Optimal. Leaf size=103 \[ \frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \, _2F_1\left (1,-\frac {i+i m-b n}{2 b n};-\frac {i (1+m)-3 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+i b n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 99, normalized size of antiderivative = 0.96, 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 = {4605, 4601,
371} \begin {gather*} \frac {2 e^{i a} x^{m+1} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i (m+1)}{b n}\right );-\frac {i (m+1)-3 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{i b n+m+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 4601
Rule 4605
Rubi steps
\begin {align*} \int x^m \sec \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sec (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (2 e^{i a} x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+i b+\frac {1+m}{n}}}{1+e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {i (1+m)}{b n}\right );-\frac {i (1+m)-3 b n}{2 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+i b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.82, size = 99, normalized size = 0.96 \begin {gather*} \frac {2 e^{i a} x^{1+m} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {-i-i m+b n}{2 b n};-\frac {i (1+m+3 i b n)}{2 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{1+m+i b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{m} \sec \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \sec {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________