Optimal. Leaf size=267 \[ -\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{2 i b c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{8}{3} b c^3 d^2 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{b c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-i b^2 c^3 d^2 \log \left (c^2 x^2+1\right )-\frac{b^2 c^2 d^2}{3 x}+2 i b^2 c^3 d^2 \log (x)-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.268309, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 14, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.56, Rules used = {37, 4874, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 4854, 2402, 2315} \[ -\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}(2,i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{2 i b c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{8}{3} b c^3 d^2 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{b c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-i b^2 c^3 d^2 \log \left (c^2 x^2+1\right )-\frac{b^2 c^2 d^2}{3 x}+2 i b^2 c^3 d^2 \log (x)-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 4874
Rule 4852
Rule 325
Rule 203
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac{4 c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x}-\frac{4 c^3 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 (i+c x)}\right ) \, dx\\ &=-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (2 b c d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (2 i b c^2 d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\frac{1}{3} \left (8 b c^3 d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx+\frac{1}{3} \left (8 b c^4 d^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{i+c x} \, dx\\ &=-\frac{b c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac{2 i b c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{8}{3} b c^3 d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )+\frac{1}{3} \left (b^2 c^2 d^2\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac{1}{3} \left (4 i b^2 c^3 d^2\right ) \int \frac{\log (1-i c x)}{x} \, dx+\frac{1}{3} \left (4 i b^2 c^3 d^2\right ) \int \frac{\log (1+i c x)}{x} \, dx+\left (2 i b^2 c^3 d^2\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx+\frac{1}{3} \left (8 b^2 c^4 d^2\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}-\frac{b c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac{2 i b c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{8}{3} b c^3 d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )-\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(i c x)+\left (i b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{3} \left (8 i b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )-\frac{1}{3} \left (b^2 c^4 d^2\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{b^2 c^2 d^2}{3 x}-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x)-\frac{b c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac{2 i b c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac{8}{3} a b c^3 d^2 \log (x)-\frac{8}{3} b c^3 d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )-\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )+\left (i b^2 c^3 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\left (i b^2 c^5 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b^2 c^2 d^2}{3 x}-\frac{1}{3} b^2 c^3 d^2 \tan ^{-1}(c x)-\frac{b c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^2}-\frac{2 i b c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-\frac{8}{3} a b c^3 d^2 \log (x)+2 i b^2 c^3 d^2 \log (x)-\frac{8}{3} b c^3 d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )-i b^2 c^3 d^2 \log \left (1+c^2 x^2\right )-\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(-i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2(i c x)+\frac{4}{3} i b^2 c^3 d^2 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )\\ \end{align*}
Mathematica [A] time = 0.65824, size = 253, normalized size = 0.95 \[ \frac{d^2 \left (4 i b^2 c^3 x^3 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+3 a^2 c^2 x^2-3 i a^2 c x-a^2-6 i a b c^2 x^2-8 a b c^3 x^3 \log (c x)+4 a b c^3 x^3 \log \left (c^2 x^2+1\right )-b \tan ^{-1}(c x) \left (a \left (6 i c^3 x^3-6 c^2 x^2+6 i c x+2\right )+b c x \left (c^2 x^2+6 i c x+1\right )+8 b c^3 x^3 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )-a b c x-b^2 c^2 x^2+6 i b^2 c^3 x^3 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+b^2 (-1-i c x)^3 \tan ^{-1}(c x)^2\right )}{3 x^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.116, size = 669, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{12 \, x^{3}{\rm integral}\left (-\frac{3 \, a^{2} c^{4} d^{2} x^{4} - 6 i \, a^{2} c^{3} d^{2} x^{3} - 6 i \, a^{2} c d^{2} x - 3 \, a^{2} d^{2} -{\left (-3 i \, a b c^{4} d^{2} x^{4} - 3 \,{\left (2 \, a b + i \, b^{2}\right )} c^{3} d^{2} x^{3} - 3 \, b^{2} c^{2} d^{2} x^{2} -{\left (6 \, a b - i \, b^{2}\right )} c d^{2} x + 3 i \, a b d^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{3 \,{\left (c^{2} x^{6} + x^{4}\right )}}, x\right ) -{\left (3 \, b^{2} c^{2} d^{2} x^{2} - 3 i \, b^{2} c d^{2} x - b^{2} d^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2}}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 )}^{2}{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]