Optimal. Leaf size=196 \[ \frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac {i a x}{c^3 d}+\frac {i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 c^4 d}+\frac {b \tan ^{-1}(c x)}{2 c^4 d}-\frac {b x}{2 c^3 d}+\frac {i b x \tan ^{-1}(c x)}{c^3 d}+\frac {i b x^2}{6 c^2 d}-\frac {2 i b \log \left (c^2 x^2+1\right )}{3 c^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4866, 4852, 266, 43, 321, 203, 4846, 260, 4854, 2402, 2315} \[ \frac {i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^4 d}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d}-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac {i a x}{c^3 d}+\frac {i b x^2}{6 c^2 d}-\frac {2 i b \log \left (c^2 x^2+1\right )}{3 c^4 d}-\frac {b x}{2 c^3 d}+\frac {i b x \tan ^{-1}(c x)}{c^3 d}+\frac {b \tan ^{-1}(c x)}{2 c^4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 203
Rule 260
Rule 266
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4866
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx &=\frac {i \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx}{c}-\frac {i \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c d}\\ &=-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}-\frac {\int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx}{c^2}+\frac {(i b) \int \frac {x^3}{1+c^2 x^2} \, dx}{3 d}+\frac {\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2 d}\\ &=\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}-\frac {i \int \frac {a+b \tan ^{-1}(c x)}{d+i c d x} \, dx}{c^3}+\frac {(i b) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{6 d}+\frac {i \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^3 d}-\frac {b \int \frac {x^2}{1+c^2 x^2} \, dx}{2 c d}\\ &=\frac {i a x}{c^3 d}-\frac {b x}{2 c^3 d}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d}+\frac {(i b) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 d}+\frac {(i b) \int \tan ^{-1}(c x) \, dx}{c^3 d}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d}-\frac {b \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d}\\ &=\frac {i a x}{c^3 d}-\frac {b x}{2 c^3 d}+\frac {i b x^2}{6 c^2 d}+\frac {b \tan ^{-1}(c x)}{2 c^4 d}+\frac {i b x \tan ^{-1}(c x)}{c^3 d}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d}-\frac {i b \log \left (1+c^2 x^2\right )}{6 c^4 d}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^4 d}-\frac {(i b) \int \frac {x}{1+c^2 x^2} \, dx}{c^2 d}\\ &=\frac {i a x}{c^3 d}-\frac {b x}{2 c^3 d}+\frac {i b x^2}{6 c^2 d}+\frac {b \tan ^{-1}(c x)}{2 c^4 d}+\frac {i b x \tan ^{-1}(c x)}{c^3 d}+\frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac {i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d}-\frac {2 i b \log \left (1+c^2 x^2\right )}{3 c^4 d}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.48, size = 166, normalized size = 0.85 \[ -\frac {i \left (\tan ^{-1}(c x) \left (6 a+b \left (2 c^3 x^3+3 i c^2 x^2-6 c x+3 i\right )+6 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+2 a c^3 x^3+3 i a c^2 x^2-3 i a \log \left (c^2 x^2+1\right )-6 a c x-b c^2 x^2+4 b \log \left (c^2 x^2+1\right )+3 b \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-3 i b c x+6 b \tan ^{-1}(c x)^2-b\right )}{6 c^4 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \log \left (-\frac {c x + i}{c x - i}\right ) - 2 i \, a x^{3}}{2 \, c d x - 2 i \, d}, x\right ) \]
