Optimal. Leaf size=74 \[ \frac {i x^3}{3}-\frac {2}{3} i x^3 \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x^2 \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 5.58, size = 229, normalized size = 3.09 \[ -\frac {x^3 \left (3 e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {3 i}{2 b d n};2-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n-3 i) \left (i \, _2F_1\left (1,-\frac {3 i}{2 b d n};1-\frac {3 i}{2 b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sin (b d n \log (x)) \csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )}{6 b d n-9 i} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \cot \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [F] time = 1.42, size = 0, normalized size = 0.00 \[ \int x^{2} \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cot \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cot {\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________