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

Optimal. Leaf size=772 \[ \text{result too large to display} \]

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x) - (d*Cos[c + d*x])/(18*b^2*x*(a + b*x^3)) + (CosIntegral[(a^(1/3)*d)/b^(1/3) + d
*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (
a^(1/3)*d)/b^(1/3)])/(54*a*b^2) - ((-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)])/(27*a^(5/3)*b^(4/3)) + (d^2*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)])/(54*a*b^2) + ((-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)])/(27*a^(5/3)*b^(4/3)) + (d^2*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)])/(54*a*b^2) + Sin[c + d*x]/(18*a*b^2*x^2) - (x*Sin[c + d*x])
/(6*b*(a + b*x^3)^2) - Sin[c + d*x]/(18*b^2*x^2*(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])/(27*a^(5/3)*b^(4/3)) - (d^2*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])/(54*a*b^2) + (Cos[c - (a^(1/3)*d)/b^(1/3)]*Si
nIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^
(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2) + ((-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])/(27*a^(5/3)*b^(4/3)) + (d^2*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])/(54*a*b^2)

________________________________________________________________________________________

Rubi [A]  time = 2.76614, antiderivative size = 772, normalized size of antiderivative = 1., number of steps used = 71, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {3343, 3331, 3345, 3297, 3303, 3299, 3302, 3333, 3346, 3344} \[ \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 )}{27 a^{5/3} b^{4/3}}-\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 )}{27 a^{5/3} 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 )}{27 a^{5/3} 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 )}{27 a^{5/3} 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 )}{27 a^{5/3} 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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a b^2}+\frac{d^2 \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 )}{54 a b^2}+\frac{d^2 \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 )}{54 a b^2}-\frac{d^2 \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 )}{54 a b^2}+\frac{d^2 \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 )}{54 a b^2}+\frac{d^2 \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 )}{54 a b^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}+\frac{\sin (c+d x)}{18 a b^2 x^2}-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac{d \cos (c+d x)}{18 a b^2 x}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x) - (d*Cos[c + d*x])/(18*b^2*x*(a + b*x^3)) + (CosIntegral[(a^(1/3)*d)/b^(1/3) + d
*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(27*a^(5/3)*b^(4/3)) + (d^2*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (
a^(1/3)*d)/b^(1/3)])/(54*a*b^2) - ((-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)])/(27*a^(5/3)*b^(4/3)) + (d^2*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)])/(54*a*b^2) + ((-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)])/(27*a^(5/3)*b^(4/3)) + (d^2*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)])/(54*a*b^2) + Sin[c + d*x]/(18*a*b^2*x^2) - (x*Sin[c + d*x])
/(6*b*(a + b*x^3)^2) - Sin[c + d*x]/(18*b^2*x^2*(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])/(27*a^(5/3)*b^(4/3)) - (d^2*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])/(54*a*b^2) + (Cos[c - (a^(1/3)*d)/b^(1/3)]*Si
nIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + (d^2*Cos[c - (a^(1/3)*d)/b^(1/3)]*SinIntegral[(a^
(1/3)*d)/b^(1/3) + d*x])/(54*a*b^2) + ((-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])/(27*a^(5/3)*b^(4/3)) + (d^2*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])/(54*a*b^2)

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 3331

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

Rule 3345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[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]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -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 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 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]

Rule 3344

Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m - n + 1)*(a + b*
x^n)^(p + 1)*Cos[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)*Cos[c + d*x], x], x] + Dist[d/(b*n*(p + 1)), Int[x^(m - n + 1)*(a + b*x^n)^(p + 1)*Sin[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]

