3.2.10 \(\int \frac {x^2 \sin (c+d x)}{(a+b x^3)^3} \, dx\) [110]

Optimal. Leaf size=777 \[ \frac {d \cos (c+d x)}{18 a b^2 x^2}-\frac {d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {\sqrt [3]{-1} 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 )}{27 a^{5/3} b^{4/3}}+\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 )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} 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 )}{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^{4/3} b^{5/3}}-\frac {(-1)^{2/3} 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^{4/3} b^{5/3}}+\frac {\sqrt [3]{-1} 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^{4/3} b^{5/3}}-\frac {\sin (c+d x)}{6 b \left (a+b x^3\right )^2}+\frac {(-1)^{2/3} 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^{4/3} b^{5/3}}-\frac {\sqrt [3]{-1} 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 )}{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^{4/3} b^{5/3}}-\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 )}{27 a^{5/3} b^{4/3}}+\frac {\sqrt [3]{-1} 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^{4/3} b^{5/3}}-\frac {(-1)^{2/3} 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 )}{27 a^{5/3} b^{4/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.93, antiderivative size = 777, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3422, 3413, 3427, 3378, 3384, 3380, 3383, 3415, 3426} \begin {gather*} -\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^{4/3} b^{5/3}}-\frac {(-1)^{2/3} 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^{4/3} b^{5/3}}+\frac {\sqrt [3]{-1} 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^{4/3} b^{5/3}}-\frac {\sqrt [3]{-1} 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 )}{27 a^{5/3} b^{4/3}}+\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 )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} 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 )}{27 a^{5/3} b^{4/3}}+\frac {(-1)^{2/3} 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^{4/3} b^{5/3}}-\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^{4/3} b^{5/3}}+\frac {\sqrt [3]{-1} 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^{4/3} b^{5/3}}-\frac {\sqrt [3]{-1} 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 )}{27 a^{5/3} b^{4/3}}-\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 )}{27 a^{5/3} b^{4/3}}-\frac {(-1)^{2/3} 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 )}{27 a^{5/3} b^{4/3}}+\frac {d \cos (c+d x)}{18 a b^2 x^2}-\frac {d \cos (c+d x)}{18 b^2 x^2 \left (a+b x^3\right )}-\frac {\sin (c+d x)}{6 b \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Cos[c + d*x])/(18*a*b^2*x^2) - (d*Cos[c + d*x])/(18*b^2*x^2*(a + b*x^3)) - ((-1)^(1/3)*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])/(27*a^(5/3)*b^(4/3)) + (d*Cos[c - (a^
(1/3)*d)/b^(1/3)]*CosIntegral[(a^(1/3)*d)/b^(1/3) + d*x])/(27*a^(5/3)*b^(4/3)) + ((-1)^(2/3)*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])/(27*a^(5/3)*b^(4/3)) - (d^2*CosInt
egral[(a^(1/3)*d)/b^(1/3) + d*x]*Sin[c - (a^(1/3)*d)/b^(1/3)])/(54*a^(4/3)*b^(5/3)) - ((-1)^(2/3)*d^2*CosInteg
ral[((-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^(4/3)*b^(5/3)) + ((-1
)^(1/3)*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^(
4/3)*b^(5/3)) - Sin[c + d*x]/(6*b*(a + b*x^3)^2) + ((-1)^(2/3)*d^2*Cos[c + ((-1)^(1/3)*a^(1/3)*d)/b^(1/3)]*Sin
Integral[((-1)^(1/3)*a^(1/3)*d)/b^(1/3) - d*x])/(54*a^(4/3)*b^(5/3)) - ((-1)^(1/3)*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])/(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^(4/3)*b^(5/3)) - (d*Sin[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)) + ((-1)^(1/3)*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^(4/3)*b^(5/3)) - ((-1)^(2/3)*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])/(27*a^(5/3)*b^(4/3))

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 3413

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

Rule 3415

Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Cos[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 3422

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

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

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
time = 0.31, size = 449, normalized size = 0.58 \begin {gather*} \frac {i d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {-2 i \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+2 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}-i d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}-i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]-i d \text {RootSum}\left [a+b \text {$\#$1}^3\&,\frac {2 i \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1}))-2 \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1})-2 \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))-2 i \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1}))+d \cos (c+d \text {$\#$1}) \text {Ci}(d (x-\text {$\#$1})) \text {$\#$1}+i d \text {Ci}(d (x-\text {$\#$1})) \sin (c+d \text {$\#$1}) \text {$\#$1}+i d \cos (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}-d \sin (c+d \text {$\#$1}) \text {Si}(d (x-\text {$\#$1})) \text {$\#$1}}{\text {$\#$1}^2}\&\right ]+\frac {6 b \cos (d x) \left (d x \left (a+b x^3\right ) \cos (c)-3 a \sin (c)\right )}{\left (a+b x^3\right )^2}-\frac {6 b \left (3 a \cos (c)+d x \left (a+b x^3\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^3\right )^2}}{108 a b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(I*d*RootSum[a + b*#1^3 & , ((-2*I)*Cos[c + d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c +
d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x - #1)] + (2*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 & ] - I*d*RootSum[a + b*#1^3 & , ((2*I)*Cos[c
+ d*#1]*CosIntegral[d*(x - #1)] - 2*CosIntegral[d*(x - #1)]*Sin[c + d*#1] - 2*Cos[c + d*#1]*SinIntegral[d*(x -
#1)] - (2*I)*Sin[c + d*#1]*SinIntegral[d*(x - #1)] + d*Cos[c + d*#1]*CosIntegral[d*(x - #1)]*#1 + I*d*CosInte
gral[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*Cos[d*x]*(d*x*(a + b*x^3)*Cos[c] - 3*a*Sin[c]))/(a + b*x^3)^2 - (6*b*(3*a*Cos
[c] + d*x*(a + b*x^3)*Sin[c])*Sin[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 = 0.19, size = 1396, normalized size = 1.80