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 [B] time = 0.10, size = 353, normalized size = 1.80 \[ \frac {i b x \arctan \left (c x \right )}{c^{3} d}-\frac {i b \arctan \left (c x \right ) x^{3}}{3 c d}+\frac {a \,x^{2}}{2 c^{2} d}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 c^{4} d}-\frac {5 i b \ln \left (c^{4} x^{4}+10 c^{2} x^{2}+9\right )}{48 c^{4} d}+\frac {i b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{4} d}+\frac {i b \,x^{2}}{6 c^{2} d}+\frac {b \arctan \left (c x \right ) x^{2}}{2 c^{2} d}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{c^{4} d}+\frac {i a x}{c^{3} d}-\frac {i a \arctan \left (c x \right )}{c^{4} d}-\frac {11 i b \ln \left (c^{2} x^{2}+1\right )}{24 c^{4} d}-\frac {b x}{2 c^{3} d}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 c^{4} d}+\frac {2 i b}{3 c^{4} d}-\frac {i a \,x^{3}}{3 c d}-\frac {5 b \arctan \left (\frac {1}{6} c^{3} x^{3}+\frac {7}{6} c x \right )}{24 c^{4} d}+\frac {5 b \arctan \left (\frac {c x}{2}\right )}{24 c^{4} d}-\frac {5 b \arctan \left (\frac {c x}{2}-\frac {i}{2}\right )}{12 c^{4} d}-\frac {i b \ln \left (c x -i\right )^{2}}{4 c^{4} d}+\frac {11 b \arctan \left (c x \right )}{12 d \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{6} \, a {\left (\frac {i \, {\left (2 \, c^{2} x^{3} + 3 i \, c x^{2} - 6 \, x\right )}}{c^{3} d} + \frac {6 \, \log \left (i \, c x + 1\right )}{c^{4} d}\right )} - \frac {-\frac {1}{2} \, {\left (-12 i \, {\left (2 \, {\left (\frac {c^{2} x^{3} - 3 \, x}{c^{7} d} + \frac {3 \, \arctan \left (c x\right )}{c^{8} d}\right )} \arctan \left (c x\right ) - \frac {c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8} d}\right )} c^{8} d - 36 \, c^{8} d \int \frac {x^{4} \log \left (c^{2} x^{2} + 1\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 9 i \, {\left (2 \, {\left (\frac {x^{2}}{c^{5} d} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{7} d}\right )} \log \left (c^{2} x^{2} + 1\right ) - \frac {2 \, c^{2} x^{2} - \log \left (c^{2} x^{2} + 1\right )^{2} - 2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{7} d}\right )} c^{7} d + 72 \, c^{7} d \int \frac {x^{3} \arctan \left (c x\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 72 \, c^{5} d \int \frac {x \arctan \left (c x\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 8 \, c^{3} x^{3} + 36 \, c^{4} d \int \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 6 i \, c^{2} x^{2} + 60 \, c x - 2 \, {\left (12 i \, c^{3} x^{3} - 18 \, c^{2} x^{2} - 36 i \, c x + 30\right )} \arctan \left (c x\right ) - 36 i \, \arctan \left (c x\right )^{2} + 2 \, {\left (6 \, c^{3} x^{3} + 9 i \, c^{2} x^{2} - 18 \, c x + 3 i\right )} \log \left (c^{2} x^{2} + 1\right ) - 9 i \, \log \left (c^{2} x^{2} + 1\right )^{2} - 36 i \, \log \left (12 \, c^{5} d x^{2} + 12 \, c^{3} d\right )\right )} b}{72 \, c^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \left (\int \frac {6 i b \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {12 a c^{4} x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac {6 b c x}{c^{2} x^{2} + 1}\, dx + \int \frac {b c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {12 i a c^{3} x^{3}}{c^{2} x^{2} + 1}\, dx + \int \frac {3 i b c^{2} x^{2}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {2 i b c^{4} x^{4}}{c^{2} x^{2} + 1}\right )\, dx + \int \left (- \frac {6 b c x \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx + \int \frac {6 b c^{3} x^{3} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\, dx + \int \left (- \frac {6 i b c^{4} x^{4} \log {\left (i c x + 1 \right )}}{c^{2} x^{2} + 1}\right )\, dx\right )}{12 c^{3} d} + \frac {\left (2 b c^{3} x^{3} + 3 i b c^{2} x^{2} - 6 b c x - 6 i b \log {\left (i c x + 1 \right )}\right ) \log {\left (- i c x + 1 \right )}}{12 c^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________