Optimal. Leaf size=90 \[ -\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^4}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4631, 3304, 3352} \[ -\frac {\sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^4}+\frac {\sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3352
Rule 4631
Rubi steps
\begin {align*} \int \frac {x^3}{\sin ^{-1}(a x)^{3/2}} \, dx &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {\cos (2 x)}{2 \sqrt {x}}-\frac {\cos (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}-\frac {\operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}+\frac {2 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^4}-\frac {2 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{a \sqrt {\sin ^{-1}(a x)}}-\frac {\sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^4}+\frac {\sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 154, normalized size = 1.71 \[ \frac {-2 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )-i \sqrt {2} \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )+i \sqrt {2} \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )+i \sqrt {-i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )-i \sqrt {i \sin ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )}{4 a^4 \sqrt {\sin ^{-1}(a x)}} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 83, normalized size = 0.92 \[ -\frac {2 \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }-4 \FresnelC \left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arcsin \left (a x \right )}\, \sqrt {\pi }+2 \sin \left (2 \arcsin \left (a x \right )\right )-\sin \left (4 \arcsin \left (a x \right )\right )}{4 a^{4} \sqrt {\arcsin \left (a x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {asin}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________