method	result	size
risch	\(-\frac {i \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} a \,d^{3}+2 i \textit {\_R1} b \,c^{3}-\textit {\_R1}^{2} b \,c^{2}-8 i \textit {\_R1}^{2} b c -a c \,d^{3}-2 i a \,d^{3}+b \,c^{4}+2 i b \,c^{3}-10 \textit {\_R1} b \,c^{2}+8 i \textit {\_R1} b c -2 b \,c^{2}\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{108 a^{2} b^{2}}-\frac {i c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}+6 i c -6 \textit {\_R1} +10\right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}-\frac {i c \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i b \textit {\_R1} \,c^{2}+\textit {\_R1}^{2} b c +4 i \textit {\_R1}^{2} b +a \,d^{3}-b \,c^{3}+2 i b \,c^{2}+2 b \textit {\_R1} c -4 i \textit {\_R1} b +6 c b \right ) {\mathrm e}^{\textit {\_R1}} \expIntegral \left (1, -i d x -i c +\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{54 a^{2} b^{2}}+\frac {i \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (i \textit {\_R1} a \,d^{3}+2 i \textit {\_R1} b \,c^{3}-\textit {\_R1}^{2} b \,c^{2}+8 i \textit {\_R1}^{2} b c -a c \,d^{3}+2 i a \,d^{3}+b \,c^{4}-2 i b \,c^{3}+10 \textit {\_R1} b \,c^{2}+8 i \textit {\_R1} b c -2 b \,c^{2}\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{108 a^{2} b^{2}}+\frac {i c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}-6 i c +6 \textit {\_R1} +10\right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{-2 i c \textit {\_R1} +\textit {\_R1}^{2}-c^{2}}\right )}{108 a^{2} b}+\frac {i c \left (\munderset {\textit {\_R1} =\RootOf \left (-3 i \textit {\_Z}^{2} b c -i a \,d^{3}+i b \,c^{3}+b \,\textit {\_Z}^{3}-3 b \,c^{2} \textit {\_Z} \right )}{\sum }\frac {\left (-2 i b \textit {\_R1} \,c^{2}+\textit {\_R1}^{2} b c -4 i \textit {\_R1}^{2} b +a \,d^{3}-b \,c^{3}-2 i b \,c^{2}-2 b \textit {\_R1} c -4 i \textit {\_R1} b +6 c b \right ) {\mathrm e}^{-\textit {\_R1}} \expIntegral \left (1, i d x +i c -\textit {\_R1} \right )}{2 i c \textit {\_R1} -\textit {\_R1}^{2}+c^{2}}\right )}{54 a^{2} b^{2}}+\frac {\left (a b \,d^{7} x^{4}+a^{2} d^{7} x \right ) \cos \left (d x +c \right )}{18 a^{2} b \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}-\frac {d^{6} \sin \left (d x +c \right )}{6 b \left (b^{2} x^{6} d^{6}+2 a b \,d^{6} x^{3}+a^{2} d^{6}\right )}\)	\(918\)
derivativedivides	\(\text {Expression too large to display}\)	\(1396\)
default	\(\text {Expression too large to display}\)	\(1396\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^3*(d^9*c^2*(-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*s
um((_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)))+1/9*sin(d*x+c)*d^3*c*(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)^2-1/9*cos(d*x+c)*d^3*c*(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)-1/27*d^3*c/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))-2/27/a^2*d^3/b*c*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))-1/18*sin(d*x+c)*
d^3*(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+1/18*cos(d*x+c)*d^3*(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)+1/54*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)*si
n(_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))+1/27*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)))

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

[Out]

