Optimal. Leaf size=943 \[ \text{result too large to display} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.75896, antiderivative size = 943, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {4980, 4978, 4864, 4856, 2402, 2315, 2447, 4984, 4884, 4920, 4854, 4858} \[ \frac{i c \sqrt{-d} \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac{i c \sqrt{-d} \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{4 \left (c^2 d-e\right ) e^{3/2}}-\frac{\text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right ) b^2}{2 e^2}+\frac{\text{PolyLog}\left (3,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{4 e^2}+\frac{\text{PolyLog}\left (3,1-\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b^2}{4 e^2}-\frac{c \sqrt{-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 ) b}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac{c \sqrt{-d} \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b}{2 \left (c^2 d-e\right ) e^{3/2}}+\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) b}{e^2}-\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (c \sqrt{-d}-i \sqrt{e}\right ) (1-i c x)}\right ) b}{2 e^2}-\frac{i \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right ) b}{2 e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )}-\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c \left (\sqrt{e} x+\sqrt{-d}\right )}{\left (\sqrt{-d} c+i \sqrt{e}\right ) (1-i c x)}\right )}{2 e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4980
Rule 4978
Rule 4864
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rule 4984
Rule 4884
Rule 4920
Rule 4854
Rule 4858
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{d x \left (a+b \tan ^{-1}(c x)\right )^2}{e \left (d+e x^2\right )^2}+\frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{d+e x^2} \, dx}{e}-\frac{d \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx}{e}\\ &=\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )^2} \, dx}{4 \sqrt{-d} e^{3/2}}-\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )^2} \, dx}{4 \sqrt{-d} e^{3/2}}+\frac{\int \left (-\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{e}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{(b c) \int \left (\frac{\sqrt{-d} e \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (-\sqrt{-d}+\sqrt{e} x\right )}+\frac{c^2 d \left (\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\sqrt{-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e^2}-\frac{(b c) \int \left (\frac{\sqrt{-d} e \left (a+b \tan ^{-1}(c x)\right )}{\left (-c^2 d+e\right ) \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{c^2 \left (d+\sqrt{-d} \sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e^2}-\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{2 e^{3/2}}+\frac{\int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 e^{3/2}}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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{\left (a+b \tan ^{-1}(c x)\right )^2 \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 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{e^2}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{b^2 \text{Li}_3\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{b^2 \text{Li}_3\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{\left (b c^3\right ) \int \frac{\left (d+\sqrt{-d} \sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac{\left (b c^3 \sqrt{-d}\right ) \int \frac{\left (\sqrt{-d}+\sqrt{e} x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}-\frac{\left (b c \sqrt{-d}\right ) \int \frac{a+b \tan ^{-1}(c x)}{-\sqrt{-d}+\sqrt{e} x} \, dx}{2 \left (c^2 d-e\right ) e}+\frac{\left (b c \sqrt{-d}\right ) \int \frac{a+b \tan ^{-1}(c x)}{\sqrt{-d}+\sqrt{e} x} \, dx}{2 \left (c^2 d-e\right ) e}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{e^2}-\frac{b c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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{b c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{e^2}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{b^2 \text{Li}_3\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{b^2 \text{Li}_3\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{\left (b c^3\right ) \int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac{\sqrt{-d} \sqrt{e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac{\left (b c^3 \sqrt{-d}\right ) \int \left (\frac{\sqrt{-d} \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}+\frac{\sqrt{e} x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac{\left (b^2 c^2 \sqrt{-d}\right ) \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 \left (c^2 d-e\right ) e^{3/2}}-\frac{\left (b^2 c^2 \sqrt{-d}\right ) \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 \left (c^2 d-e\right ) e^{3/2}}\\ &=\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{e^2}-\frac{b c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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{b c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{e^2}+\frac{i b^2 c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{i b^2 c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{b^2 \text{Li}_3\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{b^2 \text{Li}_3\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{\left (b c^3 d\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}\\ &=-\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^2}{2 \left (c^2 d-e\right ) e^2}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2}{4 e^2 \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{e^2}-\frac{b c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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{b c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}+\frac{\left (a+b \tan ^{-1}(c x)\right )^2 \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 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{e^2}+\frac{i b^2 c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{i b^2 c \sqrt{-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 \left (c^2 d-e\right ) e^{3/2}}-\frac{i b \left (a+b \tan ^{-1}(c x)\right ) \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 )}{2 e^2}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 e^2}+\frac{b^2 \text{Li}_3\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{b^2 \text{Li}_3\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 [F] time = 18.1535, size = 0, normalized size = 0. \[ \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (d+e x^2\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 9.468, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( a+b\arctan \left ( cx \right ) \right ) ^{2}}{ \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \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^{2}{\left (\frac{d}{e^{3} x^{2} + d e^{2}} + \frac{\log \left (e x^{2} + d\right )}{e^{2}}\right )} + \int \frac{b^{2} x^{3} \arctan \left (c x\right )^{2} + 2 \, a b x^{3} \arctan \left (c x\right )}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\,{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^{2} x^{3} \arctan \left (c x\right )^{2} + 2 \, a b x^{3} \arctan \left (c x\right ) + a^{2} x^{3}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (e x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]