Optimal. Leaf size=161 \[ \frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {2}{3} i b c^4 d^2 \log (x)-\frac {1}{24} i b c^4 d^2 \log (-c x+i)+\frac {17}{24} i b c^4 d^2 \log (c x+i)+\frac {3 b c^3 d^2}{4 x}-\frac {i b c^2 d^2}{3 x^2}-\frac {b c d^2}{12 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 161, 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} \[ \frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {i b c^2 d^2}{3 x^2}+\frac {3 b c^3 d^2}{4 x}-\frac {2}{3} i b c^4 d^2 \log (x)-\frac {1}{24} i b c^4 d^2 \log (-c x+i)+\frac {17}{24} i b c^4 d^2 \log (c x+i)-\frac {b c d^2}{12 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 1802
Rule 4872
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )}{x^5} \, dx &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-(b c) \int \frac {d^2 \left (-3-8 i c x+6 c^2 x^2\right )}{12 x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {1}{12} \left (b c d^2\right ) \int \frac {-3-8 i c x+6 c^2 x^2}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {1}{12} \left (b c d^2\right ) \int \left (-\frac {3}{x^4}-\frac {8 i c}{x^3}+\frac {9 c^2}{x^2}+\frac {8 i c^3}{x}+\frac {i c^4}{2 (-i+c x)}-\frac {17 i c^4}{2 (i+c x)}\right ) \, dx\\ &=-\frac {b c d^2}{12 x^3}-\frac {i b c^2 d^2}{3 x^2}+\frac {3 b c^3 d^2}{4 x}-\frac {d^2 \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {2 i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac {c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac {2}{3} i b c^4 d^2 \log (x)-\frac {1}{24} i b c^4 d^2 \log (i-c x)+\frac {17}{24} i b c^4 d^2 \log (i+c x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.07, size = 152, normalized size = 0.94 \[ \frac {d^2 \left (6 a c^2 x^2-8 i a c x-3 a-8 i b c^4 x^4 \log (x)-b c x \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )-4 i b c^2 x^2+6 b c^2 x^2 \tan ^{-1}(c x)+4 i b c^4 x^4 \log \left (c^2 x^2+1\right )+6 b c^3 x^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )-8 i b c x \tan ^{-1}(c x)-3 b \tan ^{-1}(c x)\right )}{12 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 156, normalized size = 0.97 \[ \frac {-16 i \, b c^{4} d^{2} x^{4} \log \relax (x) + 17 i \, b c^{4} d^{2} x^{4} \log \left (\frac {c x + i}{c}\right ) - i \, b c^{4} d^{2} x^{4} \log \left (\frac {c x - i}{c}\right ) + 18 \, b c^{3} d^{2} x^{3} + 4 \, {\left (3 \, a - 2 i \, b\right )} c^{2} d^{2} x^{2} + {\left (-16 i \, a - 2 \, b\right )} c d^{2} x - 6 \, a d^{2} + {\left (6 i \, b c^{2} d^{2} x^{2} + 8 \, b c d^{2} x - 3 i \, b d^{2}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{24 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 160, normalized size = 0.99 \[ -\frac {2 i c \,d^{2} a}{3 x^{3}}-\frac {d^{2} a}{4 x^{4}}+\frac {c^{2} d^{2} a}{2 x^{2}}-\frac {2 i c \,d^{2} b \arctan \left (c x \right )}{3 x^{3}}-\frac {d^{2} b \arctan \left (c x \right )}{4 x^{4}}+\frac {c^{2} d^{2} b \arctan \left (c x \right )}{2 x^{2}}-\frac {i b \,c^{2} d^{2}}{3 x^{2}}-\frac {2 i c^{4} d^{2} b \ln \left (c x \right )}{3}-\frac {b c \,d^{2}}{12 x^{3}}+\frac {3 b \,c^{3} d^{2}}{4 x}+\frac {i c^{4} d^{2} b \ln \left (c^{2} x^{2}+1\right )}{3}+\frac {3 b \,c^{4} d^{2} \arctan \left (c x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 152, normalized size = 0.94 \[ \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b c^{2} d^{2} + \frac {1}{3} i \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac {1}{x^{2}}\right )} c - \frac {2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c d^{2} + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d^{2} + \frac {a c^{2} d^{2}}{2 \, x^{2}} - \frac {2 i \, a c d^{2}}{3 \, x^{3}} - \frac {a d^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.70, size = 142, normalized size = 0.88 \[ \frac {d^2\,\left (9\,b\,c^3\,\mathrm {atan}\left (x\,\sqrt {c^2}\right )\,\sqrt {c^2}+b\,c^4\,\ln \left (c^2\,x^2+1\right )\,4{}\mathrm {i}-b\,c^4\,\ln \relax (x)\,8{}\mathrm {i}\right )}{12}-\frac {\frac {d^2\,\left (3\,a+3\,b\,\mathrm {atan}\left (c\,x\right )\right )}{12}+\frac {d^2\,x\,\left (a\,c\,8{}\mathrm {i}+b\,c+b\,c\,\mathrm {atan}\left (c\,x\right )\,8{}\mathrm {i}\right )}{12}-\frac {d^2\,x^2\,\left (6\,a\,c^2+6\,b\,c^2\,\mathrm {atan}\left (c\,x\right )-b\,c^2\,4{}\mathrm {i}\right )}{12}-\frac {3\,b\,c^3\,d^2\,x^3}{4}}{x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 16.39, size = 275, normalized size = 1.71 \[ - \frac {2 i b c^{4} d^{2} \log {\left (1485 b^{2} c^{9} d^{4} x \right )}}{3} - \frac {i b c^{4} d^{2} \log {\left (1485 b^{2} c^{9} d^{4} x - 1485 i b^{2} c^{8} d^{4} \right )}}{24} + \frac {17 i b c^{4} d^{2} \log {\left (1485 b^{2} c^{9} d^{4} x + 1485 i b^{2} c^{8} d^{4} \right )}}{24} + \frac {\left (- 6 i b c^{2} d^{2} x^{2} - 8 b c d^{2} x + 3 i b d^{2}\right ) \log {\left (i c x + 1 \right )}}{24 x^{4}} + \frac {\left (6 i b c^{2} d^{2} x^{2} + 8 b c d^{2} x - 3 i b d^{2}\right ) \log {\left (- i c x + 1 \right )}}{24 x^{4}} - \frac {3 a d^{2} - 9 b c^{3} d^{2} x^{3} + x^{2} \left (- 6 a c^{2} d^{2} + 4 i b c^{2} d^{2}\right ) + x \left (8 i a c d^{2} + b c d^{2}\right )}{12 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________