin(c)^3 + (4*a^4*b + 2*a^2*b^3 - b^5)*sin(c)^5)*cos(d*x^3 + 2*c) + (b^5*cos(d*x^3 + 2*c
)^4*cos(c) - 8*a*b^4*cos(d*x^3 + 2*c)^3*cos(c)*sin(c) + (4*a^4*b + 2*a^2*b^3 - b^5)*cos(c)^5 + 2*(4*a^4*b + 2*
a^2*b^3 - b^5)*cos(c)^3*sin(c)^2 + (4*a^4*b + 2*a^2*b^3 - b^5)*cos(c)*sin(c)^4 + 6*(a^2*b^3*cos(c)^3 + 3*a^2*b
^3*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 16*(a^3*b^2*cos(c)^3*sin(c) + a^3*b^2*cos(c)*sin(c)^3)*cos(d*x^3 + 2*
c))*sin(d*x^3 + 2*c))*sqrt(a^2 - b^2))/(b^6*cos(d*x^3 + 2*c)^6 + 6*a*b^5*cos(c)*sin(d*x^3 + 2*c)^5 + b^6*sin(d
*x^3 + 2*c)^6 - 6*a*b^5*cos(d*x^3 + 2*c)^5*sin(c) + (32*a^6 - 48*a^4*b^2 + 18*a^2*b^4 - b^6)*cos(c)^6 + 3*(32*
a^6 - 48*a^4*b^2 + 18*a^2*b^4 - b^6)*cos(c)^4*sin(c)^2 + 3*(32*a^6 - 48*a^4*b^2 + 18*a^2*b^4 - b^6)*cos(c)^2*s
in(c)^4 + (32*a^6 - 48*a^4*b^2 + 18*a^2*b^4 - b^6)*sin(c)^6 + 3*((2*a^2*b^4 - b^6)*cos(c)^2 + 5*(2*a^2*b^4 - b
^6)*sin(c)^2)*cos(d*x^3 + 2*c)^4 + 3*(b^6*cos(d*x^3 + 2*c)^2 - 2*a*b^5*cos(d*x^3 + 2*c)*sin(c) + 5*(2*a^2*b^4
- b^6)*cos(c)^2 + (2*a^2*b^4 - b^6)*sin(c)^2)*sin(d*x^3 + 2*c)^4 - 4*(3*(4*a^3*b^3 - 3*a*b^5)*cos(c)^2*sin(c)
+ 5*(4*a^3*b^3 - 3*a*b^5)*sin(c)^3)*cos(d*x^3 + 2*c)^3 + 4*(3*a*b^5*cos(d*x^3 + 2*c)^2*cos(c) + 5*(4*a^3*b^3 -
3*a*b^5)*cos(c)^3 - 6*(2*a^2*b^4 - b^6)*cos(d*x^3 + 2*c)*cos(c)*sin(c) + 3*(4*a^3*b^3 - 3*a*b^5)*cos(c)*sin(c
)^2)*sin(d*x^3 + 2*c)^3 + 3*((8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^4 + 6*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^2
*sin(c)^2 + 5*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*sin(c)^4)*cos(d*x^3 + 2*c)^2 + 3*(b^6*cos(d*x^3 + 2*c)^4 - 4*a*b^5
*cos(d*x^3 + 2*c)^3*sin(c) + 5*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^4 + 6*(8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)
^2*sin(c)^2 + (8*a^4*b^2 - 8*a^2*b^4 + b^6)*sin(c)^4 + 6*((2*a^2*b^4 - b^6)*cos(c)^2 + (2*a^2*b^4 - b^6)*sin(c
)^2)*cos(d*x^3 + 2*c)^2 - 4*(3*(4*a^3*b^3 - 3*a*b^5)*cos(c)^2*sin(c) + (4*a^3*b^3 - 3*a*b^5)*sin(c)^3)*cos(d*x
^3 + 2*c))*sin(d*x^3 + 2*c)^2 - 6*((16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)^4*sin(c) + 2*(16*a^5*b - 20*a^3*b^
3 + 5*a*b^5)*cos(c)^2*sin(c)^3 + (16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*sin(c)^5)*cos(d*x^3 + 2*c) + 6*(a*b^5*cos(d
*x^3 + 2*c)^4*cos(c) + (16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)^5 - 4*(2*a^2*b^4 - b^6)*cos(d*x^3 + 2*c)^3*cos
(c)*sin(c) + 2*(16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)^3*sin(c)^2 + (16*a^5*b - 20*a^3*b^3 + 5*a*b^5)*cos(c)*
sin(c)^4 + 2*((4*a^3*b^3 - 3*a*b^5)*cos(c)^3 + 3*(4*a^3*b^3 - 3*a*b^5)*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 4
