Optimal. Leaf size=214 \[ \frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac {14}{15} i b c^6 d^3 \log (x)-\frac {1}{120} i b c^6 d^3 \log (-c x+i)-\frac {37}{40} i b c^6 d^3 \log (c x+i)-\frac {11 b c^5 d^3}{12 x}+\frac {7 i b c^4 d^3}{15 x^2}+\frac {11 b c^3 d^3}{36 x^3}-\frac {3 i b c^2 d^3}{20 x^4}-\frac {b c d^3}{30 x^5} \]
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Rubi [A] time = 0.18, antiderivative size = 214, normalized size of antiderivative = 1.00, 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, 1802} \[ \frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac {7 i b c^4 d^3}{15 x^2}+\frac {11 b c^3 d^3}{36 x^3}-\frac {3 i b c^2 d^3}{20 x^4}-\frac {11 b c^5 d^3}{12 x}+\frac {14}{15} i b c^6 d^3 \log (x)-\frac {1}{120} i b c^6 d^3 \log (-c x+i)-\frac {37}{40} i b c^6 d^3 \log (c x+i)-\frac {b c d^3}{30 x^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 1802
Rule 4872
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )}{x^7} \, dx &=-\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 {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d^3 \left (-10-36 i c x+45 c^2 x^2+20 i c^3 x^3\right )}{60 x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\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 {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{60} \left (b c d^3\right ) \int \frac {-10-36 i c x+45 c^2 x^2+20 i c^3 x^3}{x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\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 {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {1}{60} \left (b c d^3\right ) \int \left (-\frac {10}{x^6}-\frac {36 i c}{x^5}+\frac {55 c^2}{x^4}+\frac {56 i c^3}{x^3}-\frac {55 c^4}{x^2}-\frac {56 i c^5}{x}+\frac {i c^6}{2 (-i+c x)}+\frac {111 i c^6}{2 (i+c x)}\right ) \, dx\\ &=-\frac {b c d^3}{30 x^5}-\frac {3 i b c^2 d^3}{20 x^4}+\frac {11 b c^3 d^3}{36 x^3}+\frac {7 i b c^4 d^3}{15 x^2}-\frac {11 b c^5 d^3}{12 x}-\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 {3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {14}{15} i b c^6 d^3 \log (x)-\frac {1}{120} i b c^6 d^3 \log (i-c x)-\frac {37}{40} i b c^6 d^3 \log (i+c x)\\ \end {align*}
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Mathematica [C] time = 0.12, size = 188, normalized size = 0.88 \[ \frac {d^3 \left (-2 b c x \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-c^2 x^2\right )+i \left (20 a c^3 x^3-45 i a c^2 x^2-36 a c x+10 i a+56 b c^6 x^6 \log (x)+28 b c^4 x^4+20 b c^3 x^3 \tan ^{-1}(c x)-9 b c^2 x^2-45 i b c^2 x^2 \tan ^{-1}(c x)-28 b c^6 x^6 \log \left (c^2 x^2+1\right )-15 i b c^3 x^3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )-36 b c x \tan ^{-1}(c x)+10 i b \tan ^{-1}(c x)\right )\right )}{60 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 198, normalized size = 0.93 \[ \frac {336 i \, b c^{6} d^{3} x^{6} \log \relax (x) - 333 i \, b c^{6} d^{3} x^{6} \log \left (\frac {c x + i}{c}\right ) - 3 i \, b c^{6} d^{3} x^{6} \log \left (\frac {c x - i}{c}\right ) - 330 \, b c^{5} d^{3} x^{5} + 168 i \, b c^{4} d^{3} x^{4} + {\left (120 i \, a + 110 \, b\right )} c^{3} d^{3} x^{3} + 54 \, {\left (5 \, a - i \, b\right )} c^{2} d^{3} x^{2} + {\left (-216 i \, a - 12 \, b\right )} c d^{3} x - 60 \, a d^{3} - {\left (60 \, b c^{3} d^{3} x^{3} - 135 i \, b c^{2} d^{3} x^{2} - 108 \, b c d^{3} x + 30 i \, b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{360 