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

Optimal. Leaf size=714 \[ \text{result too large to display} \]

[Out]

-((-1)^(2/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a
^(1/3)*b^(5/3)) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3))
+ ((-1)^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*
a^(1/3)*b^(5/3)) + (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) -
((-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(
2/3)*b^(4/3)) + ((-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b
^(1/3)])/(9*a^(2/3)*b^(4/3)) - (x*Sin[c + d*x])/(3*b*(a + b*x^3)) + ((-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)
/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(2/3)*d*Sin[c + ((-1)
^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(1/3)*b^(5/3)) + (Cos[c - (
a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + (d*Sin[c - (a^(1/3)*d)/b^(1/
3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + ((-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b
^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*d*Sin[c - ((-1)^(
2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3))

________________________________________________________________________________________

Rubi [A]  time = 1.07282, antiderivative size = 714, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3343, 3333, 3303, 3299, 3302, 3346} \[ -\frac{\sqrt [3]{-1} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac{\sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac{(-1)^{2/3} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac{\sqrt [3]{-1} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}+\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac{(-1)^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac{(-1)^{2/3} d \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{\sqrt [3]{-1} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{(-1)^{2/3} d \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{\sqrt [3]{-1} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{x \sin (c+d x)}{3 b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-1)^(2/3)*d*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a
^(1/3)*b^(5/3)) - (d*Cos[c - (a^(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3))
+ ((-1)^(1/3)*d*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b^(1/3)]*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*
a^(1/3)*b^(5/3)) + (CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(9*a^(2/3)*b^(4/3)) -
((-1)^(1/3)*CosIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x]*Sin[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)])/(9*a^(
2/3)*b^(4/3)) + ((-1)^(2/3)*CosIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - ((-1)^(2/3)*a^(1/3)*d)/b
^(1/3)])/(9*a^(2/3)*b^(4/3)) - (x*Sin[c + d*x])/(3*b*(a + b*x^3)) + ((-1)^(1/3)*Cos[c + ((-1)^(1/3)*a^(1/3)*d)
/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(2/3)*d*Sin[c + ((-1)
^(1/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(9*a^(1/3)*b^(5/3)) + (Cos[c - (
a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) + (d*Sin[c - (a^(1/3)*d)/b^(1/
3)]*SinIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3)) + ((-1)^(2/3)*Cos[c - ((-1)^(2/3)*a^(1/3)*d)/b
^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(2/3)*b^(4/3)) - ((-1)^(1/3)*d*Sin[c - ((-1)^(
2/3)*a^(1/3)*d)/b^(1/3)]*SinIntegral[((-1)^(2/3)*a^(1/3)*d)/b^(1/3) + d*x])/(9*a^(1/3)*b^(5/3))

Rule 3343

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(x^(m - n + 1)*(a + b*
x^n)^(p + 1)*Sin[c + d*x])/(b*n*(p + 1)), x] + (-Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*(a + b*x^n)^(p
+ 1)*Sin[c + d*x], x], x] - Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x])
/; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]

Rule 3333

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (a +
b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3346

