Optimal. Leaf size=162 \[ \frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{f}+\frac {i b \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e-c f+i f) (1-i (c+d x))}\right )}{2 f}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5048, 4857, 2402, 2315, 2447} \[ \frac {i b \text {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{2 f}-\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(1-i (c+d x)) (-c f+d e+i f)}\right )}{f}-\frac {\log \left (\frac {2}{1-i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2315
Rule 2402
Rule 2447
Rule 4857
Rule 5048
Rubi steps
\begin {align*} \int \frac {a+b \cot ^{-1}(c+d x)}{e+f x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{\frac {d e-c f}{d}+\frac {f x}{d}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1-i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{f}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )}{\left (\frac {i f}{d}+\frac {d e-c f}{d}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,c+d x\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}+\frac {i b \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}-\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i (c+d x)}\right )}{f}\\ &=-\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-i (c+d x)}\right )}{f}+\frac {\left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{f}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i (c+d x)}\right )}{2 f}+\frac {i b \text {Li}_2\left (1-\frac {2 d (e+f x)}{(d e+i f-c f) (1-i (c+d x))}\right )}{2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 304, normalized size = 1.88 \[ \frac {a \log (f (c+d x)-c f+d e)}{f}-\frac {i b \text {Li}_2\left (\frac {d e-c f+f (c+d x)}{d e+(i-c) f}\right )}{2 f}+\frac {i b \text {Li}_2\left (\frac {d e-c f+f (c+d x)}{d e-(c+i) f}\right )}{2 f}-\frac {i b \log \left (\frac {f (-c-d x+i)}{d e+(-c+i) f}\right ) \log (f (c+d x)-c f+d e)}{2 f}+\frac {i b \log \left (-\frac {-c-d x+i}{c+d x}\right ) \log (f (c+d x)-c f+d e)}{2 f}+\frac {i b \log \left (-\frac {f (c+d x+i)}{d e-(c+i) f}\right ) \log (f (c+d x)-c f+d e)}{2 f}-\frac {i b \log \left (\frac {c+d x+i}{c+d x}\right ) \log (f (c+d x)-c f+d e)}{2 f} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arccot}\left (d x + c\right ) + a}{f x + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arccot}\left (d x + c\right ) + a}{f x + e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 224, normalized size = 1.38 \[ \frac {a \ln \left (f \left (d x +c \right )-c f +d e \right )}{f}+\frac {b \ln \left (f \left (d x +c \right )-c f +d e \right ) \mathrm {arccot}\left (d x +c \right )}{f}-\frac {i b \ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )}{2 f}+\frac {i b \ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )}{2 f}-\frac {i b \dilog \left (\frac {i f -f \left (d x +c \right )}{-c f +d e +i f}\right )}{2 f}+\frac {i b \dilog \left (\frac {i f +f \left (d x +c \right )}{c f -d e +i f}\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {\arctan \left (1, d x + c\right )}{2 \, {\left (f x + e\right )}}\,{d x} + \frac {a \log \left (f x + e\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acot}\left (c+d\,x\right )}{e+f\,x} \,d x \]
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]
________________________________________________________________________________________