Optimal. Leaf size=186 \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.168029, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {5682, 5675, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{a^2 c x^2+c} \text{Erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{a^2 x^2+1}}+\frac{\sqrt{a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{a^2 x^2+1}}+\frac{1}{2} x \sqrt{a^2 c x^2+c} \sqrt{\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5682
Rule 5675
Rule 5669
Rule 5448
Rule 12
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)} \, dx &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \int \frac{\sqrt{\sinh ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{1+a^2 x^2}}-\frac{\left (a \sqrt{c+a^2 c x^2}\right ) \int \frac{x}{\sqrt{\sinh ^{-1}(a x)}} \, dx}{4 \sqrt{1+a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{1+a^2 x^2}}-\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a \sqrt{1+a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{1+a^2 x^2}}-\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{4 a \sqrt{1+a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{1+a^2 x^2}}-\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt{1+a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{1+a^2 x^2}}+\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}-\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt{1+a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{1+a^2 x^2}}+\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a \sqrt{1+a^2 x^2}}-\frac{\sqrt{c+a^2 c x^2} \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\sinh ^{-1}(a x)}\right )}{8 a \sqrt{1+a^2 x^2}}\\ &=\frac{1}{2} x \sqrt{c+a^2 c x^2} \sqrt{\sinh ^{-1}(a x)}+\frac{\sqrt{c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{3 a \sqrt{1+a^2 x^2}}+\frac{\sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{1+a^2 x^2}}-\frac{\sqrt{\frac{\pi }{2}} \sqrt{c+a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\sinh ^{-1}(a x)}\right )}{16 a \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0999308, size = 104, normalized size = 0.56 \[ \frac{\sqrt{c \left (a^2 x^2+1\right )} \left (-3 \sqrt{2} \sqrt{\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},2 \sinh ^{-1}(a x)\right )-3 \sqrt{2} \sqrt{-\sinh ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-2 \sinh ^{-1}(a x)\right )+16 \sinh ^{-1}(a x)^2\right )}{48 a \sqrt{a^2 x^2+1} \sqrt{\sinh ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{a}^{2}c{x}^{2}+c}\sqrt{{\it Arcsinh} \left ( ax \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \sqrt{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c \left (a^{2} x^{2} + 1\right )} \sqrt{\operatorname{asinh}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a^{2} c x^{2} + c} \sqrt{\operatorname{arsinh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]