Optimal. Leaf size=307 \[ -\frac{6 b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{5 c^2}+\frac{1}{10} i b c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{12 i b d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}+\frac{1}{2} b c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{6}{5} i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{5 a b d^3 x}{2 c}+\frac{3 b^2 d^3 \log \left (c^2 x^2+1\right )}{2 c^2}-\frac{13 i b^2 d^3 \tan ^{-1}(c x)}{10 c^2}-\frac{1}{30} i b^2 c d^3 x^3+\frac{13 i b^2 d^3 x}{10 c}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{1}{4} b^2 d^3 x^2 \]
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Rubi [A] time = 0.613652, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {4876, 4864, 4846, 260, 4852, 321, 203, 266, 43, 1586, 4854, 2402, 2315, 302} \[ -\frac{6 b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{5 c^2}+\frac{1}{10} i b c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{12 i b d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}+\frac{1}{2} b c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{6}{5} i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{5 a b d^3 x}{2 c}+\frac{3 b^2 d^3 \log \left (c^2 x^2+1\right )}{2 c^2}-\frac{13 i b^2 d^3 \tan ^{-1}(c x)}{10 c^2}-\frac{1}{30} i b^2 c d^3 x^3+\frac{13 i b^2 d^3 x}{10 c}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{1}{4} b^2 d^3 x^2 \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4864
Rule 4846
Rule 260
Rule 4852
Rule 321
Rule 203
Rule 266
Rule 43
Rule 1586
Rule 4854
Rule 2402
Rule 2315
Rule 302
Rubi steps
\begin{align*} \int x (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (\frac{i (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{i (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{c d}\right ) \, dx\\ &=\frac{i \int (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c}-\frac{i \int (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c d}\\ &=\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}+\frac{(2 b) \int \left (-15 d^5 \left (a+b \tan ^{-1}(c x)\right )-11 i c d^5 x \left (a+b \tan ^{-1}(c x)\right )+5 c^2 d^5 x^2 \left (a+b \tan ^{-1}(c x)\right )+i c^3 d^5 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{16 i \left (i d^5-c d^5 x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{5 c d^2}-\frac{b \int \left (-7 d^4 \left (a+b \tan ^{-1}(c x)\right )-4 i c d^4 x \left (a+b \tan ^{-1}(c x)\right )+c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{8 i \left (i d^4-c d^4 x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2}\right ) \, dx}{2 c d}\\ &=\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}-\frac{(32 i b) \int \frac{\left (i d^5-c d^5 x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c d^2}+\frac{(4 i b) \int \frac{\left (i d^4-c d^4 x\right ) \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c d}+\left (2 i b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (22 i b d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{\left (7 b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}-\frac{\left (6 b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}-\frac{1}{2} \left (b c d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (2 b c d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{5} \left (2 i b c^2 d^3\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-\frac{5 a b d^3 x}{2 c}-\frac{6}{5} i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} b c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{10} i b c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}-\frac{(32 i b) \int \frac{a+b \tan ^{-1}(c x)}{-\frac{i}{d^5}-\frac{c x}{d^5}} \, dx}{5 c d^2}+\frac{(4 i b) \int \frac{a+b \tan ^{-1}(c x)}{-\frac{i}{d^4}-\frac{c x}{d^4}} \, dx}{c d}+\frac{\left (7 b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{2 c}-\frac{\left (6 b^2 d^3\right ) \int \tan ^{-1}(c x) \, dx}{c}-\left (i b^2 c d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{1}{5} \left (11 i b^2 c d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx+\frac{1}{6} \left (b^2 c^2 d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx-\frac{1}{3} \left (2 b^2 c^2 d^3\right ) \int \frac{x^3}{1+c^2 x^2} \, dx-\frac{1}{10} \left (i b^2 c^3 d^3\right ) \int \frac{x^4}{1+c^2 x^2} \, dx\\ &=-\frac{5 a b d^3 x}{2 c}+\frac{6 i b^2 d^3 x}{5 