Optimal. Leaf size=68 \[ \frac {i x^2}{2}-i x^2 \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4594, 4592,
470, 371} \begin {gather*} \frac {i x^2}{2}-i x^2 \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 371
Rule 470
Rule 4592
Rule 4594
Rubi steps
\begin {align*} \int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int x \cot \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(219\) vs. \(2(68)=136\).
time = 6.18, size = 219, normalized size = 3.22 \begin {gather*} -\frac {x^2 \left (e^{2 i d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {i}{b d n};2-\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+(-i+b d n) \left (\cot \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\cot \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+i \, _2F_1\left (1,-\frac {i}{b d n};1-\frac {i}{b d n};e^{2 i d \left (a+b \log \left (c x^n\right )\right )}\right )+\csc \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \csc \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sin (b d n \log (x))\right )\right )}{-2 i+2 b d n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \cot \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \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 \cot {\left (a d + b d \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 x\,\mathrm {cot}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________