3.126 \(\int \frac{(a+b \tan ^{-1}(c x^3))^3}{x^7} \, dx\)

Optimal. Leaf size=146 \[ -\frac{1}{2} i b^3 c^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x^3}\right )+b^2 c^2 \log \left (2-\frac{2}{1-i c x^3}\right ) \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac{1}{2} i b c^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2-\frac{1}{6} c^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )^3-\frac{b c \left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{2 x^3}-\frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{6 x^6} \]

[Out]

(-I/2)*b*c^2*(a + b*ArcTan[c*x^3])^2 - (b*c*(a + b*ArcTan[c*x^3])^2)/(2*x^3) - (c^2*(a + b*ArcTan[c*x^3])^3)/6
- (a + b*ArcTan[c*x^3])^3/(6*x^6) + b^2*c^2*(a + b*ArcTan[c*x^3])*Log[2 - 2/(1 - I*c*x^3)] - (I/2)*b^3*c^2*Po
lyLog[2, -1 + 2/(1 - I*c*x^3)]

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Rubi [F]  time = 1.66992, 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 = {} \[ \int \frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{x^7} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcTan[c*x^3])^3/x^7,x]

[Out]

(3*a*b^2*c^2*Log[x])/4 - (b*c*(1 - I*c*x^3)*(2*a + I*b*Log[1 - I*c*x^3])^2)/(16*x^3) + (I/16)*b*c^2*Log[I*c*x^
3]*(2*a + I*b*Log[1 - I*c*x^3])^2 - (c^2*(2*a + I*b*Log[1 - I*c*x^3])^3)/48 - (2*a + I*b*Log[1 - I*c*x^3])^3/(
48*x^6) + (b^3*c*(1 + I*c*x^3)*Log[1 + I*c*x^3]^2)/(16*x^3) + (I/16)*b^3*c^2*Log[(-I)*c*x^3]*Log[1 + I*c*x^3]^
2 - (I/48)*b^3*c^2*Log[1 + I*c*x^3]^3 - ((I/48)*b^3*Log[1 + I*c*x^3]^3)/x^6 + (I/8)*b^3*c^2*PolyLog[2, (-I)*c*
x^3] - (I/8)*b^3*c^2*PolyLog[2, I*c*x^3] - (b^2*c^2*(2*a + I*b*Log[1 - I*c*x^3])*PolyLog[2, 1 - I*c*x^3])/8 +
(I/8)*b^3*c^2*Log[1 + I*c*x^3]*PolyLog[2, 1 + I*c*x^3] + (I/8)*b^3*c^2*PolyLog[3, 1 - I*c*x^3] - (I/8)*b^3*c^2
*PolyLog[3, 1 + I*c*x^3] + (I/8)*b*Defer[Subst][Defer[Int][(((-2*I)*a + b*Log[1 - I*c*x])^2*Log[1 + I*c*x])/x^
3, 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^3, x], x
, x^3]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}\left (c x^3\right )\right )^3}{x^7} \, dx &=\int \left (\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{8 x^7}+\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^7}-\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^7}+\frac{i b^3 \log ^3\left (1+i c x^3\right )}{8 x^7}\right ) \, dx\\ &=\frac{1}{8} \int \frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{x^7} \, 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^7} \, 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^7} \, dx+\frac{1}{8} \left (i b^3\right ) \int \frac{\log ^3\left (1+i c x^3\right )}{x^7} \, dx\\ &=\frac{1}{24} \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^3}{x^3} \, 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^3} \, 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^3} \, 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^3} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\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^3} \, 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^3} \, dx,x,x^3\right )+\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (1-i c x))^2}{x^2 (1-i c x)} \, dx,x,x^3\right )-\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(1+i c x)}{x^2 (1+i c x)} \, dx,x,x^3\right )\\ &=-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\frac{1}{16} (i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,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^3} \, 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^3} \, dx,x,x^3\right )+\frac{1}{16} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+i c x^3\right )\\ &=-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\frac{1}{16} (i b) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{\left (-\frac{i}{c}+\frac{i x}{c}\right )^2} \, dx,x,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^3} \, 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^3} \, dx,x,x^3\right )+\frac{1}{16} \left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\left (\frac{i}{c}-\frac{i x}{c}\right )^2} \, dx,x,1+i c x^3\right )-\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x \left (-\frac{i}{c}+\frac{i x}{c}\right )} \, dx,x,1-i c x^3\right )+\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x \left (\frac{i}{c}-\frac{i