3.2.9 \(\int \frac {x^3 \sin (c+d x)}{(a+b x^3)^3} \, dx\) [109]

Optimal. Leaf size=772 \[ \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} \]

[Out]

1/18*d*cos(d*x+c)/a/b^2/x-1/18*d*cos(d*x+c)/b^2/x/(b*x^3+a)-1/27*(-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^(5/3)/b^(4/3)+1/54*d^2*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/b^2+1/27*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a^(5/3)/b^(4/3)+1/5
4*d^2*cos(c-a^(1/3)*d/b^(1/3))*Si(a^(1/3)*d/b^(1/3)+d*x)/a/b^2+1/27*(-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^(5/3)/b^(4/3)+1/54*d^2*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/b^2+1/27*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a^(5/3)/b^(4/3)+1/
54*d^2*Ci(a^(1/3)*d/b^(1/3)+d*x)*sin(c-a^(1/3)*d/b^(1/3))/a/b^2-1/27*(-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^(5/3)/b^(4/3)+1/54*d^2*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/b^2+1/27*(-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^(5/3)/b^(4/3)+1/54*d^2*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/b^2+1/18*sin(d*x+c)/a/b^2/x^2-1/6*x*sin(d*x+c)/b/(b*x^3+a)^2-1/18*sin(d*x+c)/b^2/x^2/(b*x^3+a)

________________________________________________________________________________________

Rubi [A]
time = 1.72, antiderivative size = 772, normalized size of antiderivative = 1.00, 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 = {3424, 3412, 3426, 3378, 3384, 3380, 3383, 3414, 3427, 3425} \begin {gather*} \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 {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 {\sin (c+d x)}{18 b^2 x^2 \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} \end {gather*}

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 3378

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 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 3412

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 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 3425

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]

Rule 3426

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 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 )^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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.43, size = 457, normalized size = 0.59 \begin {gather*} \frac {i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-2 i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-2 i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d^2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}^2-i d^2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2-i d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]-i \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))+2 i \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})+2 i \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d^2 \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}^2+i d^2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}^2+i d^2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2-d^2 \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}^2}{\text {$\#$1}^2}\&\right ]+\frac {6 b x \left (d x \left (a+b x^3\right ) \cos (c+d x)+\left (-2 a+b x^3\right ) \sin (c+d x)\right )}{\left (a+b x^3\right )^2}}{108 a b^2} \end {gather*}

Antiderivative was successfully verified.

[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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.34, size = 2035, normalized size = 2.64

method	result	size
risch	\(\text {Expression too large to display}\)	\(1337\)
derivativedivides	\(\text {Expression too large to display}\)	\(2035\)
default	\(\text {Expression too large to display}\)	\(2035\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(-d^9*c^3*(-1/18*sin(d*x+c)*(8*a*c*d^3-8*a*d^3*(d*x+c)-5*b*c^4+20*b*c^3*(d*x+c)-30*b*c^2*(d*x+c)^2+20*b*
c*(d*x+c)^3-5*b*(d*x+c)^4)/a^2/d^6/(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)^2-1/18*cos(d*x+c)
*(c^2-2*(d*x+c)*c+(d*x+c)^2)/a^2/d^6/(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)-1/54/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)))-1/6*sin(d*x+c)*c^2*d^3*(a*c^2*d^3+6*c*d^
3*(d*x+c)*a-7*a*d^3*(d*x+c)^2-c^5*b+10*b*c^3*(d*x+c)^2-20*b*c^2*(d*x+c)^3+15*b*c*(d*x+c)^4-4*b*(d*x+c)^5)/a^2/
(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+1/6*cos(d*x+c)*c^2*d^3*(a*d^3-b*c^3+2*b*c^2*(d*x+c
)-b*c*(d*x+c)^2)/a^2/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)+1/18*c^2*d^3/a^2/b^2*sum((-_R
1^2*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/a^2*d^3/b*c^2*sum((c+2*_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))+1/6*sin(
d*x+c)*d^3*c*(3*a^2*d^6-6*a*b*c^3*d^3+20*a*b*c^2*d^3*(d*x+c)-14*a*b*c*d^3*(d*x+c)^2+3*b^2*c^6-20*b^2*c^5*(d*x+
c)+50*b^2*c^4*(d*x+c)^2-60*b^2*c^3*(d*x+c)^3+35*b^2*c^2*(d*x+c)^4-8*b^2*c*(d*x+c)^5)/a^2/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)^2-1/6*cos(d*x+c)*d^3*c*(a*c*d^3+a*d^3*(d*x+c)-b*c^4+2*b*c^3*(d*x+c)-b*c
^2*(d*x+c)^2)/a^2/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)-1/18*d^3*c/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*d^3*c/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=Roo
tOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/18*sin(d*x+c)*d^3*(7*a^2*c*d^6+2*a^2*d^6*(d*x+c)-14*a*b*c^4*d
^3+38*a*b*c^3*d^3*(d*x+c)-27*a*b*c^2*d^3*(d*x+c)^2+4*a*b*c*d^3*(d*x+c)^3-a*b*d^3*(d*x+c)^4+7*b^2*c^7-40*b^2*c^
6*(d*x+c)+90*b^2*c^5*(d*x+c)^2-100*b^2*c^4*(d*x+c)^3+55*b^2*c^3*(d*x+c)^4-12*b^2*c^2*(d*x+c)^5)/a^2/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+1/18*cos(d*x+c)*d^3*(a*c^2*d^3+c*d^3*(d*x+c)*a+a*d^3*(d*x+
c)^2-c^5*b+2*b*c^4*(d*x+c)-b*c^3*(d*x+c)^2)/a^2/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/
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*d^3*c/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(_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]
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)^3,x, algorithm="maxima")

