Optimal. Leaf size=105 \[ -\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac {\sqrt {\pi } \text {Erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {Erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\pi } \text {Erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {Erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5670
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {\cosh ^{-1}(a x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {x}}+\frac {\sinh (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}+\frac {\operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}-\frac {\operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac {\operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {x}} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac {\operatorname {Subst}\left (\int e^{-3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}-\frac {\operatorname {Subst}\left (\int e^{-x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\operatorname {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}+\frac {\operatorname {Subst}\left (\int e^{3 x^2} \, dx,x,\sqrt {\cosh ^{-1}(a x)}\right )}{4 a^3}\\ &=-\frac {\sqrt {\pi } \text {erf}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}-\frac {\sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}+\frac {\sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {\cosh ^{-1}(a x)}\right )}{8 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 100, normalized size = 0.95 \[ \frac {\sqrt {3} \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 \cosh ^{-1}(a x)\right )+3 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {1}{2},-\cosh ^{-1}(a x)\right )+\sqrt {\cosh ^{-1}(a x)} \left (3 \Gamma \left (\frac {1}{2},\cosh ^{-1}(a x)\right )+\sqrt {3} \Gamma \left (\frac {1}{2},3 \cosh ^{-1}(a x)\right )\right )}{24 a^3 \sqrt {\cosh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\operatorname {arcosh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\mathrm {arccosh}\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\operatorname {arcosh}\left (a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sqrt {\mathrm {acosh}\left (a\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt {\operatorname {acosh}{\left (a x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________