Optimal. Leaf size=183 \[ -\frac {2 i \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {a x-1} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {a x-1} \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {a x-1} \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 \sqrt {a x-1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.42, antiderivative size = 248, normalized size of antiderivative = 1.36, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5798, 5761, 4180, 2531, 2282, 6589} \[ -\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {2 i \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 4180
Rule 5761
Rule 5798
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (2 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (2 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (2 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (2 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (2 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {\left (2 i \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (-i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {2 i \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_3\left (i e^{\cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.19, size = 151, normalized size = 0.83 \[ \frac {i \sqrt {\frac {a x-1}{a x+1}} (a x+1) \left (-2 \cosh ^{-1}(a x) \left (\text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-\text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )\right )-2 \text {Li}_3\left (-i e^{-\cosh ^{-1}(a x)}\right )+2 \text {Li}_3\left (i e^{-\cosh ^{-1}(a x)}\right )-\left (\cosh ^{-1}(a x)^2 \left (\log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )\right )\right )}{\sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2} x^{3} - x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )^{2}}{x \sqrt {-a^{2} x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________