Optimal. Leaf size=278 \[ \frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2}{25} a e x^5-\frac{b x^4 \left (e \log \left (c^2 x^2+1\right )+d\right )}{20 c}+\frac{b x^2 \left (e \log \left (c^2 x^2+1\right )+d\right )}{10 c^3}-\frac{b \log \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{10 c^5}-\frac{77 b e x^2}{300 c^3}+\frac{b e \log ^2\left (c^2 x^2+1\right )}{20 c^5}+\frac{137 b e \log \left (c^2 x^2+1\right )}{300 c^5}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \tan ^{-1}(c x) \]
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Rubi [A] time = 0.692885, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.577, Rules used = {4852, 266, 43, 5021, 6725, 1802, 635, 203, 260, 4916, 4846, 4884, 2475, 2390, 2301} \[ \frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+\frac{2 a e x^3}{15 c^2}-\frac{2 a e x}{5 c^4}+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2}{25} a e x^5-\frac{b x^4 \left (e \log \left (c^2 x^2+1\right )+d\right )}{20 c}+\frac{b x^2 \left (e \log \left (c^2 x^2+1\right )+d\right )}{10 c^3}-\frac{b \log \left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{10 c^5}-\frac{77 b e x^2}{300 c^3}+\frac{b e \log ^2\left (c^2 x^2+1\right )}{20 c^5}+\frac{137 b e \log \left (c^2 x^2+1\right )}{300 c^5}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{9 b e x^4}{200 c}-\frac{2}{25} b e x^5 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 4852
Rule 266
Rule 43
Rule 5021
Rule 6725
Rule 1802
Rule 635
Rule 203
Rule 260
Rule 4916
Rule 4846
Rule 4884
Rule 2475
Rule 2390
Rule 2301
Rubi steps
\begin{align*} \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right ) \, dx &=\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}-\left (2 c^2 e\right ) \int \left (\frac{2 b x^3-b c^2 x^5+4 a c^3 x^6+4 b c^3 x^6 \tan ^{-1}(c x)}{20 c^3 \left (1+c^2 x^2\right )}-\frac{b x \log \left (1+c^2 x^2\right )}{10 c^5 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac{(b e) \int \frac{x \log \left (1+c^2 x^2\right )}{1+c^2 x^2} \, dx}{5 c^3}-\frac{e \int \frac{2 b x^3-b c^2 x^5+4 a c^3 x^6+4 b c^3 x^6 \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{10 c}\\ &=\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1+c^2 x\right )}{1+c^2 x} \, dx,x,x^2\right )}{10 c^3}-\frac{e \int \left (\frac{x^3 \left (2 b-b c^2 x^2+4 a c^3 x^3\right )}{1+c^2 x^2}+\frac{4 b c^3 x^6 \tan ^{-1}(c x)}{1+c^2 x^2}\right ) \, dx}{10 c}\\ &=\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,1+c^2 x^2\right )}{10 c^5}-\frac{e \int \frac{x^3 \left (2 b-b c^2 x^2+4 a c^3 x^3\right )}{1+c^2 x^2} \, dx}{10 c}-\frac{1}{5} \left (2 b c^2 e\right ) \int \frac{x^6 \tan ^{-1}(c x)}{1+c^2 x^2} \, dx\\ &=\frac{b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}-\frac{1}{5} (2 b e) \int x^4 \tan ^{-1}(c x) \, dx+\frac{1}{5} (2 b e) \int \frac{x^4 \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\frac{e \int \left (\frac{4 a}{c^3}+\frac{3 b x}{c^2}-\frac{4 a x^2}{c}-b x^3+4 a c x^4-\frac{4 a+3 b c x}{c^3 \left (1+c^2 x^2\right )}\right ) \, dx}{10 c}\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}+\frac{2 a e x^3}{15 c^2}+\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5-\frac{2}{25} b e x^5 \tan ^{-1}(c x)+\frac{b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac{e \int \frac{4 a+3 b c x}{1+c^2 x^2} \, dx}{10 c^4}+\frac{(2 b e) \int x^2 \tan ^{-1}(c x) \, dx}{5 c^2}-\frac{(2 b e) \int \frac{x^2 \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{5 c^2}+\frac{1}{25} (2 b c e) \int \frac{x^5}{1+c^2 x^2} \, dx\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}+\frac{2 a e x^3}{15 c^2}+\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)+\frac{b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac{(2 a e) \int \frac{1}{1+c^2 x^2} \, dx}{5 c^4}-\frac{(2 b e) \int \tan ^{-1}(c x) \, dx}{5 c^4}+\frac{(2 b e) \int \frac{\tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{5 c^4}+\frac{(3 b e) \int \frac{x}{1+c^2 x^2} \, dx}{10 c^3}-\frac{(2 b e) \int \frac{x^3}{1+c^2 x^2} \, dx}{15 c}+\frac{1}{25} (b c e) \operatorname{Subst}\left (\int \frac{x^2}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{5 c^4}-\frac{3 b e x^2}{20 c^3}+\frac{2 a e x^3}{15 c^2}+\frac{b e x^4}{40 c}-\frac{2}{25} a e x^5+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{3 b e \log \left (1+c^2 x^2\right )}{20 c^5}+\frac{b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}+\frac{(2 b e) \int \frac{x}{1+c^2 x^2} \, dx}{5 c^3}-\frac{(b e) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{15 c}+\frac{1}{25} (b c e) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}+\frac{x}{c^2}+\frac{1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 a e x}{5 c^4}-\frac{19 b e x^2}{100 c^3}+\frac{2 a e x^3}{15 c^2}+\frac{9 b e x^4}{200 c}-\frac{2}{25} a e x^5+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{39 b e \log \left (1+c^2 x^2\right )}{100 c^5}+\frac{b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}-\frac{(b e) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{15 c}\\ &=-\frac{2 a e x}{5 c^4}-\frac{77 b e x^2}{300 c^3}+\frac{2 a e x^3}{15 c^2}+\frac{9 b e x^4}{200 c}-\frac{2}{25} a e x^5+\frac{2 a e \tan ^{-1}(c x)}{5 c^5}-\frac{2 b e x \tan ^{-1}(c x)}{5 c^4}+\frac{2 b e x^3 \tan ^{-1}(c x)}{15 c^2}-\frac{2}{25} b e x^5 \tan ^{-1}(c x)+\frac{b e \tan ^{-1}(c x)^2}{5 c^5}+\frac{137 b e \log \left (1+c^2 x^2\right )}{300 c^5}+\frac{b e \log ^2\left (1+c^2 x^2\right )}{20 c^5}+\frac{b x^2 \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^3}-\frac{b x^4 \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 c}+\frac{1}{5} x^5 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )-\frac{b \log \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 c^5}\\ \end{align*}
Mathematica [A] time = 0.173072, size = 214, normalized size = 0.77 \[ \frac{c x \left (8 a \left (15 c^4 d x^4-2 e \left (3 c^4 x^4-5 c^2 x^2+15\right )\right )+b c x \left (e \left (27 c^2 x^2-154\right )-30 d \left (c^2 x^2-2\right )\right )\right )+\log \left (c^2 x^2+1\right ) \left (120 a c^5 e x^5+2 b e \left (-15 c^4 x^4+30 c^2 x^2+137\right )-60 b d\right )+8 \tan ^{-1}(c x) \left (30 a e+15 b c^5 d x^5-2 b c e x \left (3 c^4 x^4-5 c^2 x^2+15\right )+15 b c^5 e x^5 \log \left (c^2 x^2+1\right )\right )-30 b e \log ^2\left (c^2 x^2+1\right )+120 b e \tan ^{-1}(c x)^2}{600 c^5} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.08, size = 4941, normalized size = 17.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50644, size = 346, normalized size = 1.24 \begin{align*} \frac{1}{5} \, a d x^{5} + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{2}{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} b e \arctan \left (c x\right ) + \frac{1}{20} \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b d + \frac{1}{75} \,{\left (15 \, x^{5} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{2}{\left (\frac{3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac{15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a e + \frac{{\left (27 \, c^{4} x^{4} - 154 \, c^{2} x^{2} - 120 \, \arctan \left (c x\right )^{2} - 2 \,{\left (15 \, c^{4} x^{4} - 30 \, c^{2} x^{2} - 137\right )} \log \left (c^{2} x^{2} + 1\right ) - 30 \, \log \left (c^{2} x^{2} + 1\right )^{2}\right )} b e}{600 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37734, size = 541, normalized size = 1.95 \begin{align*} \frac{80 \, a c^{3} e x^{3} + 24 \,{\left (5 \, a c^{5} d - 2 \, a c^{5} e\right )} x^{5} - 3 \,{\left (10 \, b c^{4} d - 9 \, b c^{4} e\right )} x^{4} - 240 \, a c e x + 120 \, b e \arctan \left (c x\right )^{2} - 30 \, b e \log \left (c^{2} x^{2} + 1\right )^{2} + 2 \,{\left (30 \, b c^{2} d - 77 \, b c^{2} e\right )} x^{2} + 8 \,{\left (10 \, b c^{3} e x^{3} + 3 \,{\left (5 \, b c^{5} d - 2 \, b c^{5} e\right )} x^{5} - 30 \, b c e x + 30 \, a e\right )} \arctan \left (c x\right ) + 2 \,{\left (60 \, b c^{5} e x^{5} \arctan \left (c x\right ) + 60 \, a c^{5} e x^{5} - 15 \, b c^{4} e x^{4} + 30 \, b c^{2} e x^{2} - 30 \, b d + 137 \, b e\right )} \log \left (c^{2} x^{2} + 1\right )}{600 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 35.4914, size = 338, normalized size = 1.22 \begin{align*} \begin{cases} \frac{a d x^{5}}{5} + \frac{a e x^{5} \log{\left (c^{2} x^{2} + 1 \right )}}{5} - \frac{2 a e x^{5}}{25} + \frac{2 a e x^{3}}{15 c^{2}} - \frac{2 a e x}{5 c^{4}} + \frac{2 a e \operatorname{atan}{\left (c x \right )}}{5 c^{5}} + \frac{b d x^{5} \operatorname{atan}{\left (c x \right )}}{5} + \frac{b e x^{5} \log{\left (c^{2} x^{2} + 1 \right )} \operatorname{atan}{\left (c x \right )}}{5} - \frac{2 b e x^{5} \operatorname{atan}{\left (c x \right )}}{25} - \frac{b d x^{4}}{20 c} - \frac{b e x^{4} \log{\left (c^{2} x^{2} + 1 \right )}}{20 c} + \frac{9 b e x^{4}}{200 c} + \frac{2 b e x^{3} \operatorname{atan}{\left (c x \right )}}{15 c^{2}} + \frac{b d x^{2}}{10 c^{3}} + \frac{b e x^{2} \log{\left (c^{2} x^{2} + 1 \right )}}{10 c^{3}} - \frac{77 b e x^{2}}{300 c^{3}} - \frac{2 b e x \operatorname{atan}{\left (c x \right )}}{5 c^{4}} - \frac{b d \log{\left (c^{2} x^{2} + 1 \right )}}{10 c^{5}} - \frac{b e \log{\left (c^{2} x^{2} + 1 \right )}^{2}}{20 c^{5}} + \frac{137 b e \log{\left (c^{2} x^{2} + 1 \right )}}{300 c^{5}} + \frac{b e \operatorname{atan}^{2}{\left (c x \right )}}{5 c^{5}} & \text{for}\: c \neq 0 \\\frac{a d x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.655, size = 618, normalized size = 2.22 \begin{align*} \frac{60 \, \pi b c^{5} x^{5} e \log \left (c^{2} x^{2} + 1\right ) \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 24 \, \pi b c^{5} x^{5} e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 120 \, b c^{5} x^{5} \arctan \left (\frac{1}{c x}\right ) e \log \left (c^{2} x^{2} + 1\right ) + 120 \, b c^{5} d x^{5} \arctan \left (c x\right ) + 48 \, b c^{5} x^{5} \arctan \left (\frac{1}{c x}\right ) e + 120 \, a c^{5} x^{5} e \log \left (c^{2} x^{2} + 1\right ) + 120 \, a c^{5} d x^{5} - 48 \, a c^{5} x^{5} e - 30 \, b c^{4} x^{4} e \log \left (c^{2} x^{2} + 1\right ) + 40 \, \pi b c^{3} x^{3} e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 30 \, b c^{4} d x^{4} + 27 \, b c^{4} x^{4} e - 80 \, b c^{3} x^{3} \arctan \left (\frac{1}{c x}\right ) e + 80 \, a c^{3} x^{3} e + 60 \, b c^{2} x^{2} e \log \left (c^{2} x^{2} + 1\right ) - 120 \, \pi b c x e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) + 60 \, b c^{2} d x^{2} - 154 \, b c^{2} x^{2} e - 180 \, \pi ^{2} b e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) - 120 \, \pi b \arctan \left (\frac{1}{c x}\right ) e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) + 240 \, b c x \arctan \left (\frac{1}{c x}\right ) e - 240 \, \pi a e \mathrm{sgn}\left (c\right ) \mathrm{sgn}\left (x\right ) + 60 \, \pi ^{2} b e - 240 \, a c x e + 120 \, \pi b \arctan \left (c x\right ) e + 120 \, \pi b \arctan \left (\frac{1}{c x}\right ) e + 120 \, b \arctan \left (\frac{1}{c x}\right )^{2} e - 30 \, b e \log \left (c^{2} x^{2} + 1\right )^{2} + 240 \, a \arctan \left (c x\right ) e - 60 \, b d \log \left (c^{2} x^{2} + 1\right ) + 274 \, b e \log \left (c^{2} x^{2} + 1\right )}{600 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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