3.125 \(\int \frac{(a+b \tan ^{-1}(c x^3))^3}{x^4} \, dx\)

Optimal. Leaf size=133 \[ -i b^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac{1}{2} b^3 c \text{PolyLog}\left (3,-1+\frac{2}{1-i c x^3}\right )-\frac{1}{3} i c \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3-\frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{3 x^3}+b c \log \left (2-\frac{2}{1-i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2 \]

[Out]

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

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Rubi [F]  time = 0.860001, 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 = {} \[ \int \frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTan[c*x^3])^3/x^4,x]

[Out]

(b*c*Log[I*c*x^3]*(2*a + I*b*Log[1 - I*c*x^3])^2)/8 - ((1 - I*c*x^3)*(2*a + I*b*Log[1 - I*c*x^3])^3)/(24*x^3)
- (b^3*c*Log[(-I)*c*x^3]*Log[1 + I*c*x^3]^2)/8 - ((I/24)*b^3*(1 + I*c*x^3)*Log[1 + I*c*x^3]^3)/x^3 + (I/4)*b^2
*c*(2*a + I*b*Log[1 - I*c*x^3])*PolyLog[2, 1 - I*c*x^3] - (b^3*c*Log[1 + I*c*x^3]*PolyLog[2, 1 + I*c*x^3])/4 +
(b^3*c*PolyLog[3, 1 - I*c*x^3])/4 + (b^3*c*PolyLog[3, 1 + I*c*x^3])/4 + (I/8)*b*Defer[Subst][Defer[Int][(((-2
*I)*a + b*Log[1 - I*c*x])^2*Log[1 + I*c*x])/x^2, x], x, x^3] - (I/8)*b^2*Defer[Subst][Defer[Int][(((-2*I)*a +
b*Log[1 - I*c*x])*Log[1 + I*c*x]^2)/x^2, x], x, x^3]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{x^4} \, dx &=\int \left (\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{8 x^4}+\frac{3 i b \left (-2 i a+b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{8 x^4}-\frac{3 i b^2 \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{8 x^4}+\frac{i b^3 \log ^3\left (1+i c x^3\right )}{8 x^4}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{x^4} \, dx+\frac{1}{8} (3 i b) \int \frac{\left (-2 i a+b \log \left (1-i c x^3\right )\right )^2 \log \left (1+i c x^3\right )}{x^4} \, dx-\frac{1}{8} \left (3 i b^2\right ) \int \frac{\left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log ^2\left (1+i c x^3\right )}{x^4} \, dx+\frac{1}{8} \left (i b^3\right ) \int \frac{\log ^3\left (1+i c x^3\right )}{x^4} \, dx\\ &=\frac{1}{24} \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^3}{x^2} \, dx,x,x^3\right )+\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )+\frac{1}{24} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^3(1+i c x)}{x^2} \, dx,x,x^3\right )\\ &=-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 x^3}-\frac{i b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 x^3}+\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )+\frac{1}{8} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x} \, dx,x,x^3\right )-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x} \, dx,x,x^3\right )\\ &=\frac{1}{8} b c \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 x^3}-\frac{1}{8} b^3 c \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 x^3}+\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{4} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (i c x) (2 a+i b \log (1-i c x))}{1-i c x} \, dx,x,x^3\right )+\frac{1}{4} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (-i c x) \log (1+i c x)}{1+i c x} \, dx,x,x^3\right )\\ &=\frac{1}{8} b c \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 x^3}-\frac{1}{8} b^3 c \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 x^3}+\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{4} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x)) \log \left (i c \left (-\frac{i}{c}+\frac{i x}{c}\right )\right )}{x} \, dx,x,1-i c x^3\right )+\frac{1}{4} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-i c \left (\frac{i}{c}-\frac{i x}{c}\right )\right )}{x} \, dx,x,1+i c x^3\right )\\ &=\frac{1}{8} b c \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 x^3}-\frac{1}{8} b^3 c \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 x^3}+\frac{1}{4} i b^2 c \left (2 a+i b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (1-i c x^3\right )-\frac{1}{4} b^3 c \log \left (1+i c x^3\right ) \text{Li}_2\left (1+i c x^3\right )+\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )+\frac{1}{4} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-i c x^3\right )+\frac{1}{4} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+i c x^3\right )\\ &=\frac{1}{8} b c \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{\left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{24 x^3}-\frac{1}{8} b^3 c \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^3 \left (1+i c x^3\right ) \log ^3\left (1+i c x^3\right )}{24 x^3}+\frac{1}{4} i b^2 c \left (2 a+i b \log \left (1-i c x^3\right )\right ) \text{Li}_2\left (1-i c x^3\right )-\frac{1}{4} b^3 c \log \left (1+i c x^3\right ) \text{Li}_2\left (1+i c x^3\right )+\frac{1}{4} b^3 c \text{Li}_3\left (1-i c x^3\right )+\frac{1}{4} b^3 c \text{Li}_3\left (1+i c x^3\right )+\frac{1}{8} (i b) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x))^2 \log (1+i c x)}{x^2} \, dx,x,x^3\right )-\frac{1}{8} \left (i b^2\right ) \operatorname{Subst}\left (\int \frac{(-2 i a+b \log (1-i c x)) \log ^2(1+i c x)}{x^2} \, dx,x,x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.397328, size = 240, normalized size = 1.8 \[ a b^2 c \left (\tan ^{-1}\left (c x^3\right ) \left (\left (-\frac{1}{c x^3}-i\right ) \tan ^{-1}\left (c x^3\right )+2 \log \left (1-e^{2 i \tan ^{-1}\left (c x^3\right )}\right )\right )-i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (c x^3\right )}\right )\right )+\frac{1}{3} b^3 c \left (3 i \tan ^{-1}\left (c x^3\right ) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}\left (c x^3\right )}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}\left (c x^3\right )}\right )-\frac{\tan ^{-1}\left (c x^3\right )^3}{c x^3}+i \tan ^{-1}\left (c x^3\right )^3+3 \tan ^{-1}\left (c x^3\right )^2 \log \left (1-e^{-2 i \tan ^{-1}\left (c x^3\right )}\right )-\frac{i \pi ^3}{8}\right )-\frac{1}{2} a^2 b c \log \left (c^2 x^6+1\right )-\frac{a^2 b \tan ^{-1}\left (c x^3\right )}{x^3}+3 a^2 b c \log (x)-\frac{a^3}{3 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c*x^3])^3/x^4,x]