*((8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)^3*sin(c) + (8*a^4*b^2 - 8*a^2*b^4 + b^6)*cos(c)*sin(c)^3)*cos(d*x^3 + 2
*c))*sin(d*x^3 + 2*c) - 2*(3*b^5*cos(c)*sin(d*x^3 + 2*c)^5 - 3*b^5*cos(d*x^3 + 2*c)^5*sin(c) + (16*a^5 - 16*a^
3*b^2 + 3*a*b^4)*cos(c)^6 + 3*(16*a^5 - 16*a^3*b^2 + 3*a*b^4)*cos(c)^4*sin(c)^2 + 3*(16*a^5 - 16*a^3*b^2 + 3*a
*b^4)*cos(c)^2*sin(c)^4 + (16*a^5 - 16*a^3*b^2 + 3*a*b^4)*sin(c)^6 + 3*(a*b^4*cos(c)^2 + 5*a*b^4*sin(c)^2)*cos
(d*x^3 + 2*c)^4 + 3*(5*a*b^4*cos(c)^2 - b^5*cos(d*x^3 + 2*c)*sin(c) + a*b^4*sin(c)^2)*sin(d*x^3 + 2*c)^4 - 2*(
3*(4*a^2*b^3 - b^5)*cos(c)^2*sin(c) + 5*(4*a^2*b^3 - b^5)*sin(c)^3)*cos(d*x^3 + 2*c)^3 + 2*(3*b^5*cos(d*x^3 +
2*c)^2*cos(c) - 12*a*b^4*cos(d*x^3 + 2*c)*cos(c)*sin(c) + 5*(4*a^2*b^3 - b^5)*cos(c)^3 + 3*(4*a^2*b^3 - b^5)*c
os(c)*sin(c)^2)*sin(d*x^3 + 2*c)^3 + 6*((2*a^3*b^2 - a*b^4)*cos(c)^4 + 6*(2*a^3*b^2 - a*b^4)*cos(c)^2*sin(c)^2
+ 5*(2*a^3*b^2 - a*b^4)*sin(c)^4)*cos(d*x^3 + 2*c)^2 - 6*(b^5*cos(d*x^3 + 2*c)^3*sin(c) - 5*(2*a^3*b^2 - a*b^
4)*cos(c)^4 - 6*(2*a^3*b^2 - a*b^4)*cos(c)^2*sin(c)^2 - (2*a^3*b^2 - a*b^4)*sin(c)^4 - 3*(a*b^4*cos(c)^2 + a*b
^4*sin(c)^2)*cos(d*x^3 + 2*c)^2 + (3*(4*a^2*b^3 - b^5)*cos(c)^2*sin(c) + (4*a^2*b^3 - b^5)*sin(c)^3)*cos(d*x^3
+ 2*c))*sin(d*x^3 + 2*c)^2 - 3*((16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)^4*sin(c) + 2*(16*a^4*b - 12*a^2*b^3 + b^
5)*cos(c)^2*sin(c)^3 + (16*a^4*b - 12*a^2*b^3 + b^5)*sin(c)^5)*cos(d*x^3 + 2*c) + 3*(b^5*cos(d*x^3 + 2*c)^4*co
s(c) - 8*a*b^4*cos(d*x^3 + 2*c)^3*cos(c)*sin(c) + (16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)^5 + 2*(16*a^4*b - 12*a^
2*b^3 + b^5)*cos(c)^3*sin(c)^2 + (16*a^4*b - 12*a^2*b^3 + b^5)*cos(c)*sin(c)^4 + 2*((4*a^2*b^3 - b^5)*cos(c)^3
+ 3*(4*a^2*b^3 - b^5)*cos(c)*sin(c)^2)*cos(d*x^3 + 2*c)^2 - 16*((2*a^3*b^2 - a*b^4)*cos(c)^3*sin(c) + (2*a^3*
b^2 - a*b^4)*cos(c)*sin(c)^3)*cos(d*x^3 + 2*c))*sin(d*x^3 + 2*c))*sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*d)

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Fricas [A]  time = 1.77605, size = 458, normalized size = 8.98 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2}} \log \left (\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x^{3} + c\right )^{2} - 2 \, a b \sin \left (d x^{3} + c\right ) - a^{2} - b^{2} + 2 \,{\left (a \cos \left (d x^{3} + c\right ) \sin \left (d x^{3} + c\right ) + b \cos \left (d x^{3} + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x^{3} + c\right )^{2} - 2 \, a b \sin \left (d x^{3} + c\right ) - a^{2} - b^{2}}\right )}{6 \,{\left (a^{2} - b^{2}\right )} d}, -\frac{\arctan \left (-\frac{a \sin \left (d x^{3} + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x^{3} + c\right )}\right )}{3 \, \sqrt{a^{2} - b^{2}} d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*sqrt(-a^2 + b^2)*log(((2*a^2 - b^2)*cos(d*x^3 + c)^2 - 2*a*b*sin(d*x^3 + c) - a^2 - b^2 + 2*(a*cos(d*x^3
