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

Optimal. Leaf size=739 \[ \frac {\log (d+e x) \left (a+b \tan ^{-1}\left (c x^3\right )\right )}{e}-\frac {b c \text {Li}_2\left (\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \text {Li}_2\left (\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \text {Li}_2\left (\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \text {Li}_2\left (\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \text {Li}_2\left (\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \text {Li}_2\left (\frac {\sqrt [6]{-c^2} (d+e x)}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (\frac {e \left (1-\sqrt [6]{-c^2} x\right )}{\sqrt [6]{-c^2} d+e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d-e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+\sqrt [3]{-1}\right )}{\sqrt [6]{-c^2} d-\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (-\frac {e \left (\sqrt [6]{-c^2} x+(-1)^{2/3}\right )}{\sqrt [6]{-c^2} d-(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}+\frac {b c \log (d+e x) \log \left (\frac {(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+(-1)^{2/3} e}\right )}{2 \sqrt {-c^2} e}-\frac {b c \log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [6]{-c^2} x+1\right )}{\sqrt [6]{-c^2} d+\sqrt [3]{-1} e}\right )}{2 \sqrt {-c^2} e} \]

[Out]

(a+b*arctan(c*x^3))*ln(e*x+d)/e+1/2*b*c*ln(e*(1-(-c^2)^(1/6)*x)/((-c^2)^(1/6)*d+e))*ln(e*x+d)/e/(-c^2)^(1/2)-1
/2*b*c*ln(-e*(1+(-c^2)^(1/6)*x)/((-c^2)^(1/6)*d-e))*ln(e*x+d)/e/(-c^2)^(1/2)+1/2*b*c*ln(-e*((-1)^(1/3)+(-c^2)^
(1/6)*x)/((-c^2)^(1/6)*d-(-1)^(1/3)*e))*ln(e*x+d)/e/(-c^2)^(1/2)-1/2*b*c*ln(-e*((-1)^(2/3)+(-c^2)^(1/6)*x)/((-
c^2)^(1/6)*d-(-1)^(2/3)*e))*ln(e*x+d)/e/(-c^2)^(1/2)+1/2*b*c*ln((-1)^(2/3)*e*(1+(-1)^(1/3)*(-c^2)^(1/6)*x)/((-
c^2)^(1/6)*d+(-1)^(2/3)*e))*ln(e*x+d)/e/(-c^2)^(1/2)-1/2*b*c*ln((-1)^(1/3)*e*(1+(-1)^(2/3)*(-c^2)^(1/6)*x)/((-
c^2)^(1/6)*d+(-1)^(1/3)*e))*ln(e*x+d)/e/(-c^2)^(1/2)-1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d-e)
)/e/(-c^2)^(1/2)+1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d+e))/e/(-c^2)^(1/2)+1/2*b*c*polylog(2,(
-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d-(-1)^(1/3)*e))/e/(-c^2)^(1/2)-1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^
2)^(1/6)*d+(-1)^(1/3)*e))/e/(-c^2)^(1/2)-1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d-(-1)^(2/3)*e))
/e/(-c^2)^(1/2)+1/2*b*c*polylog(2,(-c^2)^(1/6)*(e*x+d)/((-c^2)^(1/6)*d+(-1)^(2/3)*e))/e/(-c^2)^(1/2)

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

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*}

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Mathematica [F]  time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

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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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x^{3}\right ) + a}{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.15, size = 172, normalized size = 0.23 \[ \frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctan \left (c \,x^{3}\right )}{e}-\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{6} c^{2}-6 c^{2} d \,\textit {\_Z}^{5}+15 c^{2} d^{2} \textit {\_Z}^{4}-20 c^{2} d^{3} \textit {\_Z}^{3}+15 c^{2} d^{4} \textit {\_Z}^{2}-6 c^{2} d^{5} \textit {\_Z} +c^{2} d^{6}+e^{6}\right )}{\sum }\frac {\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\dilog \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )}{\textit {\_R1}^{3}-3 \textit {\_R1}^{2} d +3 \textit {\_R1} \,d^{2}-d^{3}}\right )}{2 c} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x^3))/(d + e*x),x)

[Out]

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

[Out]

Timed out

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