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{x^3 \sin (c+d x)}{\left (a+b x^3\right )^2} \, dx &=-\frac{x \sin (c+d x)}{3 b \left (a+b x^3\right )}+\frac{\int \frac{\sin (c+d x)}{a+b x^3} \, dx}{3 b}+\frac{d \int \frac{x \cos (c+d x)}{a+b x^3} \, dx}{3 b}\\ &=-\frac{x \sin (c+d x)}{3 b \left (a+b x^3\right )}+\frac{\int \left (-\frac{\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac{\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x\right )}-\frac{\sin (c+d x)}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{3 b}+\frac{d \int \left (-\frac{\cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac{(-1)^{2/3} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac{\sqrt [3]{-1} \cos (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{3 b}\\ &=-\frac{x \sin (c+d x)}{3 b \left (a+b x^3\right )}-\frac{\int \frac{\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac{\int \frac{\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac{\int \frac{\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac{d \int \frac{\cos (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}+\frac{\left (\sqrt [3]{-1} d\right ) \int \frac{\cos (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}-\frac{\left ((-1)^{2/3} d\right ) \int \frac{\cos (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}\\ &=-\frac{x \sin (c+d x)}{3 b \left (a+b x^3\right )}-\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac{\left (d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}+\frac{\cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}+\frac{\left (\sqrt [3]{-1} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}-\frac{\cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}-\frac{\left ((-1)^{2/3} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}-\frac{\sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\cos \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{9 a^{2/3} b}+\frac{\left (d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}-\frac{\sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}+\frac{\left (\sqrt [3]{-1} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}-\frac{\sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac{\cos \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 a^{2/3} b}+\frac{\left ((-1)^{2/3} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac{\sin \left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}\\ &=-\frac{(-1)^{2/3} d \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 \sqrt [3]{a} b^{5/3}}-\frac{d \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{\sqrt [3]{-1} d \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{\text{Ci}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac{\sqrt [3]{-1} \text{Ci}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right ) \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}+\frac{(-1)^{2/3} \text{Ci}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right ) \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{9 a^{2/3} b^{4/3}}-\frac{x \sin (c+d x)}{3 b \left (a+b x^3\right )}+\frac{\sqrt [3]{-1} \cos \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 a^{2/3} b^{4/3}}-\frac{(-1)^{2/3} d \sin \left (c+\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{\cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}+\frac{d \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 \sqrt [3]{a} b^{5/3}}+\frac{(-1)^{2/3} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 a^{2/3} b^{4/3}}-\frac{\sqrt [3]{-1} d \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{9 \sqrt [3]{a} b^{5/3}}\\ \end{align*}

Mathematica [C]  time = 0.422002, size = 383, normalized size = 0.54 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{\sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{\sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i \text{$\#$1} d \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1} d \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+i \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-\text{$\#$1} d \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1} d \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-\frac{6 b x \sin (c+d x)}{a+b x^3}}{18 b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(RootSum[a + b*#1^3 & , (I*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + CosIntegral[d*(x - #1)]*Sin[c + d*#1] + Cos
[c + d*#1]*SinIntegral[d*(x - #1)] - I*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(
x - #1)]*#1 - I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1 - I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*
Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] + RootSum[a + b*#1^3 & , ((-I)*Cos[c + d*#1]*CosIntegral[d*
(x - #1)] + CosIntegral[d*(x - #1)]*Sin[c + d*#1] + Cos[c + d*#1]*SinIntegral[d*(x - #1)] + I*Sin[c + d*#1]*Si
nIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosIntegral[d*(x - #1)]*Sin[c + d*#1]
*#1 + I*d*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1 - d*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1)/#1^2 & ] - (6
*b*x*Sin[c + d*x])/(a + b*x^3))/(18*b^2)

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Maple [C]  time = 0.08, size = 1185, 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(x^3*sin(d*x+c)/(b*x^3+a)^2,x)

[Out]