Rubi steps

\begin{align*} \int \frac{x^3 \sin (c+d x)}{\left (a+b x^3\right )^3} \, dx &=-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac{\int \frac{\sin (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}+\frac{d \int \frac{x \cos (c+d x)}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\int \frac{\sin (c+d x)}{x^3 \left (a+b x^3\right )} \, dx}{9 b^2}-\frac{d^2 \int \frac{\sin (c+d x)}{x \left (a+b x^3\right )} \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\int \left (\frac{\sin (c+d x)}{a x^3}-\frac{b \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{9 b^2}-\frac{d^2 \int \left (\frac{\sin (c+d x)}{a x}-\frac{b x^2 \sin (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx}{18 b^2}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{\int \frac{\sin (c+d x)}{x^3} \, dx}{9 a b^2}+\frac{\int \frac{\sin (c+d x)}{a+b x^3} \, dx}{9 a b}-\frac{d^2 \int \frac{\sin (c+d x)}{x} \, dx}{18 a b^2}+\frac{d^2 \int \frac{x^2 \sin (c+d x)}{a+b x^3} \, dx}{18 a b}\\ &=-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\frac{\sin (c+d x)}{18 a b^2 x^2}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \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}{9 a b}-\frac{d \int \frac{\cos (c+d x)}{x^2} \, dx}{18 a b^2}+\frac{d^2 \int \left (\frac{\sin (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sin (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac{\sin (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{18 a b}-\frac{\left (d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{18 a b^2}-\frac{\left (d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{18 a b^2}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{d^2 \text{Ci}(d x) \sin (c)}{18 a b^2}+\frac{\sin (c+d x)}{18 a b^2 x^2}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d^2 \cos (c) \text{Si}(d x)}{18 a b^2}-\frac{\int \frac{\sin (c+d x)}{-\sqrt [3]{a}-\sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\int \frac{\sin (c+d x)}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}-\frac{\int \frac{\sin (c+d x)}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x} \, dx}{27 a^{5/3} b}+\frac{d^2 \int \frac{\sin (c+d x)}{x} \, dx}{18 a b^2}+\frac{d^2 \int \frac{\sin (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}+\frac{d^2 \int \frac{\sin (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}+\frac{d^2 \int \frac{\sin (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}-\frac{d^2 \text{Ci}(d x) \sin (c)}{18 a b^2}+\frac{\sin (c+d x)}{18 a b^2 x^2}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac{d^2 \cos (c) \text{Si}(d x)}{18 a b^2}+\frac{\left (d^2 \cos (c)\right ) \int \frac{\sin (d x)}{x} \, dx}{18 a b^2}-\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}{27 a^{5/3} b}+\frac{\left (d^2 \cos \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}{54 a b^{5/3}}+\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}{27 a^{5/3} b}-\frac{\left (d^2 \cos \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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\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}{27 a^{5/3} b}+\frac{\left (d^2 \cos \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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}+\frac{\left (d^2 \sin (c)\right ) \int \frac{\cos (d x)}{x} \, dx}{18 a b^2}-\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}{27 a^{5/3} b}+\frac{\left (d^2 \sin \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}{54 a b^{5/3}}-\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}{27 a^{5/3} b}+\frac{\left (d^2 \sin \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]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}-\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}{27 a^{5/3} b}+\frac{\left (d^2 \sin \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 )}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{54 a b^{5/3}}\\ &=\frac{d \cos (c+d x)}{18 a b^2 x}-\frac{d \cos (c+d x)}{18 b^2 x \left (a+b x^3\right )}+\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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a b^2}-\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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a b^2}+\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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a b^2}+\frac{\sin (c+d x)}{18 a b^2 x^2}-\frac{x \sin (c+d x)}{6 b \left (a+b x^3\right )^2}-\frac{\sin (c+d x)}{18 b^2 x^2 \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 )}{27 a^{5/3} b^{4/3}}-\frac{d^2 \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 )}{54 a b^2}+\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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a b^2}+\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 )}{27 a^{5/3} b^{4/3}}+\frac{d^2 \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 )}{54 a b^2}\\ \end{align*}

Mathematica [C]  time = 0.612753, size = 457, normalized size = 0.59 \[ \frac{i \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-2 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-i \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{i \text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\text{$\#$1}^2 d^2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \text{$\#$1}^2 d^2 \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+2 i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+2 \cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-2 \sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+2 i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+\frac{6 b x \left (\left (b x^3-2 a\right ) \sin (c+d x)+d x \left (a+b x^3\right ) \cos (c+d x)\right )}{\left (a+b x^3\right )^2}}{108 a b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(I*RootSum[a + b*#1^3 & , (2*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - (2*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#
1] - (2*I)*Cos[c + d*#1]*SinIntegral[d*(x - #1)] - 2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d^2*Cos[c + d*#1]
*CosIntegral[d*(x - #1)]*#1^2 - I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 - I*d^2*Cos[c + d*#1]*SinInte
gral[d*(x - #1)]*#1^2 - d^2*Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] - I*RootSum[a + b*#1^3 & , (2
*Cos[c + d*#1]*CosIntegral[d*(x - #1)] + (2*I)*CosIntegral[d*(x - #1)]*Sin[c + d*#1] + (2*I)*Cos[c + d*#1]*Sin
Integral[d*(x - #1)] - 2*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d^2*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1^
2 + I*d^2*CosIntegral[d*(x - #1)]*Sin[c + d*#1]*#1^2 + I*d^2*Cos[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2 - d^2*
Sin[c + d*#1]*SinIntegral[d*(x - #1)]*#1^2)/#1^2 & ] + (6*b*x*(d*x*(a + b*x^3)*Cos[c + d*x] + (-2*a + b*x^3)*S
in[c + d*x]))/(a + b*x^3)^2)/(108*a*b^2)

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Maple [C]  time = 0.129, size = 2032, normalized size = 2.6 \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)^3,x)

[Out]

