Optimal. Leaf size=109 \[ \frac {2 x \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right );\frac {1}{4} \left (1-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4572, 4580,
371} \begin {gather*} \frac {2 x \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right );\frac {1}{4} \left (1-\frac {2 i}{b n}\right );-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 )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4572
Rule 4580
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \cos ^{\frac {3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{\frac {3 i b}{2}-\frac {1}{n}} \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1-\frac {3 i b}{2}+\frac {1}{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 \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}{b n}\right );\frac {1}{4} \left (1-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 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 [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(220\) vs. \(2(109)=218\).
time = 1.04, size = 220, normalized size = 2.02 \begin {gather*} -\frac {6 i \sqrt {2} b^2 \sqrt {1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}} n^2 x \, _2F_1\left (\frac {1}{2},\frac {1}{4}-\frac {i}{2 b n};\frac {5}{4}-\frac {i}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{\sqrt {e^{-i \left (a+b \log \left (c x^n\right )\right )} \left (1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )} (-2 i+b n) (-2 i+3 b n) (2 i+3 b n)}+\frac {2 x \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \left (2 \cos \left (a+b \log \left (c x^n\right )\right )+3 b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4+9 b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \cos ^{\frac {3}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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