x}{c}\right )} \, dx,x,1+i c x^3\right )\\ &=-\frac{b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 x^3}-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}+\frac{b^3 c \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{16 x^3}-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\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^3} \, 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^3} \, dx,x,x^3\right )-\frac{1}{16} (b c) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-i c x^3\right )+\frac{1}{8} \left (i b^2 c\right ) \operatorname{Subst}\left (\int \frac{2 a+i b \log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-i c x^3\right )+\frac{1}{16} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+i c x^3\right )-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{\frac{i}{c}-\frac{i x}{c}} \, dx,x,1+i c x^3\right )-\frac{1}{16} \left (i b c^2\right ) \operatorname{Subst}\left (\int \frac{(2 a+i b \log (x))^2}{x} \, dx,x,1-i c x^3\right )-\frac{1}{16} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log ^2(x)}{x} \, dx,x,1+i c x^3\right )\\ &=\frac{3}{4} a b^2 c^2 \log (x)-\frac{b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 x^3}+\frac{1}{16} i b c^2 \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}+\frac{b^3 c \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{16 x^3}+\frac{1}{16} i b^3 c^2 \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\frac{1}{8} i b^3 c^2 \text{Li}_2\left (-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^3} \, 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^3} \, dx,x,x^3\right )-\frac{1}{8} \left (b^3 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{-\frac{i}{c}+\frac{i x}{c}} \, dx,x,1-i c x^3\right )-\frac{1}{16} c^2 \operatorname{Subst}\left (\int x^2 \, dx,x,2 a+i b \log \left (1-i c x^3\right )\right )+\frac{1}{8} \left (b^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) (2 a+i b \log (x))}{x} \, dx,x,1-i c x^3\right )-\frac{1}{16} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\log \left (1+i c x^3\right )\right )-\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x) \log (x)}{x} \, dx,x,1+i c x^3\right )\\ &=\frac{3}{4} a b^2 c^2 \log (x)-\frac{b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 x^3}+\frac{1}{16} i b c^2 \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{1}{48} c^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}+\frac{b^3 c \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{16 x^3}+\frac{1}{16} i b^3 c^2 \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{1}{48} i b^3 c^2 \log ^3\left (1+i c x^3\right )-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\frac{1}{8} i b^3 c^2 \text{Li}_2\left (-i c x^3\right )-\frac{1}{8} i b^3 c^2 \text{Li}_2\left (i c x^3\right )-\frac{1}{8} b^2 c^2 \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}{8} i b^3 c^2 \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^3} \, 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^3} \, dx,x,x^3\right )+\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-i c x^3\right )-\frac{1}{8} \left (i b^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+i c x^3\right )\\ &=\frac{3}{4} a b^2 c^2 \log (x)-\frac{b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{16 x^3}+\frac{1}{16} i b c^2 \log \left (i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac{1}{48} c^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^3-\frac{\left (2 a+i b \log \left (1-i c x^3\right )\right )^3}{48 x^6}+\frac{b^3 c \left (1+i c x^3\right ) \log ^2\left (1+i c x^3\right )}{16 x^3}+\frac{1}{16} i b^3 c^2 \log \left (-i c x^3\right ) \log ^2\left (1+i c x^3\right )-\frac{1}{48} i b^3 c^2 \log ^3\left (1+i c x^3\right )-\frac{i b^3 \log ^3\left (1+i c x^3\right )}{48 x^6}+\frac{1}{8} i b^3 c^2 \text{Li}_2\left (-i c x^3\right )-\frac{1}{8} i b^3 c^2 \text{Li}_2\left (i c x^3\right )-\frac{1}{8} b^2 c^2 \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}{8} i b^3 c^2 \log \left (1+i c x^3\right ) \text{Li}_2\left (1+i c x^3\right )+\frac{1}{8} i b^3 c^2 \text{Li}_3\left (1-i c x^3\right )-\frac{1}{8} i b^3 c^2 \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^3} \, 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^3} \, dx,x,x^3\right )\\ \end{align*}

