Optimal. Leaf size=162 \[ \frac{\left (1-a^2 x^2\right )^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac{6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\sin ^{-1}(a x)}+\frac{30 \left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac{120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{72 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{144 e^{\sin ^{-1}(a x)}}{629 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.430422, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {4836, 6688, 6720, 4435, 2194} \[ \frac{\left (1-a^2 x^2\right )^3 e^{\sin ^{-1}(a x)}}{37 a}+\frac{6}{37} x \left (1-a^2 x^2\right )^{5/2} e^{\sin ^{-1}(a x)}+\frac{30 \left (1-a^2 x^2\right )^2 e^{\sin ^{-1}(a x)}}{629 a}+\frac{120}{629} x \left (1-a^2 x^2\right )^{3/2} e^{\sin ^{-1}(a x)}+\frac{72 \left (1-a^2 x^2\right ) e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} x \sqrt{1-a^2 x^2} e^{\sin ^{-1}(a x)}+\frac{144 e^{\sin ^{-1}(a x)}}{629 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4836
Rule 6688
Rule 6720
Rule 4435
Rule 2194
Rubi steps
\begin{align*} \int e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \left (1-\sin ^2(x)\right )^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos (x) \cos ^2(x)^{5/2} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int e^x \cos ^6(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac{30 \operatorname{Subst}\left (\int e^x \cos ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{37 a}\\ &=\frac{120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac{360 \operatorname{Subst}\left (\int e^x \cos ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{629 a}\\ &=\frac{144}{629} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{72 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac{120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}+\frac{144 \operatorname{Subst}\left (\int e^x \, dx,x,\sin ^{-1}(a x)\right )}{629 a}\\ &=\frac{144 e^{\sin ^{-1}(a x)}}{629 a}+\frac{144}{629} e^{\sin ^{-1}(a x)} x \sqrt{1-a^2 x^2}+\frac{72 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )}{629 a}+\frac{120}{629} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac{30 e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^2}{629 a}+\frac{6}{37} e^{\sin ^{-1}(a x)} x \left (1-a^2 x^2\right )^{5/2}+\frac{e^{\sin ^{-1}(a x)} \left (1-a^2 x^2\right )^3}{37 a}\\ \end{align*}
Mathematica [A] time = 0.333929, size = 69, normalized size = 0.43 \[ \frac{e^{\sin ^{-1}(a x)} \left (3774 \sin \left (2 \sin ^{-1}(a x)\right )+888 \sin \left (4 \sin ^{-1}(a x)\right )+102 \sin \left (6 \sin ^{-1}(a x)\right )+1887 \cos \left (2 \sin ^{-1}(a x)\right )+222 \cos \left (4 \sin ^{-1}(a x)\right )+17 \cos \left (6 \sin ^{-1}(a x)\right )+6290\right )}{20128 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{\arcsin \left ( ax \right ) }} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}\, 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{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} e^{\left (\arcsin \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.0783, size = 176, normalized size = 1.09 \begin{align*} -\frac{{\left (17 \, a^{6} x^{6} - 81 \, a^{4} x^{4} + 183 \, a^{2} x^{2} - 6 \,{\left (17 \, a^{5} x^{5} - 54 \, a^{3} x^{3} + 61 \, a x\right )} \sqrt{-a^{2} x^{2} + 1} - 263\right )} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20448, size = 193, normalized size = 1.19 \begin{align*} \frac{6}{37} \,{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{120}{629} \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x e^{\left (\arcsin \left (a x\right )\right )} - \frac{{\left (a^{2} x^{2} - 1\right )}^{3} e^{\left (\arcsin \left (a x\right )\right )}}{37 \, a} + \frac{144}{629} \, \sqrt{-a^{2} x^{2} + 1} x e^{\left (\arcsin \left (a x\right )\right )} + \frac{30 \,{\left (a^{2} x^{2} - 1\right )}^{2} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} - \frac{72 \,{\left (a^{2} x^{2} - 1\right )} e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} + \frac{144 \, e^{\left (\arcsin \left (a x\right )\right )}}{629 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]