Optimal. Leaf size=109 \[ \frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right );\frac {1}{4} \left (9-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+5 i b n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, 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 = {4600, 4604,
371} \begin {gather*} \frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right );\frac {1}{4} \left (9-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 4600
Rule 4604
Rubi steps
\begin {align*} \int \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^{\frac {5}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {5 i b}{2}-\frac {1}{n}} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {5 i b}{2}+\frac {1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^{5/2}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i}{b n}\right );\frac {1}{4} \left (9-\frac {2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+5 i b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.67, size = 174, normalized size = 1.60 \begin {gather*} \frac {2 e^{-2 i \left (a-b n \log (x)+b \log \left (c x^n\right )\right )} x^{1-2 i b n} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (-e^{2 i a} \left (c x^n\right )^{2 i b} \left (2+b n \cot \left (a+b \log \left (c x^n\right )\right )\right )+(2+i b n) \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 )\right )}{3 b^2 n^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \csc ^{\frac {5}{2}}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________