Optimal. Leaf size=190 \[ \frac {32 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^4}-\frac {16 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^4}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4633, 4719, 4631, 3304, 3352} \[ \frac {32 \sqrt {2 \pi } \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^4}-\frac {16 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^4}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3352
Rule 4631
Rule 4633
Rule 4719
Rubi steps
\begin {align*} \int \frac {x^3}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {6 \int \frac {x^2}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac {64}{15} \int \frac {x^3}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac {8 \int \frac {x}{\sin ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^4}-\frac {128 \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 )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}-\frac {64 \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^4}+\frac {64 \operatorname {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^4}+\frac {32 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{5 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {16 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{5 a^4}-\frac {128 \operatorname {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{15 a^4}+\frac {128 \operatorname {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{15 a^4}\\ &=-\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \sin ^{-1}(a x)^{3/2}}-\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\sin ^{-1}(a x)}}+\frac {32 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^4}-\frac {16 \sqrt {\pi } C\left (\frac {2 \sqrt {\sin ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.11, size = 272, normalized size = 1.43 \[ \frac {-6 \sin \left (2 \sin ^{-1}(a x)\right )+3 \sin \left (4 \sin ^{-1}(a x)\right )+4 \sin ^{-1}(a x) \left (i e^{2 i \sin ^{-1}(a x)} \left (-4 \sin ^{-1}(a x)+i\right )-4 \sqrt {2} \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 i \sin ^{-1}(a x)\right )+e^{-2 i \sin ^{-1}(a x)} \left (4 i \sin ^{-1}(a x)-4 \sqrt {2} e^{2 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},2 i \sin ^{-1}(a x)\right )-1\right )\right )-4 \sin ^{-1}(a x) \left (i e^{4 i \sin ^{-1}(a x)} \left (-8 \sin ^{-1}(a x)+i\right )-16 \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 i \sin ^{-1}(a x)\right )+e^{-4 i \sin ^{-1}(a x)} \left (8 i \sin ^{-1}(a x)-16 e^{4 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},4 i \sin ^{-1}(a x)\right )-1\right )\right )}{60 a^4 \sin ^{-1}(a x)^{5/2}} \]
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 = 0.09, size = 139, normalized size = 0.73 \[ -\frac {-128 \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+64 \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}+64 \sin \left (4 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{2}-32 \sin \left (2 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )^{2}-8 \arcsin \left (a x \right ) \cos \left (4 \arcsin \left (a x \right )\right )+8 \arcsin \left (a x \right ) \cos \left (2 \arcsin \left (a x \right )\right )-3 \sin \left (4 \arcsin \left (a x \right )\right )+6 \sin \left (2 \arcsin \left (a x \right )\right )}{60 a^{4} \arcsin \left (a x \right )^{\frac {5}{2}}} \]
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 \[ \int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________