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 215, normalized size = 1.00 \[ -\frac {7 i c^{6} d^{3} b \ln \left (c^{2} x^{2}+1\right )}{15}+\frac {3 c^{2} d^{3} a}{4 x^{4}}-\frac {3 i c \,d^{3} b \arctan \left (c x \right )}{5 x^{5}}-\frac {d^{3} a}{6 x^{6}}+\frac {i c^{3} d^{3} b \arctan \left (c x \right )}{3 x^{3}}+\frac {3 c^{2} d^{3} b \arctan \left (c x \right )}{4 x^{4}}+\frac {i c^{3} d^{3} a}{3 x^{3}}-\frac {d^{3} b \arctan \left (c x \right )}{6 x^{6}}-\frac {3 i b \,c^{2} d^{3}}{20 x^{4}}+\frac {14 i c^{6} d^{3} b \ln \left (c x \right )}{15}+\frac {7 i b \,c^{4} d^{3}}{15 x^{2}}-\frac {b c \,d^{3}}{30 x^{5}}+\frac {11 b \,c^{3} d^{3}}{36 x^{3}}-\frac {11 b \,c^{5} d^{3}}{12 x}-\frac {3 i c \,d^{3} a}{5 x^{5}}-\frac {11 b \,c^{6} d^{3} \arctan \left (c x \right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 248, normalized size = 1.16 \[ -\frac {1}{6} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{3} d^{3} - \frac {1}{4} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c^{2} d^{3} - \frac {3}{20} i \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac {2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac {4 \, \arctan \left (c x\right )}{x^{5}}\right )} b c d^{3} - \frac {1}{90} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{3} + \frac {i \, a c^{3} d^{3}}{3 \, x^{3}} + \frac {3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac {3 i \, a c d^{3}}{5 \, x^{5}} - \frac {a d^{3}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.06, size = 192, normalized size = 0.90 \[ -\frac {\frac {d^3\,\left (30\,a+30\,b\,\mathrm {atan}\left (c\,x\right )\right )}{180}+\frac {d^3\,x\,\left (a\,c\,108{}\mathrm {i}+6\,b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,108{}\mathrm {i}\right )}{180}-\frac {d^3\,x^3\,\left (a\,c^3\,60{}\mathrm {i}+55\,b\,c^3+b\,c^3\,\mathrm {atan}\left (c\,x\right )\,60{}\mathrm {i}\right )}{180}-\frac {d^3\,x^2\,\left (135\,a\,c^2+135\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,27{}\mathrm {i}\right )}{180}+\frac {11\,b\,c^5\,d^3\,x^5}{12}-\frac {b\,c^4\,d^3\,x^4\,7{}\mathrm {i}}{15}}{x^6}-\frac {d^3\,\left (\frac {165\,b\,c^9\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )}{{\left (c^2\right )}^{3/2}}+b\,c^6\,\ln \left (c^2\,x^2+1\right )\,84{}\mathrm {i}-b\,c^6\,\ln \relax (x)\,168{}\mathrm {i}\right )}{180} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 84.08, size = 347, normalized size = 1.62 \[ \frac {14 i b c^{6} d^{3} \log {\left (1385945 b^{2} c^{13} d^{6} x \right )}}{15} - \frac {i b c^{6} d^{3} \log {\left (1385945 b^{2} c^{13} d^{6} x - 1385945 i b^{2} c^{12} d^{6} \right )}}{120} - \frac {37 i b c^{6} d^{3} \log {\left (1385945 b^{2} c^{13} d^{6} x + 1385945 i b^{2} c^{12} d^{6} \right )}}{40} + \frac {\left (- 20 b c^{3} d^{3} x^{3} + 45 i b c^{2} d^{3} x^{2} + 36 b c d^{3} x - 10 i b d^{3}\right ) \log {\left (- i c x + 1 \right )}}{120 x^{6}} + \frac {\left (20 b c^{3} d^{3} x^{3} - 45 i b c^{2} d^{3} x^{2} - 36 b c d^{3} x + 10 i b d^{3}\right ) \log {\left (i c x + 1 \right )}}{120 x^{6}} - \frac {30 a d^{3} + 165 b c^{5} d^{3} x^{5} - 84 i b c^{4} d^{3} x^{4} + x^{3} \left (- 60 i a c^{3} d^{3} - 55 b c^{3} d^{3}\right ) + x^{2} \left (- 135 a c^{2} d^{3} + 27 i b c^{2} d^{3}\right ) + x \left (108 i a c d^{3} + 6 b c d^{3}\right )}{180 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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