Optimal. Leaf size=361 \[ -\frac{i b d \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 e^2}+\frac{i b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 e^2}+\frac{d \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 e^2}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b x}{2 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.369632, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4916, 4852, 321, 203, 4980, 4856, 2402, 2315, 2447} \[ -\frac{i b d \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{i b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{4 e^2}+\frac{i b d \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{4 e^2}+\frac{d \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}-i \sqrt{e}\right )}\right )}{2 e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{(1-i c x) \left (c \sqrt{-d}+i \sqrt{e}\right )}\right )}{2 e^2}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}-\frac{b x}{2 c e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4916
Rule 4852
Rule 321
Rule 203
Rule 4980
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx &=\frac{\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{e}-\frac{d \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{d+e x^2} \, dx}{e}\\ &=\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}-\frac{(b c) \int \frac{x^2}{1+c^2 x^2} \, dx}{2 e}-\frac{d \int \left (-\frac{a+b \tan ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{a+b \tan ^{-1}(c x)}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e}\\ &=-\frac{b x}{2 c e}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}+\frac{d \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 e^{3/2}}-\frac{d \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 e^{3/2}}+\frac{b \int \frac{1}{1+c^2 x^2} \, dx}{2 c e}\\ &=-\frac{b x}{2 c e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}+\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}-2 \frac{(b c d) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 e^2}+\frac{(b c d) \int \frac{\log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 e^2}+\frac{(b c d) \int \frac{\log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 e^2}\\ &=-\frac{b x}{2 c e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}+\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}+\frac{i b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}+\frac{i b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}-2 \frac{(i b d) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{2 e^2}\\ &=-\frac{b x}{2 c e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 e}+\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2}-\frac{i b d \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{i b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}+\frac{i b d \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (c \sqrt{-d}+i \sqrt{e}\right ) (1-i c x)}\right )}{4 e^2}\\ \end{align*}
Mathematica [A] time = 0.259825, size = 503, normalized size = 1.39 \[ -\frac{i b d \text{PolyLog}\left (2,-\frac{\sqrt{e} (1-i c x)}{-\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^2}-\frac{i b d \text{PolyLog}\left (2,\frac{\sqrt{e} (1-i c x)}{\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^2}+\frac{i b d \text{PolyLog}\left (2,-\frac{\sqrt{e} (1+i c x)}{-\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^2}+\frac{i b d \text{PolyLog}\left (2,\frac{\sqrt{e} (1+i c x)}{\sqrt{e}+i c \sqrt{-d}}\right )}{4 e^2}-\frac{a d \log \left (d+e x^2\right )}{2 e^2}+\frac{a x^2}{2 e}+\frac{b \tan ^{-1}(c x)}{2 c^2 e}+\frac{i b d \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 e^2}-\frac{i b d \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}-\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^2}-\frac{i b d \log (1-i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}-i \sqrt{e}}\right )}{4 e^2}+\frac{i b d \log (1+i c x) \log \left (\frac{c \left (\sqrt{-d}+\sqrt{e} x\right )}{c \sqrt{-d}+i \sqrt{e}}\right )}{4 e^2}+\frac{b x^2 \tan ^{-1}(c x)}{2 e}-\frac{b x}{2 c e} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.211, size = 703, normalized size = 2. \begin{align*} \text{result too large to display} \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*} \frac{1}{2} \, a{\left (\frac{x^{2}}{e} - \frac{d \log \left (e x^{2} + d\right )}{e^{2}}\right )} + 2 \, b \int \frac{x^{3} \arctan \left (c x\right )}{2 \,{\left (e x^{2} + d\right )}}\,{d x} \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{b x^{3} \arctan \left (c x\right ) + a x^{3}}{e x^{2} + d}, 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*} \int \frac{x^{3} \left (a + b \operatorname{atan}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \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 (b \arctan \left (c x\right ) + a\right )} x^{3}}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]