Optimal. Leaf size=43 \[ \frac {\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(a+b x)\right )}{b (n+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3476, 364} \[ \frac {\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\coth ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 3476
Rubi steps
\begin {align*} \int \coth ^n(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^n}{-1+x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\coth ^{1+n}(a+b x) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\coth ^2(a+b x)\right )}{b (1+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 45, normalized size = 1.05 \[ \frac {\coth ^{n+1}(a+b x) \, _2F_1\left (1,\frac {n+1}{2};\frac {n+1}{2}+1;\coth ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\coth \left (b x + a\right )^{n}, x\right ) \]
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 = 0.44, size = 0, normalized size = 0.00 \[ \int \coth ^{n}\left (b x +a \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth \left (b x + a\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {coth}\left (a+b\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{n}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________