Optimal. Leaf size=193 \[ \frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\frac{176 b d^4 \log (c x+i)}{105 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3 \]
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Rubi [A] time = 0.166718, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {43, 4872, 12, 893} \[ \frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{5 i b d^4 x}{3 c^2}+\frac{176 b d^4 \log (c x+i)}{105 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3 \]
Antiderivative was successfully verified.
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Rule 43
Rule 4872
Rule 12
Rule 893
Rubi steps
\begin{align*} \int x^2 (d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-(b c) \int \frac{d^4 (i-c x)^4 \left (-1+5 i c x+15 c^2 x^2\right )}{105 c^3 (i+c x)} \, dx\\ &=\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{\left (b d^4\right ) \int \frac{(i-c x)^4 \left (-1+5 i c x+15 c^2 x^2\right )}{i+c x} \, dx}{105 c^2}\\ &=\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}-\frac{\left (b d^4\right ) \int \left (-175 i+176 c x+175 i c^2 x^2-141 c^3 x^3-70 i c^4 x^4+15 c^5 x^5-\frac{176}{i+c x}\right ) \, dx}{105 c^2}\\ &=\frac{5 i b d^4 x}{3 c^2}-\frac{88 b d^4 x^2}{105 c}-\frac{5}{9} i b d^4 x^3+\frac{47}{140} b c d^4 x^4+\frac{2}{15} i b c^2 d^4 x^5-\frac{1}{42} b c^3 d^4 x^6+\frac{i d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^3}-\frac{i d^4 (1+i c x)^6 \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{i d^4 (1+i c x)^7 \left (a+b \tan ^{-1}(c x)\right )}{7 c^3}+\frac{176 b d^4 \log (i+c x)}{105 c^3}\\ \end{align*}
Mathematica [A] time = 0.120197, size = 276, normalized size = 1.43 \[ \frac{1}{7} a c^4 d^4 x^7-\frac{2}{3} i a c^3 d^4 x^6-\frac{6}{5} a c^2 d^4 x^5+i a c d^4 x^4+\frac{1}{3} a d^4 x^3-\frac{1}{42} b c^3 d^4 x^6+\frac{2}{15} i b c^2 d^4 x^5+\frac{88 b d^4 \log \left (c^2 x^2+1\right )}{105 c^3}+\frac{1}{7} b c^4 d^4 x^7 \tan ^{-1}(c x)-\frac{2}{3} i b c^3 d^4 x^6 \tan ^{-1}(c x)-\frac{6}{5} b c^2 d^4 x^5 \tan ^{-1}(c x)+\frac{5 i b d^4 x}{3 c^2}-\frac{5 i b d^4 \tan ^{-1}(c x)}{3 c^3}+\frac{47}{140} b c d^4 x^4-\frac{88 b d^4 x^2}{105 c}+i b c d^4 x^4 \tan ^{-1}(c x)+\frac{1}{3} b d^4 x^3 \tan ^{-1}(c x)-\frac{5}{9} i b d^4 x^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 237, normalized size = 1.2 \begin{align*}{\frac{{c}^{4}{d}^{4}a{x}^{7}}{7}}+{\frac{2\,i}{15}}b{c}^{2}{d}^{4}{x}^{5}-{\frac{6\,{c}^{2}{d}^{4}a{x}^{5}}{5}}+ic{d}^{4}b\arctan \left ( cx \right ){x}^{4}+{\frac{{d}^{4}a{x}^{3}}{3}}+{\frac{{c}^{4}{d}^{4}b\arctan \left ( cx \right ){x}^{7}}{7}}-{\frac{2\,i}{3}}{c}^{3}{d}^{4}b\arctan \left ( cx \right ){x}^{6}-{\frac{6\,{c}^{2}{d}^{4}b\arctan \left ( cx \right ){x}^{5}}{5}}-{\frac{{\frac{5\,i}{3}}{d}^{4}b\arctan \left ( cx \right ) }{{c}^{3}}}+{\frac{{d}^{4}b\arctan \left ( cx \right ){x}^{3}}{3}}+{\frac{{\frac{5\,i}{3}}{d}^{4}bx}{{c}^{2}}}-{\frac{b{c}^{3}{d}^{4}{x}^{6}}{42}}-{\frac{2\,i}{3}}{c}^{3}{d}^{4}a{x}^{6}+{\frac{47\,bc{d}^{4}{x}^{4}}{140}}+ic{d}^{4}a{x}^{4}-{\frac{88\,{d}^{4}b{x}^{2}}{105\,c}}+{\frac{88\,{d}^{4}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{105\,{c}^{3}}}-{\frac{5\,i}{9}}b{d}^{4}{x}^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48581, size = 429, normalized size = 2.22 \begin{align*} \frac{1}{7} \, a c^{4} d^{4} x^{7} - \frac{2}{3} i \, a c^{3} d^{4} x^{6} - \frac{6}{5} \, a