+ c)*sin(d*x^3 + c) + b*cos(d*x^3 + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x^3 + c)^2 - 2*a*b*sin(d*x^3 + c) - a^2
- b^2))/((a^2 - b^2)*d), -1/3*arctan(-(a*sin(d*x^3 + c) + b)/(sqrt(a^2 - b^2)*cos(d*x^3 + c)))/(sqrt(a^2 - b^2
)*d)]

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Sympy [A]  time = 24.6765, size = 202, normalized size = 3.96 \begin{align*} \begin{cases} \frac{x^{3}}{3 \left (a + b \sin{\left (c \right )}\right )} & \text{for}\: d = 0 \\\frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x^{3}}{2} \right )} \right )}}{3 b d} & \text{for}\: a = 0 \\\frac{2 \sqrt{b^{2}}}{3 b^{2} d \tan{\left (\frac{c}{2} + \frac{d x^{3}}{2} \right )} - 3 b d \sqrt{b^{2}}} & \text{for}\: a = - \sqrt{b^{2}} \\- \frac{2 \sqrt{b^{2}}}{3 b^{2} d \tan{\left (\frac{c}{2} + \frac{d x^{3}}{2} \right )} + 3 b d \sqrt{b^{2}}} & \text{for}\: a = \sqrt{b^{2}} \\\frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x^{3}}{2} \right )} + \frac{b}{a} - \frac{\sqrt{- a^{2} + b^{2}}}{a} \right )}}{3 d \sqrt{- a^{2} + b^{2}}} - \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x^{3}}{2} \right )} + \frac{b}{a} + \frac{\sqrt{- a^{2} + b^{2}}}{a} \right )}}{3 d \sqrt{- a^{2} + b^{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3/(3*(a + b*sin(c))), Eq(d, 0)), (log(tan(c/2 + d*x**3/2))/(3*b*d), Eq(a, 0)), (2*sqrt(b**2)/(3*
b**2*d*tan(c/2 + d*x**3/2) - 3*b*d*sqrt(b**2)), Eq(a, -sqrt(b**2))), (-2*sqrt(b**2)/(3*b**2*d*tan(c/2 + d*x**3
/2) + 3*b*d*sqrt(b**2)), Eq(a, sqrt(b**2))), (log(tan(c/2 + d*x**3/2) + b/a - sqrt(-a**2 + b**2)/a)/(3*d*sqrt(
-a**2 + b**2)) - log(tan(c/2 + d*x**3/2) + b/a + sqrt(-a**2 + b**2)/a)/(3*d*sqrt(-a**2 + b**2)), True))

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Giac [A]  time = 1.10127, size = 86, normalized size = 1.69 \begin{align*} \frac{2 \,{\left (\pi \left \lfloor \frac{d x^{3} + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x^{3} + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{3 \, \sqrt{a^{2} - b^{2}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(pi*floor(1/2*(d*x^3 + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d*x^3 + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt
(a^2 - b^2)*d)