Optimal. Leaf size=93 \[ \frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]
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Rubi [A] time = 0.06, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3769, 3771, 2641} \[ \frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{x \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sec ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}+\frac {\operatorname {Subst}\left (\int \sqrt {\sec (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}+\frac {\left (\sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\cos (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\sec \left (a+b \log \left (c x^n\right )\right )}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 72, normalized size = 0.77 \[ \frac {\sqrt {\sec \left (a+b \log \left (c x^n\right )\right )} \left (\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 \sqrt {\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 247, normalized size = 2.66 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}\, \left (4 \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )-2 \left (\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right ) \cos \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{3 n \sqrt {-2 \left (\sin ^{4}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )+\sin ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}\, \sin \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )-1}\, b} \]
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 \[ \int \frac {1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x\,{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x \sec ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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