Optimal. Leaf size=115 \[ x \left (-e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^{-p} F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.02, 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 = {} \[ \int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \coth ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 3.97, size = 387, normalized size = 3.37 \[ \frac {x (2 b d n+1) \left (\frac {e^{2 a d} \left (c x^n\right )^{2 b d}+1}{e^{2 a d} \left (c x^n\right )^{2 b d}-1}\right )^p F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d n p e^{2 a d} \left (c x^n\right )^{2 b d} F_1\left (1+\frac {1}{2 b d n};p,1-p;2+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+2 b d n p e^{2 a d} \left (c x^n\right )^{2 b d} F_1\left (1+\frac {1}{2 b d n};p+1,-p;2+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+(2 b d n+1) F_1\left (\frac {1}{2 b d n};p,-p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (b d \log \left (c x^{n}\right ) + a d\right )^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \coth ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {coth}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{p}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________