[Out]

-a^3/(3*x^3) - (a^2*b*ArcTan[c*x^3])/x^3 + 3*a^2*b*c*Log[x] - (a^2*b*c*Log[1 + c^2*x^6])/2 + a*b^2*c*(ArcTan[c
*x^3]*((-I - 1/(c*x^3))*ArcTan[c*x^3] + 2*Log[1 - E^((2*I)*ArcTan[c*x^3])]) - I*PolyLog[2, E^((2*I)*ArcTan[c*x
^3])]) + (b^3*c*((-I/8)*Pi^3 + I*ArcTan[c*x^3]^3 - ArcTan[c*x^3]^3/(c*x^3) + 3*ArcTan[c*x^3]^2*Log[1 - E^((-2*
I)*ArcTan[c*x^3])] + (3*I)*ArcTan[c*x^3]*PolyLog[2, E^((-2*I)*ArcTan[c*x^3])] + (3*PolyLog[3, E^((-2*I)*ArcTan
[c*x^3])])/2))/3

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Maple [F]  time = 0.381, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( c{x}^{3} \right ) \right ) ^{3}}{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^3))^3/x^4,x)

[Out]

int((a+b*arctan(c*x^3))^3/x^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (c{\left (\log \left (c^{2} x^{6} + 1\right ) - \log \left (x^{6}\right )\right )} + \frac{2 \, \arctan \left (c x^{3}\right )}{x^{3}}\right )} a^{2} b - \frac{a^{3}}{3 \, x^{3}} - \frac{\frac{15}{2} \, b^{3} \arctan \left (c x^{3}\right )^{3} - \frac{21}{8} \, b^{3} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right )^{2} - 3 \, x^{3} \int -\frac{84 \, b^{3} c^{2} x^{6} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right ) - 196 \,{\left (b^{3} c^{2} x^{6} + b^{3}\right )} \arctan \left (c x^{3}\right )^{3} - 12 \,{\left (64 \, a b^{2} c^{2} x^{6} + 15 \, b^{3} c x^{3} + 64 \, a b^{2}\right )} \arctan \left (c x^{3}\right )^{2} + 21 \,{\left (b^{3} c x^{3} -{\left (b^{3} c^{2} x^{6} + b^{3}\right )} \arctan \left (c x^{3}\right )\right )} \log \left (c^{2} x^{6} + 1\right )^{2}}{8 \,{\left (c^{2} x^{10} + x^{4}\right )}}\,{d x}}{96 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))^3/x^4,x, algorithm="maxima")

[Out]

-1/2*(c*(log(c^2*x^6 + 1) - log(x^6)) + 2*arctan(c*x^3)/x^3)*a^2*b - 1/3*a^3/x^3 - 1/96*(4*b^3*arctan(c*x^3)^3
- 3*b^3*arctan(c*x^3)*log(c^2*x^6 + 1)^2 - 96*x^3*integrate(-1/32*(12*b^3*c^2*x^6*arctan(c*x^3)*log(c^2*x^6 +
1) - 28*(b^3*c^2*x^6 + b^3)*arctan(c*x^3)^3 - 12*(8*a*b^2*c^2*x^6 + b^3*c*x^3 + 8*a*b^2)*arctan(c*x^3)^2 + 3*
(b^3*c*x^3 - (b^3*c^2*x^6 + b^3)*arctan(c*x^3))*log(c^2*x^6 + 1)^2)/(c^2*x^10 + x^4), x))/x^3

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \arctan \left (c x^{3}\right )^{3} + 3 \, a b^{2} \arctan \left (c x^{3}\right )^{2} + 3 \, a^{2} b \arctan \left (c x^{3}\right ) + a^{3}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))^3/x^4,x, algorithm="fricas")

[Out]

integral((b^3*arctan(c*x^3)^3 + 3*a*b^2*arctan(c*x^3)^2 + 3*a^2*b*arctan(c*x^3) + a^3)/x^4, 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))**3/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))^3/x^4,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x^3) + a)^3/x^4, x)