c^{2} d^{4} x^{5} + \frac{1}{84} \,{\left (12 \, x^{7} \arctan \left (c x\right ) - c{\left (\frac{2 \, c^{4} x^{6} - 3 \, c^{2} x^{4} + 6 \, x^{2}}{c^{6}} - \frac{6 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8}}\right )}\right )} b c^{4} d^{4} + i \, a c d^{4} x^{4} - \frac{2}{45} i \,{\left (15 \, x^{6} \arctan \left (c x\right ) - c{\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 c^{3} d^{4} - \frac{3}{10} \,{\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 c^{2} d^{4} + \frac{1}{3} \, a d^{4} x^{3} + \frac{1}{3} i \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c d^{4} + \frac{1}{6} \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b d^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.31694, size = 525, normalized size = 2.72 \begin{align*} \frac{180 \, a c^{7} d^{4} x^{7} +{\left (-840 i \, a - 30 \, b\right )} c^{6} d^{4} x^{6} - 168 \,{\left (9 \, a - i \, b\right )} c^{5} d^{4} x^{5} +{\left (1260 i \, a + 423 \, b\right )} c^{4} d^{4} x^{4} + 140 \,{\left (3 \, a - 5 i \, b\right )} c^{3} d^{4} x^{3} - 1056 \, b c^{2} d^{4} x^{2} + 2100 i \, b c d^{4} x + 2106 \, b d^{4} \log \left (\frac{c x + i}{c}\right ) + 6 \, b d^{4} \log \left (\frac{c x - i}{c}\right ) +{\left (90 i \, b c^{7} d^{4} x^{7} + 420 \, b c^{6} d^{4} x^{6} - 756 i \, b c^{5} d^{4} x^{5} - 630 \, b c^{4} d^{4} x^{4} + 210 i \, b c^{3} d^{4} x^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{1260 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.20341, size = 325, normalized size = 1.68 \begin{align*} \frac{a c^{4} d^{4} x^{7}}{7} - \frac{88 b d^{4} x^{2}}{105 c} + \frac{5 i b d^{4} x}{3 c^{2}} + \frac{b d^{4} \left (\frac{\log{\left (x - \frac{i}{c} \right )}}{210} + \frac{117 \log{\left (x + \frac{i}{c} \right )}}{70}\right )}{c^{3}} + x^{6} \left (- \frac{2 i a c^{3} d^{4}}{3} - \frac{b c^{3} d^{4}}{42}\right ) + x^{5} \left (- \frac{6 a c^{2} d^{4}}{5} + \frac{2 i b c^{2} d^{4}}{15}\right ) + x^{4} \left (i a c d^{4} + \frac{47 b c d^{4}}{140}\right ) + x^{3} \left (\frac{a d^{4}}{3} - \frac{5 i b d^{4}}{9}\right ) + \left (- \frac{i b c^{4} d^{4} x^{7}}{14} - \frac{b c^{3} d^{4} x^{6}}{3} + \frac{3 i b c^{2} d^{4} x^{5}}{5} + \frac{b c d^{4} x^{4}}{2} - \frac{i b d^{4} x^{3}}{6}\right ) \log{\left (i c x + 1 \right )} + \left (\frac{i b c^{4} d^{4} x^{7}}{14} + \frac{b c^{3} d^{4} x^{6}}{3} - \frac{3 i b c^{2} d^{4} x^{5}}{5} - \frac{b c d^{4} x^{4}}{2} + \frac{i b d^{4} x^{3}}{6}\right ) \log{\left (- i c x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20647, size = 333, normalized size = 1.73 \begin{align*} \frac{180 \, b c^{7} d^{4} x^{7} \arctan \left (c x\right ) + 180 \, a c^{7} d^{4} x^{7} - 840 \, b c^{6} d^{4} i x^{6} \arctan \left (c x\right ) - 840 \, a c^{6} d^{4} i x^{6} - 30 \, b c^{6} d^{4} x^{6} + 168 \, b c^{5} d^{4} i x^{5} - 1512 \, b c^{5} d^{4} x^{5} \arctan \left (c x\right ) - 1512 \, a c^{5} d^{4} x^{5} + 1260 \, b c^{4} d^{4} i x^{4} \arctan \left (c x\right ) + 1260 \, a c^{4} d^{4} i x^{4} + 423 \, b c^{4} d^{4} x^{4} - 700 \, b c^{3} d^{4} i x^{3} + 420 \, b c^{3} d^{4} x^{3} \arctan \left (c x\right ) + 420 \, a c^{3} d^{4} x^{3} - 1056 \, b c^{2} d^{4} x^{2} + 2100 \, b c d^{4} i x + 2106 \, b d^{4} \log \left (c x + i\right ) + 6 \, b d^{4} \log \left (c x - i\right )}{1260 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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