Mathematica [A]  time = 0.329347, size = 196, normalized size = 1.34 \[ -\frac{3 i b^3 c^2 x^6 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}\left (c x^3\right )}\right )+a \left (a \left (a+3 b c x^3\right )-6 b^2 c^2 x^6 \log \left (\frac{c x^3}{\sqrt{c^2 x^6+1}}\right )\right )+3 b^2 \tan ^{-1}\left (c x^3\right )^2 \left (a c^2 x^6+a+b c x^3 \left (1+i c x^3\right )\right )+3 b \tan ^{-1}\left (c x^3\right ) \left (a \left (a c^2 x^6+a+2 b c x^3\right )-2 b^2 c^2 x^6 \log \left (1-e^{2 i \tan ^{-1}\left (c x^3\right )}\right )\right )+b^3 \left (c^2 x^6+1\right ) \tan ^{-1}\left (c x^3\right )^3}{6 x^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c*x^3])^3/x^7,x]

[Out]

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

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Maple [F]  time = 0.557, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( c{x}^{3} \right ) \right ) ^{3}}{{x}^{7}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x^3))^3/x^7,x)

[Out]

int((a+b*arctan(c*x^3))^3/x^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left ({\left (c \arctan \left (c x^{3}\right ) + \frac{1}{x^{3}}\right )} c + \frac{\arctan \left (c x^{3}\right )}{x^{6}}\right )} a^{2} b + \frac{1}{2} \,{\left ({\left (\arctan \left (c x^{3}\right )^{2} - \log \left (c^{2} x^{6} + 1\right ) + 6 \, \log \left (x\right )\right )} c^{2} - 2 \,{\left (c \arctan \left (c x^{3}\right ) + \frac{1}{x^{3}}\right )} c \arctan \left (c x^{3}\right )\right )} a b^{2} - \frac{a b^{2} \arctan \left (c x^{3}\right )^{2}}{2 \, x^{6}} + \frac{\frac{3}{8} \,{\left (8 \, x^{6} \int -\frac{84 \, c^{2} x^{6} \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right ) - 180 \, c x^{3} \arctan \left (c x^{3}\right )^{2} - 392 \,{\left (c^{2} x^{6} + 1\right )} \arctan \left (c x^{3}\right )^{3} + 21 \,{\left (c x^{3} - 2 \,{\left (c^{2} x^{6} + 1\right )} \arctan \left (c x^{3}\right )\right )} \log \left (c^{2} x^{6} + 1\right )^{2}}{8 \,{\left (c^{2} x^{13} + x^{7}\right )}}\,{d x} - 20 \, \arctan \left (c x^{3}\right )^{3} + 7 \, \arctan \left (c x^{3}\right ) \log \left (c^{2} x^{6} + 1\right )^{2}\right )} b^{3}}{192 \, x^{6}} - \frac{a^{3}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))^3/x^7,x, algorithm="maxima")

[Out]

-1/2*((c*arctan(c*x^3) + 1/x^3)*c + arctan(c*x^3)/x^6)*a^2*b + 1/2*((arctan(c*x^3)^2 - log(c^2*x^6 + 1) + 6*lo
g(x))*c^2 - 2*(c*arctan(c*x^3) + 1/x^3)*c*arctan(c*x^3))*a*b^2 - 1/2*a*b^2*arctan(c*x^3)^2/x^6 + 1/192*(192*x^
6*integrate(-1/64*(12*c^2*x^6*arctan(c*x^3)*log(c^2*x^6 + 1) - 12*c*x^3*arctan(c*x^3)^2 - 56*(c^2*x^6 + 1)*arc
tan(c*x^3)^3 + 3*(c*x^3 - 2*(c^2*x^6 + 1)*arctan(c*x^3))*log(c^2*x^6 + 1)^2)/(c^2*x^13 + x^7), x) - 4*arctan(c
*x^3)^3 + 3*arctan(c*x^3)*log(c^2*x^6 + 1)^2)*b^3/x^6 - 1/6*a^3/x^6

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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^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))^3/x^7,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^7, 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**7,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^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x^3))^3/x^7,x, algorithm="giac")

[Out]

integrate((b*arctan(c*x^3) + a)^3/x^7, x)