3.2.2 \(\int \frac {x^3 \sin (c+d x)}{(a+b x^3)^2} \, dx\) [102]

Optimal. Leaf size=714 \[ -\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}} \]

[Out]

-1/9*d*Ci(a^(1/3)*d/b^(1/3)+d*x)*cos(c-a^(1/3)*d/b^(1/3))/a^(1/3)/b^(5/3)-1/9*(-1)^(2/3)*d*Ci((-1)^(1/3)*a^(1/
3)*d/b^(1/3)-d*x)*cos(c+(-1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(1/3)/b^(5/3)+1/9*(-1)^(1/3)*d*Ci((-1)^(2/3)*a^(1/3)*d
/b^(1/3)+d*x)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))/a^(1/3)/b^(5/3)-1/9*(-1)^(1/3)*cos(c+(-1)^(1/3)*a^(1/3)*d/b^
(1/3))*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)/a^(2/3)/b^(4/3)+1/9*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)
+d*x)/a^(2/3)/b^(4/3)+1/9*(-1)^(2/3)*cos(c-(-1)^(2/3)*a^(1/3)*d/b^(1/3))*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)/
a^(2/3)/b^(4/3)+1/9*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)+1/9*d*Si(a^(1/3)*d/b^(1
/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(1/3)/b^(5/3)-1/9*(-1)^(1/3)*Ci((-1)^(1/3)*a^(1/3)*d/b^(1/3)-d*x)*sin(c+(-
1)^(1/3)*a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)+1/9*(-1)^(2/3)*d*Si(-(-1)^(1/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c+(-1)^
(1/3)*a^(1/3)*d/b^(1/3))/a^(1/3)/b^(5/3)+1/9*(-1)^(2/3)*Ci((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*
a^(1/3)*d/b^(1/3))/a^(2/3)/b^(4/3)-1/9*(-1)^(1/3)*d*Si((-1)^(2/3)*a^(1/3)*d/b^(1/3)+d*x)*sin(c-(-1)^(2/3)*a^(1
/3)*d/b^(1/3))/a^(1/3)/b^(5/3)-1/3*x*sin(d*x+c)/b/(b*x^3+a)

________________________________________________________________________________________

Rubi [A]
time = 0.66, antiderivative size = 714, normalized size of antiderivative = 1.00, 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 = {3424, 3414, 3384, 3380, 3383, 3427} \begin {gather*} -\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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*((-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])/
(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 3380

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 3383

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 3384

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 3414

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 3424

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 3427

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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.15, size = 383, normalized size = 0.54 \begin {gather*} \frac {\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {i \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+\text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+\cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-i d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+\text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-i \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+\text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+\cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}+i d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}+i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-\frac {6 b x \sin (c+d x)}{a+b x^3}}{18 b^2} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.43, size = 1184, normalized size = 1.66

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1184\)
default	\(\text {Expression too large to display}\)	\(1184\)
risch	\(\text {Expression too large to display}\)	\(1379\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(-d^6*c^3*(sin(d*x+c)*(1/3/a/d^3*(d*x+c)-1/3*c/a/d^3)/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*
x+c)^3)+2/9/a/d^3/b*sum(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)))+sin(d*x+c)*(c^2*d^3/a*(d*x+c)^2-c^3*d^3/a*(
d*x+c))/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)+1/3*c^2*d^3/a/b*sum((c+_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^2*d^3/a/b*sum(_RR1/(-_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))+sin(d*x+c)*(-2*c^2*d^3/a*(d*x+c)^2+3*c^3*d^3/a*(d*x+c)+d^3*c*(a*d^3-b*c^3)/a/b)
/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^3)-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*d^3*c/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)^2-1/3*d^3*(a
*d^3+5*b*c^3)/a/b*(d*x+c)-2/3*d^3*c*(a*d^3-b*c^3)/a/b)/(a*d^3-b*c^3+3*b*c^2*(d*x+c)-3*b*c*(d*x+c)^2+b*(d*x+c)^
3)+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+_RR1-c)*sin(_RR1)+Ci(d*x-_RR1+c)*cos(_RR1)),_RR1=R
ootOf(_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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

