3.30 \(\int \frac{a+b \tan ^{-1}(c x^3)}{d+e x} \, dx\)

Optimal. Leaf size=739 \[ -\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \text{PolyLog}\left (2,\frac{\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt{-c^2} e}+\frac{\log (d+e x) \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{e}+\frac{b c \log (d+e x) \log \left (\frac{e \left (1-\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left (-\frac{e \left (\sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left (-\frac{e \left (\sqrt [6]{-c^2} x+\sqrt [3]{-1}\right )}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left (-\frac{e \left (\sqrt [6]{-c^2} x+(-1)^{2/3}\right )}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt{-c^2} e}+\frac{b c \log (d+e x) \log \left (\frac{(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt{-c^2} e}-\frac{b c \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt{-c^2} e} \]

[Out]

((a + b*ArcTan[c*x^3])*Log[d + e*x])/e + (b*c*Log[(e*(1 - (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + e)]*Log[d + e*x])
/(2*Sqrt[-c^2]*e) - (b*c*Log[-((e*(1 + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e)
+ (b*c*Log[-((e*((-1)^(1/3) + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - (-1)^(1/3)*e))]*Log[d + e*x])/(2*Sqrt[-c^2]*e
) - (b*c*Log[-((e*((-1)^(2/3) + (-c^2)^(1/6)*x))/((-c^2)^(1/6)*d - (-1)^(2/3)*e))]*Log[d + e*x])/(2*Sqrt[-c^2]
*e) + (b*c*Log[((-1)^(2/3)*e*(1 + (-1)^(1/3)*(-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + (-1)^(2/3)*e)]*Log[d + e*x])/(
2*Sqrt[-c^2]*e) - (b*c*Log[((-1)^(1/3)*e*(1 + (-1)^(2/3)*(-c^2)^(1/6)*x))/((-c^2)^(1/6)*d + (-1)^(1/3)*e)]*Log
[d + e*x])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d - e)])/(2*Sqrt[-c^2]*e)
+ (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + e)])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^
(1/6)*(d + e*x))/((-c^2)^(1/6)*d - (-1)^(1/3)*e)])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))
/((-c^2)^(1/6)*d + (-1)^(1/3)*e)])/(2*Sqrt[-c^2]*e) - (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d
- (-1)^(2/3)*e)])/(2*Sqrt[-c^2]*e) + (b*c*PolyLog[2, ((-c^2)^(1/6)*(d + e*x))/((-c^2)^(1/6)*d + (-1)^(2/3)*e)
])/(2*Sqrt[-c^2]*e)

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Rubi [F]  time = 0.0625405, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \tan ^{-1}\left (c x^3\right )}{d+e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(a*Log[d + e*x])/e + b*Defer[Int][ArcTan[c*x^3]/(d + e*x), x]

Rubi steps

\begin{align*} \int \frac{a+b \tan ^{-1}\left (c x^3\right )}{d+e x} \, dx &=\int \left (\frac{a}{d+e x}+\frac{b \tan ^{-1}\left (c x^3\right )}{d+e x}\right ) \, dx\\ &=\frac{a \log (d+e x)}{e}+b \int \frac{\tan ^{-1}\left (c x^3\right )}{d+e x} \, dx\\ \end{align*}

Mathematica [F]  time = 180.003, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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Maple [C]  time = 0.128, size = 172, normalized size = 0.2 \begin{align*}{\frac{a\ln \left ( ex+d \right ) }{e}}+{\frac{b\ln \left ( ex+d \right ) \arctan \left ( c{x}^{3} \right ) }{e}}-{\frac{b{e}^{2}}{2\,c}\sum _{{\it \_R1}={\it RootOf} \left ({{\it \_Z}}^{6}{c}^{2}-6\,{c}^{2}d{{\it \_Z}}^{5}+15\,{c}^{2}{d}^{2}{{\it \_Z}}^{4}-20\,{c}^{2}{d}^{3}{{\it \_Z}}^{3}+15\,{c}^{2}{d}^{4}{{\it \_Z}}^{2}-6\,{c}^{2}{d}^{5}{\it \_Z}+{c}^{2}{d}^{6}+{e}^{6} \right ) }{\frac{1}{{{\it \_R1}}^{3}-3\,{{\it \_R1}}^{2}d+3\,{\it \_R1}\,{d}^{2}-{d}^{3}} \left ( \ln \left ( ex+d \right ) \ln \left ({\frac{-ex+{\it \_R1}-d}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-ex+{\it \_R1}-d}{{\it \_R1}}} \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*ln(e*x+d)/e+b*ln(e*x+d)/e*arctan(c*x^3)-1/2*b*e^2/c*sum(1/(_R1^3-3*_R1^2*d+3*_R1*d^2-d^3)*(ln(e*x+d)*ln((-e*
x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1)),_R1=RootOf(_Z^6*c^2-6*_Z^5*c^2*d+15*_Z^4*c^2*d^2-20*_Z^3*c^2*d^3+15*_Z^
2*c^2*d^4-6*_Z*c^2*d^5+c^2*d^6+e^6))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \, b \int \frac{\arctan \left (c x^{3}\right )}{2 \,{\left (e x + d\right )}}\,{d x} + \frac{a \log \left (e x + d\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*integrate(1/2*arctan(c*x^3)/(e*x + d), x) + a*log(e*x + d)/e

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \arctan \left (c x^{3}\right ) + a}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctan(c*x^3) + a)/(e*x + d), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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