Optimal. Leaf size=382 \[ -\frac{6 i b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac{11 b^3 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c}+\frac{3 b^3 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right )}{c}-\frac{1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{11 i b^2 d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}-3 a b^2 d^3 x+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{6 b d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac{3 b^3 d^3 \log \left (c^2 x^2+1\right )}{2 c}-\frac{i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)+\frac{1}{4} i b^3 d^3 x \]
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Rubi [A] time = 0.706987, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 1586, 4992, 6610} \[ -\frac{6 i b^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac{11 b^3 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c}+\frac{3 b^3 d^3 \text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right )}{c}-\frac{1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{11 i b^2 d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}-3 a b^2 d^3 x+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{6 b d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac{3 b^3 d^3 \log \left (c^2 x^2+1\right )}{2 c}-\frac{i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)+\frac{1}{4} i b^3 d^3 x \]
Antiderivative was successfully verified.
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Rule 4864
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4916
Rule 260
Rule 4884
Rule 321
Rule 203
Rule 1586
Rule 4992
Rule 6610
Rubi steps
\begin{align*} \int (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{(3 i b) \int \left (-7 d^4 \left (a+b \tan ^{-1}(c x)\right )^2-4 i c d^4 x \left (a+b \tan ^{-1}(c x)\right )^2+c^2 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{8 i \left (i d^4-c d^4 x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{4 d}\\ &=-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{(6 b) \int \frac{\left (i d^4-c d^4 x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{d}-\frac{1}{4} \left (21 i b d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (3 b c d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\frac{1}{4} \left (3 i b c^2 d^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{(6 b) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{-\frac{i}{d^4}-\frac{c x}{d^4}} \, dx}{d}+\frac{1}{2} \left (21 i b^2 c d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{2} \left (i b^2 c^3 d^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{21 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c}-\frac{1}{2} \left (21 i b^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx-\left (3 b^2 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (3 b^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\left (12 b^2 d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\frac{1}{2} \left (i b^2 c d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{2} \left (i b^2 c d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-3 a b^2 d^3 x-\frac{1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c}-\frac{21 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c}-\frac{6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c}-\frac{1}{2} \left (i b^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx+\left (6 i b^3 d^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx+\frac{1}{2} \left (21 i b^3 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 b^3 d^3\right ) \int \tan ^{-1}(c x) \, dx+\frac{1}{4} \left (i b^3 c^2 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx\\ &=-3 a b^2 d^3 x+\frac{1}{4} i b^3 d^3 x-3 b^3 d^3 x \tan ^{-1}(c x)-\frac{1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c}-\frac{11 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c}-\frac{6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c}+\frac{3 b^3 d^3 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{c}-\frac{1}{4} \left (i b^3 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx+\frac{1}{2} \left (i b^3 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\frac{\left (21 b^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{2 c}+\left (3 b^3 c d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx\\ &=-3 a b^2 