Optimal. Leaf size=109 \[ \frac {2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right );\frac {1}{4} \left (7-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.07, 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 = {4484, 4492, 364} \[ \frac {2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right );\frac {1}{4} \left (7-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4484
Rule 4492
Rubi steps
\begin {align*} \int \frac {1}{\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 ) \operatorname {Subst}\left (\int \frac {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}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}\right ) \operatorname {Subst}\left (\int \frac {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 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac {2 x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i}{b n}\right );\frac {1}{4} \left (7-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [B] time = 3.71, size = 431, normalized size = 3.95 \[ \frac {x \left ((3 b n-2 i) x^{-i b n} \left (2 x^{i b n} \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} (b n \cos (b n \log (x))-2 \sin (b n \log (x)))-(b n-2 i) \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {b n+2 i}{4 b n};\frac {3}{4}-\frac {i}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}\right )-\left (b^2 n^2+4\right ) x^{i b n} \sqrt {2+2 e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n};\frac {7}{4}-\frac {i}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )}\right )}{b n (3 b n-2 i) \sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \left (b n \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-2 \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )} \]
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 \[ \int \frac {1}{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {1}{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\cos ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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