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

Optimal. Leaf size=523 \[ -\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e}-\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac{b \text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}+\frac{\log (d+e x) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{e}+\frac{b \log (d+e x) \log \left (\frac{e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right )}{2 e}-\frac{b \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac{b \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{c} x+\sqrt [3]{-1}\right )}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac{b \log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{c} x+(-1)^{2/3}\right )}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac{b \log (d+e x) \log \left (\frac{(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}-\frac{b \log (d+e x) \log \left (\frac{\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e} \]

[Out]

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

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Rubi [F]  time = 0.0615003, 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 = {} \[ \int \frac{a+b \tanh ^{-1}\left (c x^3\right )}{d+e x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [C]  time = 99.2102, size = 515, normalized size = 0.98 \[ \frac{a \log (d+e x)}{e}+\frac{b \left (-\text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )+\text{PolyLog}\left (2,\frac{\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-i \sqrt{3} e-e}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-i \sqrt{3} e+e}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i \sqrt{3} e-e}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i \sqrt{3} e+e}\right )+2 \tanh ^{-1}\left (c x^3\right ) \log (d+e x)-\log (d+e x) \log \left (\frac{e \left (-2 \sqrt [3]{c} x-i \sqrt{3}+1\right )}{2 \sqrt [3]{c} d-i \sqrt{3} e+e}\right )+\log (d+e x) \log \left (\frac{e \left (-2 i \sqrt [3]{c} x+\sqrt{3}-i\right )}{2 i \sqrt [3]{c} d+\left (\sqrt{3}-i\right ) e}\right )+\log (d+e x) \log \left (\frac{e \left (2 i \sqrt [3]{c} x+\sqrt{3}+i\right )}{\left (\sqrt{3}+i\right ) e-2 i \sqrt [3]{c} d}\right )-\log (d+e x) \log \left (-\frac{e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )-\log (d+e x) \log \left (-\frac{e \left (2 \sqrt [3]{c} x-i \sqrt{3}-1\right )}{2 \sqrt [3]{c} d+i \sqrt{3} e+e}\right )+\log (d+e x) \log \left (\frac{e-\sqrt [3]{c} e x}{\sqrt [3]{c} d+e}\right )\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Log[d + e*x])/e + (b*(2*ArcTanh[c*x^3]*Log[d + e*x] - Log[(e*(1 - I*Sqrt[3] - 2*c^(1/3)*x))/(2*c^(1/3)*d +
e - I*Sqrt[3]*e)]*Log[d + e*x] + Log[(e*(-I + Sqrt[3] - (2*I)*c^(1/3)*x))/((2*I)*c^(1/3)*d + (-I + Sqrt[3])*e)
]*Log[d + e*x] + Log[(e*(I + Sqrt[3] + (2*I)*c^(1/3)*x))/((-2*I)*c^(1/3)*d + (I + Sqrt[3])*e)]*Log[d + e*x] -
Log[-((e*(1 + c^(1/3)*x))/(c^(1/3)*d - e))]*Log[d + e*x] - Log[-((e*(-1 - I*Sqrt[3] + 2*c^(1/3)*x))/(2*c^(1/3)
*d + e + I*Sqrt[3]*e))]*Log[d + e*x] + Log[d + e*x]*Log[(e - c^(1/3)*e*x)/(c^(1/3)*d + e)] - PolyLog[2, (c^(1/
3)*(d + e*x))/(c^(1/3)*d - e)] + PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d + e)] + PolyLog[2, (2*c^(1/3)*(d +
e*x))/(2*c^(1/3)*d - e - I*Sqrt[3]*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d + e - I*Sqrt[3]*e)] + P
olyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d - e + I*Sqrt[3]*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)
*d + e + I*Sqrt[3]*e)]))/(2*e)

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Maple [C]  time = 0.171, size = 182, normalized size = 0.4 \begin{align*}{\frac{a\ln \left ( ex+d \right ) }{e}}+{\frac{b\ln \left ( ex+d \right ){\it Artanh} \left ( c{x}^{3} \right ) }{e}}+{\frac{b}{2\,e}\sum _{{\it \_R1}={\it RootOf} \left ( c{{\it \_Z}}^{3}-3\,cd{{\it \_Z}}^{2}+3\,c{d}^{2}{\it \_Z}-c{d}^{3}-{e}^{3} \right ) }\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 ) }-{\frac{b}{2\,e}\sum _{{\it \_R1}={\it RootOf} \left ( c{{\it \_Z}}^{3}-3\,cd{{\it \_Z}}^{2}+3\,c{d}^{2}{\it \_Z}-c{d}^{3}+{e}^{3} \right ) }\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*ln(e*x+d)/e+b*ln(e*x+d)/e*arctanh(c*x^3)+1/2*b/e*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1),
_R1=RootOf(_Z^3*c-3*_Z^2*c*d+3*_Z*c*d^2-c*d^3-e^3))-1/2*b/e*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1
-d)/_R1),_R1=RootOf(_Z^3*c-3*_Z^2*c*d+3*_Z*c*d^2-c*d^3+e^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b \int \frac{\log \left (c x^{3} + 1\right ) - \log \left (-c x^{3} + 1\right )}{e x + d}\,{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*arctanh(c*x^3))/(e*x+d),x, algorithm="maxima")

[Out]

1/2*b*integrate((log(c*x^3 + 1) - log(-c*x^3 + 1))/(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 \operatorname{artanh}\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*arctanh(c*x^3))/(e*x+d),x, algorithm="fricas")

[Out]

integral((b*arctanh(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*atanh(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 \operatorname{artanh}\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*arctanh(c*x^3))/(e*x+d),x, algorithm="giac")

[Out]

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