Optimal. Leaf size=86 \[ \frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right );\frac {3}{2} \left (1-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+3 i} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 86, 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 = {4510, 4506, 364} \[ \frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right );\frac {3}{2} \left (1-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-b n+3 i} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 4506
Rule 4510
Rubi steps
\begin {align*} \int x^2 \csc \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {3}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=-\frac {\left (2 i e^{i a} x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+i b+\frac {3}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n}\\ &=\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2} \left (1-\frac {3 i}{b n}\right );\frac {3}{2} \left (1-\frac {i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i-b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.57, size = 82, normalized size = 0.95 \[ -\frac {2 e^{i a} x^3 \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac {1}{2}-\frac {3 i}{2 b n};\frac {3}{2}-\frac {3 i}{2 b n};e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{b n-3 i} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int x^{2} \csc \left (a +b \ln \left (c \,x^{n}\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \csc \left (b \log \left (c x^{n}\right ) + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \csc {\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________