Optimal. Leaf size=96 \[ \frac {e^{\text {ArcSin}(a x)} x}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\text {ArcSin}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {\left (\frac {2}{3}-\frac {4 i}{3}\right ) e^{(1+2 i) \text {ArcSin}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcSin}(a x)}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.20, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4920, 6820,
6852, 4533, 4536} \begin {gather*} \frac {x e^{\text {ArcSin}(a x)}}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\text {ArcSin}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {\left (\frac {2}{3}-\frac {4 i}{3}\right ) e^{(1+2 i) \text {ArcSin}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcSin}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4533
Rule 4536
Rule 4920
Rule 6820
Rule 6852
Rubi steps
\begin {align*} \int \frac {e^{\sin ^{-1}(a x)}}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^x \cos (x)}{\left (1-\sin ^2(x)\right )^{5/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \frac {e^x \cos (x)}{\cos ^2(x)^{5/2}} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int e^x \sec ^4(x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {e^{\sin ^{-1}(a x)} x}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\sin ^{-1}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {5 \text {Subst}\left (\int e^x \sec ^2(x) \, dx,x,\sin ^{-1}(a x)\right )}{6 a}\\ &=\frac {e^{\sin ^{-1}(a x)} x}{3 \left (1-a^2 x^2\right )^{3/2}}-\frac {e^{\sin ^{-1}(a x)}}{6 a \left (1-a^2 x^2\right )}+\frac {\left (\frac {2}{3}-\frac {4 i}{3}\right ) e^{(1+2 i) \sin ^{-1}(a x)} \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \sin ^{-1}(a x)}\right )}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 84, normalized size = 0.88 \begin {gather*} \frac {e^{\text {ArcSin}(a x)} \left (-1+\frac {2 a x}{\sqrt {1-a^2 x^2}}+(1-2 i) \left (1+e^{2 i \text {ArcSin}(a x)}\right )^2 \, _2F_1\left (1-\frac {i}{2},2;2-\frac {i}{2};-e^{2 i \text {ArcSin}(a x)}\right )\right )}{6 \left (a-a^3 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________