-1/2*((cos(c)^2 + sin(c)^2)*d*x^2*cos(d*x + c) + 7*(cos(c)^2 + sin(c)^2)*x*sin(d*x + c) + ((d*x^2*cos(c) - 7*x
*sin(c))*cos(d*x + c)^2 + (d*x^2*cos(c) - 7*x*sin(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + 2*(((b^3*cos(c)^2 + b^3
*sin(c)^2)*d^2*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2*x^3
+ (a^3*cos(c)^2 + a^3*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^9 + 3*(a*b^2*cos(c
)^2 + a*b^2*sin(c)^2)*d^2*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^
2)*sin(d*x + c)^2)*integrate(-1/2*(9*a*d*x*cos(d*x + c) - 7*(8*b*x^3 - a)*sin(d*x + c))/(b^4*d^2*x^12 + 4*a*b^
3*d^2*x^9 + 6*a^2*b^2*d^2*x^6 + 4*a^3*b*d^2*x^3 + a^4*d^2), x) + 2*(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^9 + 3
*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2*x^3 + (a^3*cos(c)^2 + a^3
*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*
d^2*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^2)*sin(d*x + c)^2)*int
egrate(-1/2*(9*a*d*x*cos(d*x + c) - 7*(8*b*x^3 - a)*sin(d*x + c))/((b^4*d^2*x^12 + 4*a*b^3*d^2*x^9 + 6*a^2*b^2
*d^2*x^6 + 4*a^3*b*d^2*x^3 + a^4*d^2)*cos(d*x + c)^2 + (b^4*d^2*x^12 + 4*a*b^3*d^2*x^9 + 6*a^2*b^2*d^2*x^6 + 4
*a^3*b*d^2*x^3 + a^4*d^2)*sin(d*x + c)^2), x) + ((d*x^2*sin(c) + 7*x*cos(c))*cos(d*x + c)^2 + (d*x^2*sin(c) +
7*x*cos(c))*sin(d*x + c)^2)*sin(d*x + 2*c))/(((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^9 + 3*(a*b^2*cos(c)^2 + a*b^
2*sin(c)^2)*d^2*x^6 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^2*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^2)*cos(d*x
+ c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^2*x^9 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^2*x^6 + 3*(a^2*b*cos(
c)^2 + a^2*b*sin(c)^2)*d^2*x^3 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^2)*sin(d*x + c)^2)

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Fricas [C] Result contains complex when optimal does not.
time = 0.42, size = 935, normalized size = 1.20 \begin {gather*} \frac {{\left ({\left (-i \, b^{2} x^{6} - 2 i \, a b x^{3} - i \, a^{2} - \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 2 \, {\left (i \, b^{2} x^{6} + 2 i \, a b x^{3} + i \, a^{2} - \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\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 ({\left (i \, b^{2} x^{6} + 2 i \, a b x^{3} + i \, a^{2} + \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 2 \, {\left (-i \, b^{2} x^{6} - 2 i \, a b x^{3} - i \, a^{2} + \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\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 ({\left (-i \, b^{2} x^{6} - 2 i \, a b x^{3} - i \, a^{2} + \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 2 \, {\left (i \, b^{2} x^{6} + 2 i \, a b x^{3} + i \, a^{2} + \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\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 ({\left (i \, b^{2} x^{6} + 2 i \, a b x^{3} + i \, a^{2} - \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} - 2 \, {\left (-i \, b^{2} x^{6} - 2 i \, a b x^{3} - i \, a^{2} - \sqrt {3} {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\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 )} - 2 \, {\left ({\left (i \, b^{2} x^{6} + 2 i \, a b x^{3} + i \, a^{2}\right )} \left (-\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 2 \, {\left (i \, b^{2} x^{6} + 2 i \, a b x^{3} + i \, a^{2}\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 (-i \, b^{2} x^{6} - 2 i \, a b x^{3} - i \, a^{2}\right )} \left (\frac {i \, a d^{3}}{b}\right )^{\frac {2}{3}} + 2 \, {\left (-i \, b^{2} x^{6} - 2 i \, a b x^{3} - i \, a^{2}\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 \, a^{2} \sin \left (d x + c\right ) + 12 \, {\left (a b d x^{4} + a^{2} d x\right )} \cos \left (d x + c\right )}{216 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*(((-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(I*a*d^3/b)^(2/3) - 2*(I*b^2*
x^6 + 2*I*a*b*x^3 + I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(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*b^2*x^6 + 2*I*a*b*x^3 + I*a^2
+ sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(-I*a*d^3/b)^(2/3) - 2*(-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 + sqrt(3)*(b^2
*x^6 + 2*a*b*x^3 + a^2))*(-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*b^2*x^6 - 2*I*a*b*x^3 - I*a^2 + sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2)
)*(I*a*d^3/b)^(2/3) - 2*(I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 + sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(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
*b^2*x^6 + 2*I*a*b*x^3 + I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(-I*a*d^3/b)^(2/3) - 2*(-I*b^2*x^6 - 2*I
*a*b*x^3 - I*a^2 - sqrt(3)*(b^2*x^6 + 2*a*b*x^3 + a^2))*(-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*((I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2)*(-I*
a*d^3/b)^(2/3) + 2*(I*b^2*x^6 + 2*I*a*b*x^3 + I*a^2)*(-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*((-I*b^2*x^6 - 2*I*a*b*x^3 - I*a^2)*(I*a*d^3/b)^(2/3) + 2*(-I*b^2*x^6 - 2*I*a*b*x^3
- I*a^2)*(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)) - 36*a^2*sin(d*x + c)
+ 12*(a*b*d*x^4 + a^2*d*x)*cos(d*x + c))/(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)

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

[Out]

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

[Out]

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

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