c}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{6}{5} i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} b c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{10} i b c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}-\frac{12 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{5 c^2}-\frac{1}{2} \left (7 b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx+\left (6 b^2 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx+\frac{\left (i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{c}-\frac{\left (11 i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{5 c}-\frac{\left (4 i b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{c}+\frac{\left (32 i b^2 d^3\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{5 c}+\frac{1}{12} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )-\frac{1}{3} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )-\frac{1}{10} \left (i b^2 c^3 d^3\right ) \int \left (-\frac{1}{c^4}+\frac{x^2}{c^2}+\frac{1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{5 a b d^3 x}{2 c}+\frac{13 i b^2 d^3 x}{10 c}-\frac{1}{30} i b^2 c d^3 x^3-\frac{6 i b^2 d^3 \tan ^{-1}(c x)}{5 c^2}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{6}{5} i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} b c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{10} i b c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}-\frac{12 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{5 c^2}+\frac{5 b^2 d^3 \log \left (1+c^2 x^2\right )}{4 c^2}+\frac{\left (4 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{c^2}-\frac{\left (32 b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{5 c^2}-\frac{\left (i b^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{10 c}+\frac{1}{12} \left (b^2 c^2 d^3\right ) \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 )-\frac{1}{3} \left (b^2 c^2 d^3\right ) \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 )\\ &=-\frac{5 a b d^3 x}{2 c}+\frac{13 i b^2 d^3 x}{10 c}-\frac{1}{4} b^2 d^3 x^2-\frac{1}{30} i b^2 c d^3 x^3-\frac{13 i b^2 d^3 \tan ^{-1}(c x)}{10 c^2}-\frac{5 b^2 d^3 x \tan ^{-1}(c x)}{2 c}-\frac{6}{5} i b d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} b c d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{10} i b c^2 d^3 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )^2}{5 c^2}-\frac{12 i b d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{5 c^2}+\frac{3 b^2 d^3 \log \left (1+c^2 x^2\right )}{2 c^2}-\frac{6 b^2 d^3 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{5 c^2}\\ \end{align*}
Mathematica [A] time = 1.28737, size = 325, normalized size = 1.06 \[ \frac{d^3 \left (-72 b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-12 i a^2 c^5 x^5-45 a^2 c^4 x^4+60 i a^2 c^3 x^3+30 a^2 c^2 x^2+6 i a b c^4 x^4+30 a b c^3 x^3-72 i a b c^2 x^2+72 i a b \log \left (c^2 x^2+1\right )+6 b \tan ^{-1}(c x) \left (a \left (-4 i c^5 x^5-15 c^4 x^4+20 i c^3 x^3+10 c^2 x^2+25\right )+b \left (i c^4 x^4+5 c^3 x^3-12 i c^2 x^2-25 c x-13 i\right )-24 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-150 a b c x-18 i a b-2 i b^2 c^3 x^3-15 b^2 c^2 x^2+90 b^2 \log \left (c^2 x^2+1\right )+78 i b^2 c x+3 b^2 (1-4 i c x) (c x-i)^4 \tan ^{-1}(c x)^2-15 b^2\right )}{60 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.1, size = 656, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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*} \frac{1}{80} \,{\left (4 i \, b^{2} c^{3} d^{3} x^{5} + 15 \, b^{2} c^{2} d^{3} x^{4} - 20 i \, b^{2} c d^{3} x^{3} - 10 \, b^{2} d^{3} x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\rm integral}\left (\frac{-20 i \, a^{2} c^{5} d^{3} x^{6} - 60 \, a^{2} c^{4} d^{3} x^{5} + 40 i \, a^{2} c^{3} d^{3} x^{4} - 40 \, a^{2} c^{2} d^{3} x^{3} + 60 i \, a^{2} c d^{3} x^{2} + 20 \, a^{2} d^{3} x +{\left (20 \, a b c^{5} d^{3} x^{6} +{\left (-60 i \, a b - 4 \, b^{2}\right )} c^{4} d^{3} x^{5} - 5 \,{\left (8 \, a b - 3 i \, b^{2}\right )} c^{3} d^{3} x^{4} +{\left (-40 i \, a b + 20 \, b^{2}\right )} c^{2} d^{3} x^{3} - 10 \,{\left (6 \, a b + i \, b^{2}\right )} c d^{3} x^{2} + 20 i \, a b d^{3} x\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{20 \,{\left (c^{2} x^{2} + 1\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, c d x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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