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

Optimal. Leaf size=357 \[ -\frac{\sqrt [3]{a} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{a} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{a} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{\cos (c+d x)}{b d} \]

[Out]

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

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Rubi [A]  time = 0.675749, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {3345, 2638, 3333, 3303, 3299, 3302} \[ -\frac{\sqrt [3]{a} \sin \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}+\frac{\sqrt [3]{-1} \sqrt [3]{a} \sin \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{CosIntegral}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \sin \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{CosIntegral}\left (\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}+d x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{-1} \sqrt [3]{a} \cos \left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}+c\right ) \text{Si}\left (\frac{\sqrt [3]{-1} \sqrt [3]{a} d}{\sqrt [3]{b}}-d x\right )}{3 b^{4/3}}-\frac{\sqrt [3]{a} \cos \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{(-1)^{2/3} \sqrt [3]{a} \cos \left (c-\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \text{Si}\left (x d+\frac{(-1)^{2/3} \sqrt [3]{a} d}{\sqrt [3]{b}}\right )}{3 b^{4/3}}-\frac{\cos (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

Int[(x^3*Sin[c + d*x])/(a + b*x^3),x]

[Out]

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

Rule 3345

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3333

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

Rule 3303

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

Rule 3299

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

Rule 3302

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

Rubi steps

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

Mathematica [C]  time = 0.350349, size = 216, normalized size = 0.61 \[ -\frac{i a d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{-i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))-i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]-i a d \text{RootSum}\left [\text{$\#$1}^3 b+a\& ,\frac{i \sin (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))+\cos (\text{$\#$1} d+c) \text{CosIntegral}(d (x-\text{$\#$1}))-\sin (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))+i \cos (\text{$\#$1} d+c) \text{Si}(d (x-\text{$\#$1}))}{\text{$\#$1}^2}\& \right ]+6 b \cos (c+d x)}{6 b^2 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.019, size = 392, normalized size = 1.1 \begin{align*}{\frac{1}{{d}^{4}} \left ( -{\frac{{d}^{3}\cos \left ( dx+c \right ) }{b}}+{\frac{{d}^{3}}{3\,{b}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{ \left ( 3\,{{\it \_R1}}^{2}bc-3\,{\it \_R1}\,b{c}^{2}-a{d}^{3}+{c}^{3}b \right ) \left ( -{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{c{d}^{3}}{b}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{{{\it \_R1}}^{2} \left ( -{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}+{\frac{{c}^{2}{d}^{3}}{b}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{{\it \_R1}\, \left ( -{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}}-{\frac{{c}^{3}{d}^{3}}{3\,b}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{3}b-3\,{{\it \_Z}}^{2}bc+3\,{\it \_Z}\,b{c}^{2}+a{d}^{3}-{c}^{3}b \right ) }{\frac{-{\it Si} \left ( -dx+{\it \_R1}-c \right ) \cos \left ({\it \_R1} \right ) +{\it Ci} \left ( dx-{\it \_R1}+c \right ) \sin \left ({\it \_R1} \right ) }{{{\it \_R1}}^{2}-2\,{\it \_R1}\,c+{c}^{2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(-d^3/b*cos(d*x+c)+1/3/b^2*d^3*sum((3*_R1^2*b*c-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))-c*d^3/b*sum(_R1^2
/(_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))+c^2*d^3/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=Roo
tOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^2+a*d^3-b*c^3))-1/3*c^3*d^3/b*sum(1/(_R1^2-2*_R1*c+c^2)*(-Si(-d*x+_R1-c)*cos(_R
1)+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)))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [C]  time = 2.22818, size = 1006, normalized size = 2.82 \begin{align*} \frac{\left (\frac{i \, a d^{3}}{b}\right )^{\frac{1}{3}}{\left (-i \, \sqrt{3} - 1\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 (-\frac{i \, a d^{3}}{b}\right )^{\frac{1}{3}}{\left (-i \, \sqrt{3} - 1\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 (\frac{i \, a d^{3}}{b}\right )^{\frac{1}{3}}{\left (i \, \sqrt{3} - 1\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 (-\frac{i \, a d^{3}}{b}\right )^{\frac{1}{3}}{\left (i \, \sqrt{3} - 1\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 (-\frac{i \, a d^{3}}{b}\right )^{\frac{1}{3}}{\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 (\frac{i \, a d^{3}}{b}\right )^{\frac{1}{3}}{\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 )} - 12 \, \cos \left (d x + c\right )}{12 \, b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*((I*a*d^3/b)^(1/3)*(-I*sqrt(3) - 1)*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*d^3/b)^(1/3)*(-I*sqrt(3) - 1)*Ei(I*d*x + 1/2*(-I*a*d^3/b)^(1/3)*(-I*sqr
t(3) - 1))*e^(1/2*(-I*a*d^3/b)^(1/3)*(I*sqrt(3) + 1) + I*c) + (I*a*d^3/b)^(1/3)*(I*sqrt(3) - 1)*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*d^3/b)^(1/3)*(I*
sqrt(3) - 1)*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*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*a*d^3/b)^(1/3)*E
i(-I*d*x + (I*a*d^3/b)^(1/3))*e^(-I*c - (I*a*d^3/b)^(1/3)) - 12*cos(d*x + c))/(b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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