Optimal. Leaf size=255 \[ -\frac{23 i b^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{30 c^6}+\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}-\frac{23 b^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^6}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}-\frac{23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}-\frac{b^3 x^3}{60 c^3}+\frac{19 b^3 x}{60 c^5}-\frac{19 b^3 \tan ^{-1}(c x)}{60 c^6} \]
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Rubi [A] time = 0.94766, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {4852, 4916, 302, 203, 321, 4920, 4854, 2402, 2315, 4846, 4884} \[ -\frac{23 i b^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{30 c^6}+\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}-\frac{23 b^2 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{15 c^6}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}-\frac{23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}-\frac{b^3 x^3}{60 c^3}+\frac{19 b^3 x}{60 c^5}-\frac{19 b^3 \tan ^{-1}(c x)}{60 c^6} \]
Antiderivative was successfully verified.
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Rule 4852
Rule 4916
Rule 302
Rule 203
Rule 321
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4846
Rule 4884
Rubi steps
\begin{align*} \int x^5 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{1}{2} (b c) \int \frac{x^6 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{b \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c}+\frac{b \int \frac{x^4 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c}\\ &=-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3+\frac{1}{5} b^2 \int \frac{x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{b \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c^3}-\frac{b \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c^3}\\ &=\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{b \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c^5}+\frac{b \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c^5}+\frac{b^2 \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac{b^2 \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac{b^2 \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{b^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^4}+\frac{b^2 \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^4}-\frac{b^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^4}+\frac{b^2 \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^4}+\frac{b^2 \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^4}-\frac{b^3 \int \frac{x^4}{1+c^2 x^2} \, dx}{20 c}\\ &=-\frac{4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{b^2 \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^5}-\frac{b^2 \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^5}-\frac{b^2 \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^5}+\frac{b^3 \int \frac{x^2}{1+c^2 x^2} \, dx}{10 c^3}+\frac{b^3 \int \frac{x^2}{1+c^2 x^2} \, dx}{6 c^3}-\frac{b^3 \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{20 c}\\ &=\frac{19 b^3 x}{60 c^5}-\frac{b^3 x^3}{60 c^3}-\frac{4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{23 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^6}-\frac{b^3 \int \frac{1}{1+c^2 x^2} \, dx}{20 c^5}-\frac{b^3 \int \frac{1}{1+c^2 x^2} \, dx}{10 c^5}-\frac{b^3 \int \frac{1}{1+c^2 x^2} \, dx}{6 c^5}+\frac{b^3 \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^5}+\frac{b^3 \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^5}+\frac{b^3 \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^5}\\ &=\frac{19 b^3 x}{60 c^5}-\frac{b^3 x^3}{60 c^3}-\frac{19 b^3 \tan ^{-1}(c x)}{60 c^6}-\frac{4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{23 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^6}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{5 c^6}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^6}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c^6}\\ &=\frac{19 b^3 x}{60 c^5}-\frac{b^3 x^3}{60 c^3}-\frac{19 b^3 \tan ^{-1}(c x)}{60 c^6}-\frac{4 b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{15 c^4}+\frac{b^2 x^4 \left (a+b \tan ^{-1}(c x)\right )}{20 c^2}-\frac{23 i b \left (a+b \tan ^{-1}(c x)\right )^2}{30 c^6}-\frac{b x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^5}+\frac{b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^3}-\frac{b x^5 \left (a+b \tan ^{-1}(c x)\right )^2}{10 c}+\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{6 c^6}+\frac{1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^3-\frac{23 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{15 c^6}-\frac{23 i b^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{30 c^6}\\ \end{align*}
Mathematica [A] time = 0.764416, size = 291, normalized size = 1.14 \[ \frac{46 i b^3 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+b \tan ^{-1}(c x) \left (30 a^2 \left (c^6 x^6+1\right )-4 a b c x \left (3 c^4 x^4-5 c^2 x^2+15\right )+b^2 \left (3 c^4 x^4-16 c^2 x^2-19\right )-92 b^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-6 a^2 b c^5 x^5+10 a^2 b c^3 x^3-30 a^2 b c x+10 a^3 c^6 x^6+3 a b^2 c^4 x^4-16 a b^2 c^2 x^2+46 a b^2 \log \left (c^2 x^2+1\right )+2 b^2 \tan ^{-1}(c x)^2 \left (15 a \left (c^6 x^6+1\right )+b \left (-3 c^5 x^5+5 c^3 x^3-15 c x+23 i\right )\right )-19 a b^2-b^3 c^3 x^3+10 b^3 \left (c^6 x^6+1\right ) \tan ^{-1}(c x)^3+19 b^3 c x}{60 c^6} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.017, size = 528, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x^{5} \arctan \left (c x\right )^{3} + 3 \, a b^{2} x^{5} \arctan \left (c x\right )^{2} + 3 \, a^{2} b x^{5} \arctan \left (c x\right ) + a^{3} x^{5}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \left (a + b \operatorname{atan}{\left (c x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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