Optimal. Leaf size=111 \[ \frac {2 x^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 i}{b n}\right );\frac {1}{4} \left (3-\frac {4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4493, 4491, 364} \[ \frac {2 x^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 i}{b n}\right );\frac {1}{4} \left (3-\frac {4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 4491
Rule 4493
Rubi steps
\begin {align*} \int x \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {2}{n}} \sqrt {\sin (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^2 \left (c x^n\right )^{\frac {i b}{2}-\frac {2}{n}} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname {Subst}\left (\int x^{-1-\frac {i b}{2}+\frac {2}{n}} \sqrt {1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ &=\frac {2 x^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4} \left (-1-\frac {4 i}{b n}\right );\frac {1}{4} \left (3-\frac {4 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{(4-i b n) \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.39, size = 94, normalized size = 0.85 \[ \frac {2 x^2 \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \, _2F_1\left (1,\frac {5}{4}-\frac {i}{b n};\frac {3}{4}-\frac {i}{b n};e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{-4+i b n} \]
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 \[ \int x \sqrt {\sin \left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int x \left (\sqrt {\sin }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\sin \left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\sqrt {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sqrt {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________