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

Optimal. Leaf size=523 \[ \frac {\log (d+e x) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (\frac {e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+\sqrt [3]{-1}\right )}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+(-1)^{2/3}\right )}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (\frac {(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e} \]

[Out]

(a+b*arctanh(c*x^3))*ln(e*x+d)/e+1/2*b*ln(e*(1-c^(1/3)*x)/(c^(1/3)*d+e))*ln(e*x+d)/e-1/2*b*ln(-e*(1+c^(1/3)*x)
/(c^(1/3)*d-e))*ln(e*x+d)/e+1/2*b*ln(-e*((-1)^(1/3)+c^(1/3)*x)/(c^(1/3)*d-(-1)^(1/3)*e))*ln(e*x+d)/e-1/2*b*ln(
-e*((-1)^(2/3)+c^(1/3)*x)/(c^(1/3)*d-(-1)^(2/3)*e))*ln(e*x+d)/e+1/2*b*ln((-1)^(2/3)*e*(1+(-1)^(1/3)*c^(1/3)*x)
/(c^(1/3)*d+(-1)^(2/3)*e))*ln(e*x+d)/e-1/2*b*ln((-1)^(1/3)*e*(1+(-1)^(2/3)*c^(1/3)*x)/(c^(1/3)*d+(-1)^(1/3)*e)
)*ln(e*x+d)/e-1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d-e))/e+1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d+e))/
e+1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d-(-1)^(1/3)*e))/e-1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d+(-1)^
(1/3)*e))/e-1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d-(-1)^(2/3)*e))/e+1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/
3)*d+(-1)^(2/3)*e))/e

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Rubi [F]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [C]  time = 100.90, size = 515, normalized size = 0.98 \[ \frac {a \log (d+e x)}{e}+\frac {b \left (-\text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )+\text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )+\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-i \sqrt {3} e-e}\right )-\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-i \sqrt {3} e+e}\right )+\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i \sqrt {3} e-e}\right )-\text {Li}_2\left (\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+i \sqrt {3} e+e}\right )+2 \tanh ^{-1}\left (c x^3\right ) \log (d+e x)-\log (d+e x) \log \left (\frac {e \left (-2 \sqrt [3]{c} x-i \sqrt {3}+1\right )}{2 \sqrt [3]{c} d-i \sqrt {3} e+e}\right )+\log (d+e x) \log \left (\frac {e \left (-2 i \sqrt [3]{c} x+\sqrt {3}-i\right )}{2 i \sqrt [3]{c} d+\left (\sqrt {3}-i\right ) e}\right )+\log (d+e x) \log \left (\frac {e \left (2 i \sqrt [3]{c} x+\sqrt {3}+i\right )}{\left (\sqrt {3}+i\right ) e-2 i \sqrt [3]{c} d}\right )-\log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )-\log (d+e x) \log \left (-\frac {e \left (2 \sqrt [3]{c} x-i \sqrt {3}-1\right )}{2 \sqrt [3]{c} d+i \sqrt {3} e+e}\right )+\log (d+e x) \log \left (\frac {e-\sqrt [3]{c} e x}{\sqrt [3]{c} d+e}\right )\right )}{2 e} \]

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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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d}\,{d x} \]

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)

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maple [C]  time = 0.15, size = 182, normalized size = 0.35 \[ \frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctanh \left (c \,x^{3}\right )}{e}-\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3} c -3 c d \,\textit {\_Z}^{2}+3 c \,d^{2} \textit {\_Z} -c \,d^{3}+e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3} c -3 c d \,\textit {\_Z}^{2}+3 c \,d^{2} \textit {\_Z} -c \,d^{3}-e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, b \int \frac {\log \left (c x^{3} + 1\right ) - \log \left (-c x^{3} + 1\right )}{e x + d}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atanh}\left (c\,x^3\right )}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^3))/(d + e*x),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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