1/d^4*(1/18*sin(d*x+c)*d^3*(12*b^2*c^2*(d*x+c)^5+(d*x+c)^4*a*b*d^3-55*(d*x+c)^4*b^2*c^3-4*(d*x+c)^3*a*b*c*d^3+
100*(d*x+c)^3*b^2*c^4+27*(d*x+c)^2*a*b*c^2*d^3-90*(d*x+c)^2*b^2*c^5-2*(d*x+c)*a^2*d^6-38*(d*x+c)*a*b*c^3*d^3+4
0*(d*x+c)*b^2*c^6-7*a^2*c*d^6+14*a*b*c^4*d^3-7*b^2*c^7)/a^2/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+1/18*cos(d*x+c)*d^3*((d*x+c)^2*a*d^3-(d*x+c)^2*b*c^3+(d*x+c)*a*c*d^3+2*(d*x+c)*b*c^4+a*c^2*d^3-c^5
*b)/a^2/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)+1/54*d^3/a^2/b^2*sum((_R1^2*a*d^3-_R1^2*b*
c^3+_R1*a*c*d^3+2*_R1*b*c^4+a*c^2*d^3-b*c^5+12*_R1*b*c^2+2*a*d^3-2*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*c*d^3/a^2/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(_R
R1)),_RR1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/6*sin(d*x+c)*c*d^3*(8*b^2*c*(d*x+c)^5-35*b^2*c^2
*(d*x+c)^4+60*b^2*c^3*(d*x+c)^3+14*(d*x+c)^2*a*b*c*d^3-50*(d*x+c)^2*b^2*c^4-20*(d*x+c)*a*b*c^2*d^3+20*(d*x+c)*
b^2*c^5-3*a^2*d^6+6*a*b*c^3*d^3-3*b^2*c^6)/a^2/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+1
/6*cos(d*x+c)*c*d^3*(c^2*(d*x+c)^2*b-(d*x+c)*a*d^3-2*(d*x+c)*b*c^3-a*c*d^3+c^4*b)/a^2/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)+1/18*c*d^3/a^2/b^2*sum((_R1^2*b*c^2-_R1*a*d^3-2*_R1*b*c^3-a*c*d^3+b*c^4-8*
_R1*b*c-2*b*c^2)/(_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*c*d^3/a^2/b^2*sum((4*_RR1^2*b*c-5*_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
))+1/6*sin(d*x+c)*c^2*d^3*(4*b*(d*x+c)^5-15*b*c*(d*x+c)^4+20*b*c^2*(d*x+c)^3+7*(d*x+c)^2*a*d^3-10*(d*x+c)^2*b*
c^3-6*(d*x+c)*a*c*d^3-a*c^2*d^3+c^5*b)/a^2/((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)^2-1/6*cos
(d*x+c)*c^2*d^3*(c*(d*x+c)^2*b-2*(d*x+c)*b*c^2-a*d^3+c^3*b)/a^2/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)-1/18*c^2*d^3/a^2/b^2*sum((_R1^2*b*c-2*_R1*b*c^2-a*d^3+b*c^3-4*_R1*b-6*b*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))-1/9*c^2
*d^3/a^2/b*sum((2*_RR1+c)/(_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))-d^9*c^3*(1/18*sin(d*x+c)*(5*(d*x+c)^4*b-20*c*(d*x+c)^3*b+30*c^2*(d*x+c)^2*b+8
*(d*x+c)*a*d^3-20*(d*x+c)*b*c^3-8*a*c*d^3+5*c^4*b)/a^2/d^6/((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)^2-1/18*cos(d*x+c)*((d*x+c)^2-2*(d*x+c)*c+c^2)/a^2/d^6/((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)-1/54/a^2/d^6/b*sum((_R1^2-2*_R1*c+c^2-10)/(_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^2/d^6/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)^3,x, algorithm="maxima")

[Out]

Timed out

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Fricas [C]  time = 2.70271, size = 2021, normalized size = 2.62 \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)^3,x, algorithm="fricas")

[Out]

1/108*((I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3 + I*a^3*d^3 + (b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(3)*(I*b^3*x^6
+ 2*I*a*b^2*x^3 + I*a^2*b))*(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) + (-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3 + (b^3*x^6 + 2*a*b^2*
x^3 + a^2*b + sqrt(3)*(I*b^3*x^6 + 2*I*a*b^2*x^3 + I*a^2*b))*(-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) + (I*a*b^2*d^3*x^6 + 2*I*a^2*b*d^3*x^3
+ I*a^3*d^3 + (b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(3)*(-I*b^3*x^6 - 2*I*a*b^2*x^3 - I*a^2*b))*(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) + (-I
*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3 + (b^3*x^6 + 2*a*b^2*x^3 + a^2*b + sqrt(3)*(-I*b^3*x^6 - 2*I*a*
b^2*x^3 - I*a^2*b))*(-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) + (-I*a*b^2*d^3*x^6 - 2*I*a^2*b*d^3*x^3 - I*a^3*d^3 - 2*(b^3*x^6 + 2*a*b^2*x^3
+ a^2*b)*(-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)) + (I*a*b^2*d^3*x^6 +
2*I*a^2*b*d^3*x^3 + I*a^3*d^3 - 2*(b^3*x^6 + 2*a*b^2*x^3 + a^2*b)*(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)) + 6*(a*b^2*d^2*x^5 + a^2*b*d^2*x^2)*cos(d*x + c) + 6*(a*b^2*d*x^4 - 2*a^2*b
*d*x)*sin(d*x + c))/(a^2*b^4*d*x^6 + 2*a^3*b^3*d*x^3 + a^4*b^2*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)**3,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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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