d^3 x+\frac{1}{4} i b^3 d^3 x-\frac{i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)-\frac{1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c}-\frac{11 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c}+\frac{3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac{6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c}+\frac{21 b^3 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{4 c}+\frac{3 b^3 d^3 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{c}+\frac{\left (b^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{2 c}\\ &=-3 a b^2 d^3 x+\frac{1}{4} i b^3 d^3 x-\frac{i b^3 d^3 \tan ^{-1}(c x)}{4 c}-3 b^3 d^3 x \tan ^{-1}(c x)-\frac{1}{4} i b^2 c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{7 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{21}{4} i b d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{3}{2} b c d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{4} i b c^2 d^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{i d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 c}+\frac{6 b d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c}-\frac{11 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c}+\frac{3 b^3 d^3 \log \left (1+c^2 x^2\right )}{2 c}-\frac{6 i b^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c}+\frac{11 b^3 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c}+\frac{3 b^3 d^3 \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{c}\\ \end{align*}
Mathematica [A] time = 1.66321, size = 693, normalized size = 1.81 \[ -\frac{i d^3 \left (2 b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right ) \left (12 a+12 b \tan ^{-1}(c x)-11 i b\right )+12 i b^3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )-a^2 b c^3 x^3+6 i a^2 b c^2 x^2-12 i a^2 b \log \left (c^2 x^2+1\right )+3 a^2 b c^4 x^4 \tan ^{-1}(c x)-12 i a^2 b c^3 x^3 \tan ^{-1}(c x)-18 a^2 b c^2 x^2 \tan ^{-1}(c x)+21 a^2 b c x+12 i a^2 b c x \tan ^{-1}(c x)-21 a^2 b \tan ^{-1}(c x)+a^3 c^4 x^4-4 i a^3 c^3 x^3-6 a^3 c^2 x^2+4 i a^3 c x+a b^2 c^2 x^2-22 a b^2 \log \left (c^2 x^2+1\right )+3 a b^2 c^4 x^4 \tan ^{-1}(c x)^2-12 i a b^2 c^3 x^3 \tan ^{-1}(c x)^2-2 a b^2 c^3 x^3 \tan ^{-1}(c x)-18 a b^2 c^2 x^2 \tan ^{-1}(c x)^2+12 i a b^2 c^2 x^2 \tan ^{-1}(c x)-12 i a b^2 c x+12 i a b^2 c x \tan ^{-1}(c x)^2+42 a b^2 c x \tan ^{-1}(c x)+3 a b^2 \tan ^{-1}(c x)^2+12 i a b^2 \tan ^{-1}(c x)+48 i a b^2 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+a b^2+6 i b^3 \log \left (c^2 x^2+1\right )+b^3 c^4 x^4 \tan ^{-1}(c x)^3-4 i b^3 c^3 x^3 \tan ^{-1}(c x)^3-b^3 c^3 x^3 \tan ^{-1}(c x)^2-6 b^3 c^2 x^2 \tan ^{-1}(c x)^3+6 i b^3 c^2 x^2 \tan ^{-1}(c x)^2+b^3 c^2 x^2 \tan ^{-1}(c x)-b^3 c x+4 i b^3 c x \tan ^{-1}(c x)^3+21 b^3 c x \tan ^{-1}(c x)^2-12 i b^3 c x \tan ^{-1}(c x)+b^3 \tan ^{-1}(c x)^3-16 i b^3 \tan ^{-1}(c x)^2+b^3 \tan ^{-1}(c x)+24 i b^3 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+44 b^3 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{4 c} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 5.163, size = 2004, normalized size = 5.3 \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}{32} \,{\left (b^{3} c^{3} d^{3} x^{4} - 4 i \, b^{3} c^{2} d^{3} x^{3} - 6 \, b^{3} c d^{3} x^{2} + 4 i \, b^{3} d^{3} x\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{3} +{\rm integral}\left (\frac{-16 i \, a^{3} c^{5} d^{3} x^{5} - 48 \, a^{3} c^{4} d^{3} x^{4} + 32 i \, a^{3} c^{3} d^{3} x^{3} - 32 \, a^{3} c^{2} d^{3} x^{2} + 48 i \, a^{3} c d^{3} x + 16 \, a^{3} d^{3} +{\left (12 i \, a b^{2} c^{5} d^{3} x^{5} + 3 \,{\left (12 \, a b^{2} - i \, b^{3}\right )} c^{4} d^{3} x^{4} +{\left (-24 i \, a b^{2} - 12 \, b^{3}\right )} c^{3} d^{3} x^{3} + 6 \,{\left (4 \, a b^{2} + 3 i \, b^{3}\right )} c^{2} d^{3} x^{2} - 12 \, a b^{2} d^{3} +{\left (-36 i \, a b^{2} + 12 \, b^{3}\right )} c d^{3} x\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\left (24 \, a^{2} b c^{5} d^{3} x^{5} - 72 i \, a^{2} b c^{4} d^{3} x^{4} - 48 \, a^{2} b c^{3} d^{3} x^{3} - 48 i \, a^{2} b c^{2} d^{3} x^{2} - 72 \, a^{2} b c d^{3} x + 24 i \, a^{2} b d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{16 \,{\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 )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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