Optimal. Leaf size=232 \[ -\frac {e \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^2}-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}+\frac {b c \log (x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {4876, 4852, 266, 36, 29, 31, 4848, 2391, 4856, 2402, 2315, 2447} \[ -\frac {i b e \text {PolyLog}(2,-i c x)}{2 d^2}+\frac {i b e \text {PolyLog}(2,i c x)}{2 d^2}+\frac {i b e \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 d^2}-\frac {e \log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d^2}-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}+\frac {b c \log (x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 31
Rule 36
Rule 266
Rule 2315
Rule 2391
Rule 2402
Rule 2447
Rule 4848
Rule 4852
Rule 4856
Rule 4876
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^2 (d+e x)} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 x}+\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{x} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{d^2}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}-\frac {(i b e) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^2}+\frac {(i b e) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^2}+\frac {(b c e) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d^2}-\frac {(b c e) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}+\frac {(i b e) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{d^2}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}+\frac {b c \log (x)}{d}-\frac {a e \log (x)}{d^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d^2}+\frac {e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\frac {i b e \text {Li}_2(-i c x)}{2 d^2}+\frac {i b e \text {Li}_2(i c x)}{2 d^2}+\frac {i b e \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d^2}-\frac {i b e \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 223, normalized size = 0.96 \[ -\frac {-2 a e x \log (d+e x)+2 a d+2 a e x \log (x)+b c d x \log \left (c^2 x^2+1\right )-i b e x \text {Li}_2\left (\frac {e (1-i c x)}{i c d+e}\right )+i b e x \text {Li}_2\left (-\frac {e (c x-i)}{c d+i e}\right )-i b e x \log (1-i c x) \log \left (\frac {c (d+e x)}{c d-i e}\right )+i b e x \log (1+i c x) \log \left (\frac {c (d+e x)}{c d+i e}\right )-2 b c d x \log (x)+2 b d \tan ^{-1}(c x)+i b e x \text {Li}_2(-i c x)-i b e x \text {Li}_2(i c x)}{2 d^2 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e x^{3} + d x^{2}}, x\right ) \]
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 [A] time = 0.06, size = 321, normalized size = 1.38 \[ -\frac {a}{d x}-\frac {a e \ln \left (c x \right )}{d^{2}}+\frac {a e \ln \left (c e x +d c \right )}{d^{2}}-\frac {b \arctan \left (c x \right )}{d x}-\frac {b \arctan \left (c x \right ) e \ln \left (c x \right )}{d^{2}}+\frac {b \arctan \left (c x \right ) e \ln \left (c e x +d c \right )}{d^{2}}+\frac {i b e \dilog \left (\frac {-c e x +i e}{d c +i e}\right )}{2 d^{2}}+\frac {i b e \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 d^{2}}-\frac {i b e \dilog \left (i c x +1\right )}{2 d^{2}}-\frac {i b e \dilog \left (\frac {c e x +i e}{-d c +i e}\right )}{2 d^{2}}+\frac {c b \ln \left (c x \right )}{d}-\frac {b c \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i b e \dilog \left (-i c x +1\right )}{2 d^{2}}-\frac {i b e \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 d^{2}}-\frac {i b e \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +i e}{-d c +i e}\right )}{2 d^{2}}+\frac {i b e \ln \left (c e x +d c \right ) \ln \left (\frac {-c e x +i e}{d c +i e}\right )}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ a {\left (\frac {e \log \left (e x + d\right )}{d^{2}} - \frac {e \log \relax (x)}{d^{2}} - \frac {1}{d x}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e x^{3} + d x^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________