Optimal. Leaf size=61 \[ -\frac {\text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 d}+\frac {\text {sech}^{-1}(a+b x)^2}{2 d}-\frac {\text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6319, 12, 6281, 5660, 3718, 2190, 2279, 2391} \[ -\frac {\text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )}{2 d}+\frac {\text {sech}^{-1}(a+b x)^2}{2 d}-\frac {\text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 6281
Rule 6319
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b \text {sech}^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\text {sech}^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\cosh ^{-1}(x)}{x} \, dx,x,\frac {1}{a+b x}\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (\frac {1}{a+b x}\right )\right )}{d}\\ &=\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (\frac {1}{a+b x}\right )\right )}{d}\\ &=\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac {1}{a+b x}\right )}\right )}{d}+\frac {\operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac {1}{a+b x}\right )\right )}{d}\\ &=\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac {1}{a+b x}\right )}\right )}{d}+\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (\frac {1}{a+b x}\right )}\right )}{2 d}\\ &=\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right )^2}{2 d}-\frac {\cosh ^{-1}\left (\frac {1}{a+b x}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac {1}{a+b x}\right )}\right )}{d}-\frac {\text {Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac {1}{a+b x}\right )}\right )}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 52, normalized size = 0.85 \[ \frac {\text {Li}_2\left (-e^{-2 \text {sech}^{-1}(a+b x)}\right )-\text {sech}^{-1}(a+b x) \left (\text {sech}^{-1}(a+b x)+2 \log \left (e^{-2 \text {sech}^{-1}(a+b x)}+1\right )\right )}{2 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsech}\left (b x + a\right )}{b d x + a d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )}{d x + \frac {a d}{b}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 104, normalized size = 1.70 \[ \frac {\mathrm {arcsech}\left (b x +a \right )^{2}}{2 d}-\frac {\mathrm {arcsech}\left (b x +a \right ) \ln \left (1+\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{d}-\frac {\polylog \left (2, -\left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )^{2}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, \log \left (\sqrt {b x + a + 1} \sqrt {-b x - a + 1} b x + \sqrt {b x + a + 1} \sqrt {-b x - a + 1} a + b x + a\right ) \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )^{2}}{2 \, d} - \frac {\log \left (b x + a + 1\right ) \log \left (b x + a\right ) + {\rm Li}_2\left (-b x - a\right )}{2 \, d} - \frac {\log \left (b x + a\right ) \log \left (-b x - a + 1\right ) + {\rm Li}_2\left (b x + a\right )}{2 \, d} + \int \frac {{\left (b^{2} x + a b\right )} \log \left (b x + a\right )}{b^{2} d x^{2} + 2 \, a b d x + a^{2} d + {\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d - d\right )} \sqrt {b x + a + 1} \sqrt {-b x - a + 1} - d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{d\,x+\frac {a\,d}{b}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {b \int \frac {\operatorname {asech}{\left (a + b x \right )}}{a + b x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________