Optimal. Leaf size=513 \[ -\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,i c x)+\frac{37}{40} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )+\frac{14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac{3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac{28}{15} i a b c^6 d^3 \log (x)-\frac{11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}+\frac{37}{20} i b c^6 d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{60} i b c^6 d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}+\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{i b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left (c^2 x^2+1\right )+\frac{37 i b^2 c^5 d^3}{30 x}+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x) \]
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Rubi [A] time = 0.516808, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {43, 4874, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391, 4854, 2402, 2315} \[ -\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{PolyLog}(2,i c x)+\frac{37}{40} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )+\frac{14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}-\frac{3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac{28}{15} i a b c^6 d^3 \log (x)-\frac{11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}+\frac{37}{20} i b c^6 d^3 \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{60} i b c^6 d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}+\frac{61 b^2 c^4 d^3}{180 x^2}-\frac{i b^2 c^3 d^3}{10 x^3}-\frac{b^2 c^2 d^3}{60 x^4}-\frac{113}{90} b^2 c^6 d^3 \log \left (c^2 x^2+1\right )+\frac{37 i b^2 c^5 d^3}{30 x}+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 43
Rule 4874
Rule 4852
Rule 266
Rule 44
Rule 325
Rule 203
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^7} \, dx &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}-(2 b c) \int \left (-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{11 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{12 x^4}+\frac{14 i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^3}-\frac{11 c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{12 x^2}-\frac{14 i c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x}+\frac{i c^6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{120 (-i+c x)}+\frac{37 i c^6 d^3 \left (a+b \tan ^{-1}(c x)\right )}{40 (i+c x)}\right ) \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{1}{3} \left (b c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^6} \, dx+\frac{1}{5} \left (6 i b c^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^5} \, dx-\frac{1}{6} \left (11 b c^3 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^4} \, dx-\frac{1}{15} \left (28 i b c^4 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+\frac{1}{6} \left (11 b c^5 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx+\frac{1}{15} \left (28 i b c^6 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx-\frac{1}{60} \left (i b c^7 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{-i+c x} \, dx-\frac{1}{20} \left (37 i b c^7 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i+c x} \, dx\\ &=-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac{3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac{11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac{14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} i a b c^6 d^3 \log (x)+\frac{37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )+\frac{1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )+\frac{1}{15} \left (b^2 c^2 d^3\right ) \int \frac{1}{x^5 \left (1+c^2 x^2\right )} \, dx+\frac{1}{10} \left (3 i b^2 c^3 d^3\right ) \int \frac{1}{x^4 \left (1+c^2 x^2\right )} \, dx-\frac{1}{18} \left (11 b^2 c^4 d^3\right ) \int \frac{1}{x^3 \left (1+c^2 x^2\right )} \, dx-\frac{1}{15} \left (14 i b^2 c^5 d^3\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac{1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac{\log (1-i c x)}{x} \, dx+\frac{1}{15} \left (14 b^2 c^6 d^3\right ) \int \frac{\log (1+i c x)}{x} \, dx+\frac{1}{6} \left (11 b^2 c^6 d^3\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx-\frac{1}{60} \left (i b^2 c^7 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\frac{1}{20} \left (37 i b^2 c^7 d^3\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{i b^2 c^3 d^3}{10 x^3}+\frac{14 i b^2 c^5 d^3}{15 x}-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac{3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac{11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac{14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} i a b c^6 d^3 \log (x)+\frac{37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )+\frac{1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(i c x)+\frac{1}{30} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{36} \left (11 b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{10} \left (3 i b^2 c^5 d^3\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac{1}{60} \left (b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )+\frac{1}{12} \left (11 b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\frac{1}{20} \left (37 b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )+\frac{1}{15} \left (14 i b^2 c^7 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx\\ &=-\frac{i b^2 c^3 d^3}{10 x^3}+\frac{37 i b^2 c^5 d^3}{30 x}+\frac{14}{15} i b^2 c^6 d^3 \tan ^{-1}(c x)-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac{3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac{11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac{14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} i a b c^6 d^3 \log (x)+\frac{37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )+\frac{1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(i c x)+\frac{37}{40} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+\frac{1}{30} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{c^2}{x^2}+\frac{c^4}{x}-\frac{c^6}{1+c^2 x}\right ) \, dx,x,x^2\right )-\frac{1}{36} \left (11 b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )+\frac{1}{12} \left (11 b^2 c^6 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{10} \left (3 i b^2 c^7 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx-\frac{1}{12} \left (11 b^2 c^8 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b^2 c^2 d^3}{60 x^4}-\frac{i b^2 c^3 d^3}{10 x^3}+\frac{61 b^2 c^4 d^3}{180 x^2}+\frac{37 i b^2 c^5 d^3}{30 x}+\frac{37}{30} i b^2 c^6 d^3 \tan ^{-1}(c x)-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^5}-\frac{3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{10 x^4}+\frac{11 b c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{18 x^3}+\frac{14 i b c^4 d^3 \left (a+b \tan ^{-1}(c x)\right )}{15 x^2}-\frac{11 b c^5 d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x}-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{6 x^6}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{5 x^5}+\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 x^4}+\frac{i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{3 x^3}+\frac{28}{15} i a b c^6 d^3 \log (x)+\frac{113}{45} b^2 c^6 d^3 \log (x)+\frac{37}{20} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1-i c x}\right )+\frac{1}{60} i b c^6 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-\frac{113}{90} b^2 c^6 d^3 \log \left (1+c^2 x^2\right )-\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(-i c x)+\frac{14}{15} b^2 c^6 d^3 \text{Li}_2(i c x)+\frac{37}{40} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1-i c x}\right )-\frac{1}{120} b^2 c^6 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )\\ \end{align*}
Mathematica [A] time = 1.46127, size = 401, normalized size = 0.78 \[ \frac{d^3 \left (168 b^2 c^6 x^6 \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+60 i a^2 c^3 x^3+135 a^2 c^2 x^2-108 i a^2 c x-30 a^2-330 a b c^5 x^5+168 i a b c^4 x^4+110 a b c^3 x^3-54 i a b c^2 x^2+336 i a b c^6 x^6 \log (c x)-168 i a b c^6 x^6 \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (-3 a \left (55 c^6 x^6-20 i c^3 x^3-45 c^2 x^2+36 i c x+10\right )+b c x \left (111 i c^5 x^5-165 c^4 x^4+84 i c^3 x^3+55 c^2 x^2-27 i c x-6\right )+168 i b c^6 x^6 \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )-12 a b c x+64 b^2 c^6 x^6+222 i b^2 c^5 x^5+61 b^2 c^4 x^4-18 i b^2 c^3 x^3-3 b^2 c^2 x^2+452 b^2 c^6 x^6 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+3 b^2 (c x-i)^4 \left (c^2 x^2+4 i c x-10\right ) \tan ^{-1}(c x)^2\right )}{180 x^6} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.115, size = 853, normalized size = 1.7 \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{240 \, x^{6}{\rm integral}\left (\frac{-60 i \, a^{2} c^{5} d^{3} x^{5} - 180 \, a^{2} c^{4} d^{3} x^{4} + 120 i \, a^{2} c^{3} d^{3} x^{3} - 120 \, a^{2} c^{2} d^{3} x^{2} + 180 i \, a^{2} c d^{3} x + 60 \, a^{2} d^{3} +{\left (60 \, a b c^{5} d^{3} x^{5} +{\left (-180 i \, a b + 20 \, b^{2}\right )} c^{4} d^{3} x^{4} - 15 \,{\left (8 \, a b + 3 i \, b^{2}\right )} c^{3} d^{3} x^{3} +{\left (-120 i \, a b - 36 \, b^{2}\right )} c^{2} d^{3} x^{2} - 10 \,{\left (18 \, a b - i \, b^{2}\right )} c d^{3} x + 60 i \, a b d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{60 \,{\left (c^{2} x^{9} + x^{7}\right )}}, x\right ) +{\left (-20 i \, b^{2} c^{3} d^{3} x^{3} - 45 \, b^{2} c^{2} d^{3} x^{2} + 36 i \, b^{2} c d^{3} x + 10 \, b^{2} d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2}}{240 \, x^{6}} \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 \frac{{\left (i \, c d x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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