Optimal. Leaf size=191 \[ \frac {2 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^3}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.35, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4633, 4719, 4631, 3305, 3351, 4621, 4723} \[ \frac {2 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^3}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\sin ^{-1}(a x)}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3305
Rule 3351
Rule 4621
Rule 4631
Rule 4633
Rule 4719
Rule 4723
Rubi steps
\begin {align*} \int \frac {x^2}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-\frac {1}{5} (6 a) \int \frac {x^3}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac {12}{5} \int \frac {x^2}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac {8 \int \frac {1}{\sin ^{-1}(a x)^{3/2}} \, dx}{15 a^2}\\ &=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\sin ^{-1}(a x)}}-\frac {24 \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{4 \sqrt {x}}+\frac {3 \sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}-\frac {16 \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {\sin ^{-1}(a x)}} \, dx}{15 a}\\ &=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\sin ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a^3}+\frac {6 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}-\frac {18 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^3}\\ &=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\sin ^{-1}(a x)}}-\frac {32 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{15 a^3}+\frac {12 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{5 a^3}-\frac {36 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{5 a^3}\\ &=-\frac {2 x^2 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {8 x}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^3}{5 \sin ^{-1}(a x)^{3/2}}-\frac {16 \sqrt {1-a^2 x^2}}{15 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {24 x^2 \sqrt {1-a^2 x^2}}{5 a \sqrt {\sin ^{-1}(a x)}}+\frac {2 \sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^3}-\frac {6 \sqrt {6 \pi } S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.34, size = 280, normalized size = 1.47 \[ \frac {3 e^{3 i \sin ^{-1}(a x)} \left (-12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+1\right )+e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )-4 \sqrt {-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )+36 \sqrt {3} \sqrt {-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},-3 i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+4 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )-3\right )-3 e^{-3 i \sin ^{-1}(a x)} \left (12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+12 \sqrt {3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},3 i \sin ^{-1}(a x)\right )-1\right )}{60 a^3 \sin ^{-1}(a x)^{5/2}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\arcsin \left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 154, normalized size = 0.81 \[ -\frac {36 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-4 \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-4 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+36 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-2 a x \arcsin \left (a x \right )+6 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}-3 \cos \left (3 \arcsin \left (a x \right )\right )}{30 a^{3} \arcsin \left (a x \right )^{\frac {5}{2}}} \]
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 \[ \int \frac {x^2}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________