Optimal. Leaf size=411 \[ \frac{\sqrt{\pi } c^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\sqrt{\pi } c^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} c e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{3}} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{2}} c e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{3}} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.856515, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5865, 5805, 6741, 6742, 5299, 2205, 2204, 5298, 5618} \[ \frac{\sqrt{\pi } c^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\sqrt{\pi } c^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{2}} c e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{3}} e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}-\frac{\sqrt{\frac{\pi }{2}} c e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{\sqrt{\pi } e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}+\frac{\sqrt{\frac{\pi }{3}} e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5865
Rule 5805
Rule 6741
Rule 6742
Rule 5299
Rule 2205
Rule 2204
Rule 5298
Rule 5618
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-\frac{c}{d}+\frac{x}{d}\right )^2}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \left (-\frac{c}{d}+\frac{\sinh (x)}{d}\right )^2}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \cosh \left (\frac{a-x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \left (c+\sinh \left (\frac{a-x^2}{b}\right )\right )^2 \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (c^2 \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right )+c \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right )+\cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \sinh ^2\left (\frac{a}{b}-\frac{x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \sinh ^2\left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac{(2 c) \operatorname{Subst}\left (\int \sinh \left (\frac{2 a}{b}-\frac{2 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (\frac{1}{4} \cosh \left (\frac{3 a}{b}-\frac{3 x^2}{b}\right )-\frac{1}{4} \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right )\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac{c \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}-\frac{c \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}+\frac{c^2 \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{b d^3}\\ &=\frac{c^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{c e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{c^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{c e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\operatorname{Subst}\left (\int \cosh \left (\frac{3 a}{b}-\frac{3 x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 b d^3}-\frac{\operatorname{Subst}\left (\int \cosh \left (\frac{a}{b}-\frac{x^2}{b}\right ) \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{2 b d^3}\\ &=\frac{c^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{c e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{c^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{c e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{\operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}-\frac{\operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}-\frac{\operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}+\frac{\operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 b d^3}\\ &=-\frac{e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}+\frac{c^2 e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{c e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}-\frac{e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}+\frac{c^2 e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}-\frac{c e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{2 \sqrt{b} d^3}+\frac{e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 \sqrt{b} d^3}\\ \end{align*}
Mathematica [A] time = 1.04065, size = 471, normalized size = 1.15 \[ \frac{\sqrt{\pi } \left (3 \left (4 c^2-1\right ) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-12 c^2 \sinh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+12 c^2 \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{3} \sinh \left (\frac{3 a}{b}\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+6 \sqrt{2} c \left (\sinh \left (\frac{2 a}{b}\right )+\cosh \left (\frac{2 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{3} \cosh \left (\frac{3 a}{b}\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+6 \sqrt{2} c \sinh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+3 \sinh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-\sqrt{3} \sinh \left (\frac{3 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-6 \sqrt{2} c \cosh \left (\frac{2 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )-3 \cosh \left (\frac{a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{3} \cosh \left (\frac{3 a}{b}\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )\right )}{24 \sqrt{b} d^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.264, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt{a+b{\it Arcsinh} \left ( dx+c \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 \frac{x^{2}}{\sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}}\,{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 \frac{x^{2}}{\sqrt{a + b \operatorname{asinh}{\left (c + d 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 \frac{x^{2}}{\sqrt{b \operatorname{arsinh}\left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]