Optimal. Leaf size=87 \[ -\frac {8 e^{3 i a} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3+\frac {i}{b n}\right );\frac {1}{2} \left (5+\frac {i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (1-3 i b n)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4509, 4505, 364} \[ -\frac {8 e^{3 i a} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3+\frac {i}{b n}\right );\frac {1}{2} \left (5+\frac {i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (1-3 i b n)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 4505
Rule 4509
Rubi steps
\begin {align*} \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int x^{-1-\frac {1}{n}} \sec ^3(a+b \log (x)) \, dx,x,c x^n\right )}{n x}\\ &=\frac {\left (8 e^{3 i a} \left (c x^n\right )^{\frac {1}{n}}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+3 i b-\frac {1}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )}{n x}\\ &=-\frac {8 e^{3 i a} \left (c x^n\right )^{3 i b} \, _2F_1\left (3,\frac {1}{2} \left (3+\frac {i}{b n}\right );\frac {1}{2} \left (5+\frac {i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-3 i b n) x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.61, size = 123, normalized size = 1.41 \[ \frac {\left (b n \tan \left (a+b \log \left (c x^n\right )\right )+1\right ) \sec \left (a+b \log \left (c x^n\right )\right )-2 i e^{i a} (b n-i) \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}+\frac {i}{2 b n};\frac {3}{2}+\frac {i}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b^2 n^2 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.63, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^2\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________