Optimal. Leaf size=109 \[ \frac {2 x \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right );\frac {1}{4} \left (1-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
[Out]
________________________________________________________________________________________
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 = {4503, 4507, 364} \[ \frac {2 x \, _2F_1\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i}{b n}\right );\frac {1}{4} \left (1-\frac {2 i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 4503
Rule 4507
Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\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}}}{\sec ^{\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}}\right ) \operatorname {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} \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\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 )}{(2-3 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.61, size = 168, normalized size = 1.54 \[ \frac {2 x \left (3 b^2 n^2 \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \, _2F_1\left (1,\frac {3}{4}-\frac {i}{2 b n};\frac {5}{4}-\frac {i}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \sec ^2\left (a+b \log \left (c x^n\right )\right )+(2+i b n) \left (3 b n \tan \left (a+b \log \left (c x^n\right )\right )+2\right )\right )}{(2+3 i b n) (b n-2 i) (3 b n+2 i) \sec ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________