Optimal. Leaf size=189 \[ \frac{1}{2} b c^3 d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} b c^3 d^3 \text{PolyLog}(2,i c x)+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-i a c^3 d^3 \log (x)+\frac{5}{3} b c^3 d^3 \log \left (c^2 x^2+1\right )-\frac{3 i b c^2 d^3}{2 x}-\frac{10}{3} b c^3 d^3 \log (x)-\frac{3}{2} i b c^3 d^3 \tan ^{-1}(c x)-\frac{b c d^3}{6 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.203295, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {4876, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391} \[ \frac{1}{2} b c^3 d^3 \text{PolyLog}(2,-i c x)-\frac{1}{2} b c^3 d^3 \text{PolyLog}(2,i c x)+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-i a c^3 d^3 \log (x)+\frac{5}{3} b c^3 d^3 \log \left (c^2 x^2+1\right )-\frac{3 i b c^2 d^3}{2 x}-\frac{10}{3} b c^3 d^3 \log (x)-\frac{3}{2} i b c^3 d^3 \tan ^{-1}(c x)-\frac{b c d^3}{6 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4876
Rule 4852
Rule 266
Rule 44
Rule 325
Rule 203
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx &=\int \left (\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^4}+\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^3 \int \frac{a+b \tan ^{-1}(c x)}{x^4} \, dx+\left (3 i c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx-\left (3 c^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (i c^3 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 \log (x)+\frac{1}{3} \left (b c d^3\right ) \int \frac{1}{x^3 \left (1+c^2 x^2\right )} \, dx+\frac{1}{2} \left (3 i b c^2 d^3\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx+\frac{1}{2} \left (b c^3 d^3\right ) \int \frac{\log (1-i c x)}{x} \, dx-\frac{1}{2} \left (b c^3 d^3\right ) \int \frac{\log (1+i c x)}{x} \, dx-\left (3 b c^3 d^3\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{3 i b c^2 d^3}{2 x}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 \log (x)+\frac{1}{2} b c^3 d^3 \text{Li}_2(-i c x)-\frac{1}{2} b c^3 d^3 \text{Li}_2(i c x)+\frac{1}{6} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} \left (3 b c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} \left (3 i b c^4 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{3 i b c^2 d^3}{2 x}-\frac{3}{2} i b c^3 d^3 \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 \log (x)+\frac{1}{2} b c^3 d^3 \text{Li}_2(-i c x)-\frac{1}{2} b c^3 d^3 \text{Li}_2(i c x)+\frac{1}{6} \left (b c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )-\frac{1}{2} \left (3 b c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} \left (3 b c^5 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^3}{6 x^2}-\frac{3 i b c^2 d^3}{2 x}-\frac{3}{2} i b c^3 d^3 \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 \log (x)-\frac{10}{3} b c^3 d^3 \log (x)+\frac{5}{3} b c^3 d^3 \log \left (1+c^2 x^2\right )+\frac{1}{2} b c^3 d^3 \text{Li}_2(-i c x)-\frac{1}{2} b c^3 d^3 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [C] time = 0.094224, size = 170, normalized size = 0.9 \[ \frac{d^3 \left (-9 i b c^2 x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )+3 b c^3 x^3 \text{PolyLog}(2,-i c x)-3 b c^3 x^3 \text{PolyLog}(2,i c x)+18 a c^2 x^2-6 i a c^3 x^3 \log (x)-9 i a c x-2 a-20 b c^3 x^3 \log (x)+10 b c^3 x^3 \log \left (c^2 x^2+1\right )+18 b c^2 x^2 \tan ^{-1}(c x)-b c x-9 i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)\right )}{6 x^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.051, size = 255, normalized size = 1.4 \begin{align*}{\frac{-{\frac{3\,i}{2}}c{d}^{3}b\arctan \left ( cx \right ) }{{x}^{2}}}+3\,{\frac{{c}^{2}{d}^{3}a}{x}}-{\frac{{d}^{3}a}{3\,{x}^{3}}}-i{c}^{3}{d}^{3}b\arctan \left ( cx \right ) \ln \left ( cx \right ) -{\frac{{\frac{3\,i}{2}}b{c}^{2}{d}^{3}}{x}}+3\,{\frac{b{c}^{2}{d}^{3}\arctan \left ( cx \right ) }{x}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{{\frac{3\,i}{2}}c{d}^{3}a}{{x}^{2}}}+{\frac{{c}^{3}{d}^{3}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) }{2}}-{\frac{{c}^{3}{d}^{3}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) }{2}}+{\frac{{c}^{3}{d}^{3}b{\it dilog} \left ( 1+icx \right ) }{2}}-{\frac{{c}^{3}{d}^{3}b{\it dilog} \left ( 1-icx \right ) }{2}}+{\frac{5\,b{c}^{3}{d}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3}}-i{c}^{3}{d}^{3}a\ln \left ( cx \right ) -{\frac{3\,i}{2}}b{c}^{3}{d}^{3}\arctan \left ( cx \right ) -{\frac{bc{d}^{3}}{6\,{x}^{2}}}-{\frac{10\,{c}^{3}{d}^{3}b\ln \left ( cx \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -i \, b c^{3} d^{3} \int \frac{\arctan \left (c x\right )}{x}\,{d x} - i \, a c^{3} d^{3} \log \left (x\right ) + \frac{3}{2} \,{\left (c{\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \arctan \left (c x\right )}{x}\right )} b c^{2} d^{3} - \frac{3}{2} i \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b c d^{3} + \frac{1}{6} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d^{3} + \frac{3 \, a c^{2} d^{3}}{x} - \frac{3 i \, a c d^{3}}{2 \, x^{2}} - \frac{a d^{3}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{-2 i \, a c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{3} x^{2} + 6 i \, a c d^{3} x + 2 \, a d^{3} +{\left (b c^{3} d^{3} x^{3} - 3 i \, b c^{2} d^{3} x^{2} - 3 \, b c d^{3} x + i \, b d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{2 \, x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{3} \left (\int \frac{a}{x^{4}}\, dx + \int - \frac{3 a c^{2}}{x^{2}}\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{3 i a c}{x^{3}}\, dx + \int - \frac{i a c^{3}}{x}\, dx + \int - \frac{3 b c^{2} \operatorname{atan}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{3 i b c \operatorname{atan}{\left (c x \right )}}{x^{3}}\, dx + \int - \frac{i b c^{3} \operatorname{atan}{\left (c x \right )}}{x}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, c d x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]