1/d^4*(sin(d*x+c)*(c^2*d^3/a*(d*x+c)^2-1/3*d^3*(a*d^3+5*b*c^3)/a/b*(d*x+c)-2/3*c*d^3*(a*d^3-b*c^3)/a/b)/((d*x+
c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+c)*b*c^2+a*d^3-c^3*b)+1/9*d^3/a/b^2*sum((3*_R1*b*c^2+a*d^3-b*c^3)/(_R1^2-2*_R1*c
+c^2)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-
1/9*d^3/a/b^2*sum((3*_RR1^2*b*c^2-_RR1*a*d^3-5*_RR1*b*c^3-2*a*c*d^3+2*b*c^4)/(_RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_R
R1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+sin(d*x+c)*(-
2*c^2*d^3/a*(d*x+c)^2+3*c^3*d^3/a*(d*x+c)+c*d^3*(a*d^3-b*c^3)/a/b)/((d*x+c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+c)*b*c^
2+a*d^3-c^3*b)-2/3*c^2*d^3/a/b*sum(_R1/(_R1^2-2*_R1*c+c^2)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_
R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/3*c*d^3/a/b^2*sum((2*_RR1^2*b*c-3*_RR1*b*c^2-a*d^3+b*c^
3)/(_RR1^2-2*_RR1*c+c^2)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*
_Z*b*c^2+a*d^3-b*c^3))+sin(d*x+c)*(c^2*d^3/a*(d*x+c)^2-c^3*d^3/a*(d*x+c))/((d*x+c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+
c)*b*c^2+a*d^3-c^3*b)+1/3*c^2*d^3/a/b*sum((_R1+c)/(_R1^2-2*_R1*c+c^2)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*
sin(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/3*c^2*d^3/a/b*sum(_RR1/(_RR1-c)*(Si(-d*x+_RR
1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-d^6*c^3*(sin(d
*x+c)*(1/3/a/d^3*(d*x+c)-1/3*c/a/d^3)/((d*x+c)^3*b-3*c*(d*x+c)^2*b+3*(d*x+c)*b*c^2+a*d^3-c^3*b)+2/9/a/d^3/b*su
m(1/(_R1^2-2*_R1*c+c^2)*(-Si(-d*x+_R1-c)*cos(_R1)+Ci(d*x-_R1+c)*sin(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*
c^2+a*d^3-b*c^3))-1/9/a/d^3/b*sum(1/(_RR1-c)*(Si(-d*x+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=RootOf(
_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))))

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

[Out]

Timed out

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Fricas [C]  time = 2.4791, size = 1613, normalized size = 2.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(12*a*d*x*sin(d*x + c) + ((b*x^3 - sqrt(3)*(I*b*x^3 + I*a) + a)*(I*a*d^3/b)^(2/3) - (b*x^3 + sqrt(3)*(I*
b*x^3 + I*a) + a)*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(I*a*d^3/b)^(1
/3)*(I*sqrt(3) + 1) - I*c) + ((b*x^3 - sqrt(3)*(I*b*x^3 + I*a) + a)*(-I*a*d^3/b)^(2/3) - (b*x^3 + sqrt(3)*(I*b
*x^3 + I*a) + a)*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)^(
1/3)*(I*sqrt(3) + 1) + I*c) + ((b*x^3 - sqrt(3)*(-I*b*x^3 - I*a) + a)*(I*a*d^3/b)^(2/3) - (b*x^3 + sqrt(3)*(-I
*b*x^3 - I*a) + a)*(I*a*d^3/b)^(1/3))*Ei(-I*d*x + 1/2*(I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(I*a*d^3/b)^(1
/3)*(-I*sqrt(3) + 1) - I*c) + ((b*x^3 - sqrt(3)*(-I*b*x^3 - I*a) + a)*(-I*a*d^3/b)^(2/3) - (b*x^3 + sqrt(3)*(-
I*b*x^3 - I*a) + a)*(-I*a*d^3/b)^(1/3))*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1))*e^(1/2*(-I*a*d^3/b)
^(1/3)*(-I*sqrt(3) + 1) + I*c) - 2*((b*x^3 + a)*(-I*a*d^3/b)^(2/3) - (b*x^3 + a)*(-I*a*d^3/b)^(1/3))*Ei(I*d*x
+ (-I*a*d^3/b)^(1/3))*e^(I*c - (-I*a*d^3/b)^(1/3)) - 2*((b*x^3 + a)*(I*a*d^3/b)^(2/3) - (b*x^3 + a)*(I*a*d^3/b
)^(1/3))*Ei(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)))/(a*b^2*d*x^3 + a^2*b*d)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \sin \left (d x + c\right )}{{\left (b x^{3} + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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