Optimal. Leaf size=130 \[ \frac {2 x^{m+1} \, _2F_1\left (-\frac {3}{2},-\frac {2 i m+3 b n+2 i}{4 b n};-\frac {2 i m-b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(-3 i b n+2 m+2) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4494, 4492, 364} \[ \frac {2 x^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-3\right );-\frac {2 i m-b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{(-3 i b n+2 m+2) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4492
Rule 4494
Rubi steps
\begin {align*} \int x^m \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1+m}{n}} \cos ^{\frac {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^{1+m} \left (c x^n\right )^{\frac {3 i b}{2}-\frac {1+m}{n}} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname {Subst}\left (\int x^{-1-\frac {3 i b}{2}+\frac {1+m}{n}} \left (1+e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ &=\frac {2 x^{1+m} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i (1+m)}{b n}\right );-\frac {2 i+2 i m-b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 2.03, size = 204, normalized size = 1.57 \[ \frac {x^{m+1} \left (6 b^2 n^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \, _2F_1\left (1,-\frac {2 i m-3 b n+2 i}{4 b n};-\frac {2 i m-5 b n+2 i}{4 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(i b n+2 m+2) \left (4 (m+1) \cos ^2\left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )\right )}{(i b n+2 m+2) (-3 i b n+2 m+2) (3 i b n+2 m+2) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}} \]
Warning: Unable to verify antiderivative.
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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.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cos ^{\frac {3}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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