Optimal. Leaf size=138 \[ \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e}-\frac {i b \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}+\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4856, 2402, 2315, 2447} \[ -\frac {i b \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 e}+\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 e}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{e}-\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2402
Rule 2447
Rule 4856
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx &=-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {\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 )}{e}+\frac {(b c) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{e}-\frac {(b c) \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}{e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {\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 )}{e}-\frac {i b \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{e}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{e}+\frac {\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 )}{e}+\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 e}-\frac {i b \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 138, normalized size = 1.00 \[ \frac {2 a \log (d+e x)+i b \text {Li}_2\left (\frac {e (1-i c x)}{i c d+e}\right )-i b \text {Li}_2\left (-\frac {e (c x-i)}{c d+i e}\right )+i b \log (1-i c x) \log \left (\frac {c (d+e x)}{c d-i e}\right )-i b \log (1+i c x) \log \left (\frac {c (d+e x)}{c d+i e}\right )}{2 e} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e x + d}, x\right ) \]
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 [A] time = 0.06, size = 168, normalized size = 1.22 \[ \frac {a \ln \left (c e x +d c \right )}{e}+\frac {b \ln \left (c e x +d c \right ) \arctan \left (c x \right )}{e}+\frac {i b \ln \left (c e x +d c \right ) \ln \left (\frac {-c e x +i e}{d c +i e}\right )}{2 e}-\frac {i b \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +i e}{-d c +i e}\right )}{2 e}+\frac {i b \dilog \left (\frac {-c e x +i e}{d c +i e}\right )}{2 e}-\frac {i b \dilog \left (\frac {c e x +i e}{-d c +i e}\right )}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e x + d\right )}}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________