[Out]

-1/2*(6*(cos(c)^2 + sin(c)^2)*d*x^2*sin(d*x + c) + ((d^2*x^3*cos(c) - 6*d*x^2*sin(c) - 42*x*cos(c))*cos(d*x +
c)^2 + (d^2*x^3*cos(c) - 6*d*x^2*sin(c) - 42*x*cos(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + ((cos(c)^2 + sin(c)^2)
*d^2*x^3 - 42*(cos(c)^2 + sin(c)^2)*x)*cos(d*x + c) - 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^9 + 3*(a*b^2*cos
(c)^2 + a*b^2*sin(c)^2)*d^3*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*
d^3)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x^6 + 3
*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(3/2*
(18*a*d*x*sin(d*x + c) + (3*a*d^2*x^2 + 112*b*x^3 - 14*a)*cos(d*x + c))/(b^4*d^3*x^12 + 4*a*b^3*d^3*x^9 + 6*a^
2*b^2*d^3*x^6 + 4*a^3*b*d^3*x^3 + a^4*d^3), x) - 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^9 + 3*(a*b^2*cos(c)^2
+ a*b^2*sin(c)^2)*d^3*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*
cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x^6 + 3*(a^2
*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(3/2*(18*a
*d*x*sin(d*x + c) + (3*a*d^2*x^2 + 112*b*x^3 - 14*a)*cos(d*x + c))/((b^4*d^3*x^12 + 4*a*b^3*d^3*x^9 + 6*a^2*b^
2*d^3*x^6 + 4*a^3*b*d^3*x^3 + a^4*d^3)*cos(d*x + c)^2 + (b^4*d^3*x^12 + 4*a*b^3*d^3*x^9 + 6*a^2*b^2*d^3*x^6 +
4*a^3*b*d^3*x^3 + a^4*d^3)*sin(d*x + c)^2), x) + ((d^2*x^3*sin(c) + 6*d*x^2*cos(c) - 42*x*sin(c))*cos(d*x + c)
^2 + (d^2*x^3*sin(c) + 6*d*x^2*cos(c) - 42*x*sin(c))*sin(d*x + c)^2)*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*sin
(c)^2)*d^3*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^3 + (
a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^9 + 3*(a*b^2*cos(c)^2
+ a*b^2*sin(c)^2)*d^3*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*s
in(d*x + c)^2)

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Fricas [C] Result contains complex when optimal does not.
time = 0.42, size = 890, normalized size = 1.15 \begin {gather*} \frac {{\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} + {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b + \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\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 (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} + {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b + \sqrt {3} {\left (i \, b^{3} x^{6} + 2 i \, a b^{2} x^{3} + i \, a^{2} b\right )}\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 (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} + {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b + \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\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 (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} + {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b + \sqrt {3} {\left (-i \, b^{3} x^{6} - 2 i \, a b^{2} x^{3} - i \, a^{2} b\right )}\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 (-i \, a b^{2} d^{3} x^{6} - 2 i \, a^{2} b d^{3} x^{3} - i \, a^{3} d^{3} - 2 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\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 )} + {\left (i \, a b^{2} d^{3} x^{6} + 2 i \, a^{2} b d^{3} x^{3} + i \, a^{3} d^{3} - 2 \, {\left (b^{3} x^{6} + 2 \, a b^{2} x^{3} + a^{2} b\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 )} + 6 \, {\left (a b^{2} d^{2} x^{5} + a^{2} b d^{2} x^{2}\right )} \cos \left (d x + c\right ) + 6 \, {\left (a b^{2} d x^{4} - 2 \, a^{2} b d x\right )} \sin \left (d x + c\right )}{108 \, {\left (a^{2} b^{4} d x^{6} + 2 \, a^{3} b^{3} d x^{3} + a^{4} b^{2} 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)^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)] 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)**3,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)^3,x, algorithm="giac")

[Out]

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

[Out]

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

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