-1/2*(3*(cos(c)^2 + sin(c)^2)*d*x^2*sin(d*x + c) + ((d^2*x^3*cos(c) - 3*d*x^2*sin(c) - 12*x*cos(c))*cos(d*x +
c)^2 + (d^2*x^3*cos(c) - 3*d*x^2*sin(c) - 12*x*cos(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + ((cos(c)^2 + sin(c)^2)
*d^2*x^3 - 12*(cos(c)^2 + sin(c)^2)*x)*cos(d*x + c) - 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^6 + 2*(a*b*cos(c
)^2 + a*b*sin(c)^2)*d^3*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^
2)*d^3*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*sin(d*x + c)^2)*inte
grate(3*(3*a*d*x*sin(d*x + c) + (a*d^2*x^2 + 10*b*x^3 - 2*a)*cos(d*x + c))/(b^3*d^3*x^9 + 3*a*b^2*d^3*x^6 + 3*
a^2*b*d^3*x^3 + a^3*d^3), x) - 2*(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3
*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^6 + 2*(a*b*cos
(c)^2 + a*b*sin(c)^2)*d^3*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(3*(3*a*d*x*sin(d*
x + c) + (a*d^2*x^2 + 10*b*x^3 - 2*a)*cos(d*x + c))/((b^3*d^3*x^9 + 3*a*b^2*d^3*x^6 + 3*a^2*b*d^3*x^3 + a^3*d^
3)*cos(d*x + c)^2 + (b^3*d^3*x^9 + 3*a*b^2*d^3*x^6 + 3*a^2*b*d^3*x^3 + a^3*d^3)*sin(d*x + c)^2), x) + ((d^2*x^
3*sin(c) + 3*d*x^2*cos(c) - 12*x*sin(c))*cos(d*x + c)^2 + (d^2*x^3*sin(c) + 3*d*x^2*cos(c) - 12*x*sin(c))*sin(
d*x + c)^2)*sin(d*x + 2*c))/(((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^6 + 2*(a*b*cos(c)^2 + a*b*sin(c)^2)*d^3*x^3
+ (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^2*cos(c)^2 + b^2*sin(c)^2)*d^3*x^6 + 2*(a*b*cos(c)^2
+ a*b*sin(c)^2)*d^3*x^3 + (a^2*cos(c)^2 + a^2*sin(c)^2)*d^3)*sin(d*x + c)^2)

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Fricas [C] Result contains complex when optimal does not.
time = 0.40, size = 670, normalized size = 0.94 \begin {gather*} -\frac {12 \, a d x \sin \left (d x + c\right ) + {\left ({\left (b x^{3} - \sqrt {3} {\left (i \, b x^{3} + i \, a\right )} + a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - {\left (b x^{3} + \sqrt {3} {\left (i \, b x^{3} + i \, a\right )} + a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left ({\left (b x^{3} - \sqrt {3} {\left (i \, b x^{3} + i \, a\right )} + a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - {\left (b x^{3} + \sqrt {3} {\left (i \, b x^{3} + i \, a\right )} + a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} + 1\right )} + i \, c\right )} + {\left ({\left (b x^{3} - \sqrt {3} {\left (-i \, b x^{3} - i \, a\right )} + a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - {\left (b x^{3} + \sqrt {3} {\left (-i \, b x^{3} - i \, a\right )} + a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} - i \, c\right )} + {\left ({\left (b x^{3} - \sqrt {3} {\left (-i \, b x^{3} - i \, a\right )} + a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - {\left (b x^{3} + \sqrt {3} {\left (-i \, b x^{3} - i \, a\right )} + a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} - 1\right )}\right ) e^{\left (\frac {1}{2} \, \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} + 1\right )} + i \, c\right )} - 2 \, {\left ({\left (b x^{3} + a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (i \, d x + \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (i \, c - \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} - 2 \, {\left ({\left (b x^{3} + a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )} {\rm Ei}\left (-i \, d x + \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right ) e^{\left (-i \, c - \left (\frac {i \, a d^{3}}{b}\right )^{\frac {1}{3}}\right )}}{36 \, {\left (a b^{2} d x^{3} + a^{2} b d\right )}} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,\sin \left (c+d\,x\right )}{{\left (b\,x^3+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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