Optimal. Leaf size=81 \[ \frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\log \left (1-e^{-2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e}-\frac {b \text {Li}_2\left (e^{-2 \sinh ^{-1}(c+d x)}\right )}{2 d e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5865, 12, 5659, 3716, 2190, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right ) \left (a+b \sinh ^{-1}(c+d x)\right )}{d e} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 5865
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{c e+d e x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e x} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}\\ &=-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 b d e}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )}{d e}+\frac {b \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )}{2 d e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 70, normalized size = 0.86 \[ \frac {b^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c+d x)}\right )-\left (a+b \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)-2 b \log \left (1-e^{2 \sinh ^{-1}(c+d x)}\right )\right )}{2 b d e} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{d e x + c e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{d e x + c e}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 159, normalized size = 1.96 \[ \frac {a \ln \left (d x +c \right )}{d e}-\frac {b \arcsinh \left (d x +c \right )^{2}}{2 d e}+\frac {b \arcsinh \left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{d e}+\frac {b \polylog \left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{d e}+\frac {b \arcsinh \left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{d e}+\frac {b \polylog \left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} + 1}\right )}{d e x + c e}\,{d x} + \frac {a \log \left (d e x + c e\right )}{d e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{c\,e+d\,e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{c + d x}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________