Optimal. Leaf size=131 \[ \text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {a^2+1}}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{2} \sinh ^{-1}(a+b x)^2 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.24, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5865, 5799, 5561, 2190, 2279, 2391} \[ \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {a^2+1}}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}+a}\right )-\frac {1}{2} \sinh ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2279
Rule 2391
Rule 5561
Rule 5799
Rule 5865
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a+b x)}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sinh ^{-1}(x)}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x \cosh (x)}{-\frac {a}{b}+\frac {\sinh (x)}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1}{2} \sinh ^{-1}(a+b x)^2+\frac {\operatorname {Subst}\left (\int \frac {e^x x}{-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}+\frac {e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}+\frac {\operatorname {Subst}\left (\int \frac {e^x x}{-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}+\frac {e^x}{b}} \, dx,x,\sinh ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1}{2} \sinh ^{-1}(a+b x)^2+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )-\operatorname {Subst}\left (\int \log \left (1+\frac {e^x}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )-\operatorname {Subst}\left (\int \log \left (1+\frac {e^x}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=-\frac {1}{2} \sinh ^{-1}(a+b x)^2+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )-\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{\left (-\frac {a}{b}-\frac {\sqrt {1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )-\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{\left (-\frac {a}{b}+\frac {\sqrt {1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{\sinh ^{-1}(a+b x)}\right )\\ &=-\frac {1}{2} \sinh ^{-1}(a+b x)^2+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\sinh ^{-1}(a+b x) \log \left (1-\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )+\text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a-\sqrt {1+a^2}}\right )+\text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {1+a^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 153, normalized size = 1.17 \[ \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(a+b x)}}{\sqrt {a^2+1}-a}\right )+\text {Li}_2\left (\frac {e^{\sinh ^{-1}(a+b x)}}{a+\sqrt {a^2+1}}\right )+\sinh ^{-1}(a+b x) \log \left (\frac {e^{\sinh ^{-1}(a+b x)}}{b \left (-\frac {\sqrt {a^2+1}}{b}-\frac {a}{b}\right )}+1\right )+\sinh ^{-1}(a+b x) \log \left (\frac {e^{\sinh ^{-1}(a+b x)}}{b \left (\frac {\sqrt {a^2+1}}{b}-\frac {a}{b}\right )}+1\right )-\frac {1}{2} \sinh ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (b x + a\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.17, size = 388, normalized size = 2.96 \[ -\frac {\arcsinh \left (b x +a \right )^{2}}{2}+\frac {\left (a^{2}+1+\sqrt {a^{2}+1}\, a \right ) \arcsinh \left (b x +a \right ) \left (2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) a^{2}+\ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )-2 \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right ) \sqrt {a^{2}+1}\, a +\ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )\right )}{a^{2}+1}+\dilog \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )+\dilog \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )+\frac {a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}-b x -\sqrt {1+\left (b x +a \right )^{2}}}{a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}}-\frac {a \arcsinh \left (b x +a \right ) \ln \left (\frac {\sqrt {a^{2}+1}+b x +\sqrt {1+\left (b x +a \right )^{2}}}{-a +\sqrt {a^{2}+1}}\right )}{\sqrt {a^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsinh}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asinh}\left (a+b\,x\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________