Optimal. Leaf size=263 \[ -\frac {3 i b^2 c \text {Li}_2\left (\frac {2}{1-i c x}-1\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac {3 i b^2 c \text {Li}_3\left (\frac {2}{i c x+1}-1\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d}+\frac {3 b c \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac {3 b c \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d}-\frac {i c \log \left (2-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{d}+\frac {3 b^3 c \text {Li}_3\left (\frac {2}{1-i c x}-1\right )}{2 d}-\frac {3 b^3 c \text {Li}_4\left (\frac {2}{i c x+1}-1\right )}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.60, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4870, 4852, 4924, 4868, 4884, 4992, 6610, 4994, 4998} \[ -\frac {3 i b^2 c \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac {3 i b^2 c \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 d}+\frac {3 b c \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 d}+\frac {3 b^3 c \text {PolyLog}\left (3,-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 b^3 c \text {PolyLog}\left (4,-1+\frac {2}{1+i c x}\right )}{4 d}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac {3 b c \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{d}-\frac {i c \log \left (2-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4852
Rule 4868
Rule 4870
Rule 4884
Rule 4924
Rule 4992
Rule 4994
Rule 4998
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{x^2 (d+i c d x)} \, dx &=-\left ((i c) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{x (d+i c d x)} \, dx\right )+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{x^2} \, dx}{d}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {(3 b c) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (3 i b c^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {(3 i b c) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x (i+c x)} \, dx}{d}-\frac {\left (3 b^2 c^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {\left (6 b^2 c^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d}+\frac {\left (3 i b^3 c^2\right ) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d}\\ &=-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )}{d}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 b^3 c \text {Li}_4\left (-1+\frac {2}{1+i c x}\right )}{4 d}+\frac {\left (3 i b^3 c^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d x}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1-i c x}\right )}{d}-\frac {i c \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (2-\frac {2}{1+i c x}\right )}{d}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )}{d}+\frac {3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {3 b^3 c \text {Li}_3\left (-1+\frac {2}{1-i c x}\right )}{2 d}-\frac {3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {3 b^3 c \text {Li}_4\left (-1+\frac {2}{1+i c x}\right )}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.49, size = 436, normalized size = 1.66 \[ -\frac {-i a^3 c \log \left (c^2 x^2+1\right )+2 i a^3 c \log (x)+2 a^3 c \tan ^{-1}(c x)+\frac {2 a^3}{x}+3 a^2 b c \left (2 \left (-\log \left (\frac {c x}{\sqrt {c^2 x^2+1}}\right )+\tan ^{-1}(c x)^2+\tan ^{-1}(c x) \left (\frac {1}{c x}+i \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )\right )+\text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )\right )+6 i a b^2 c \left (i \tan ^{-1}(c x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(c x)}\right )+\text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{-2 i \tan ^{-1}(c x)}\right )-\frac {i \tan ^{-1}(c x)^2}{c x}+\tan ^{-1}(c x)^2+\tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )+2 i \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )-\frac {i \pi ^3}{24}\right )+2 i b^3 c \left (\frac {3}{2} i \left (\tan ^{-1}(c x)+2 i\right ) \tan ^{-1}(c x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(c x)}\right )+\frac {3}{2} \left (\tan ^{-1}(c x)+i\right ) \text {Li}_3\left (e^{-2 i \tan ^{-1}(c x)}\right )-\frac {3}{4} i \text {Li}_4\left (e^{-2 i \tan ^{-1}(c x)}\right )-\frac {i \tan ^{-1}(c x)^3}{c x}-\tan ^{-1}(c x)^3+\tan ^{-1}(c x)^3 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )+3 i \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-\frac {i \pi ^4}{64}+\frac {\pi ^3}{8}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 6 i \, a b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 12 \, a^{2} b \log \left (-\frac {c x + i}{c x - i}\right ) + 8 i \, a^{3}}{8 \, c d x^{3} - 8 i \, d x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.32, size = 11233, normalized size = 42.71 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{x^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \left (\int \frac {a^{3}}{c x^{3} - i x^{2}}\, dx + \int \frac {b^{3} \operatorname {atan}^{3}{\left (c x \right )}}{c x^{3} - i x^{2}}\, dx + \int \frac {3 a b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{c x^{3} - i x^{2}}\, dx + \int \frac {3 a^{2} b \operatorname {atan}{\left (c x \right )}}{c x^{3} - i x^{2}}\, dx\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________