Optimal. Leaf size=63 \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]
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Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2635, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{5 b n} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \cos ^{\frac {5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n}+\frac {3 \operatorname {Subst}\left (\int \sqrt {\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n}+\frac {2 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 58, normalized size = 0.92 \[ \frac {6 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}}{5 b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}, x\right ) \]
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 \[ \int \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 280, normalized size = 4.44 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (-8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{6}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+8 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right ) \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{5 n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]
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 \[ \int \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 65, normalized size = 1.03 \[ -\frac {2\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{7/2}\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{7\,b\,n\,\sqrt {{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2}} \]
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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