Optimal. Leaf size=54 \[ \frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {m+2}{4};\frac {m+6}{4};\coth ^2(c+d x)\right )}{d (m+2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {m+2}{4};\frac {m+6}{4};\coth ^2(c+d x)\right )}{d (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \sqrt {b \coth ^m(c+d x)} \, dx &=\left (\coth ^{-\frac {m}{2}}(c+d x) \sqrt {b \coth ^m(c+d x)}\right ) \int \coth ^{\frac {m}{2}}(c+d x) \, dx\\ &=-\frac {\left (\coth ^{-\frac {m}{2}}(c+d x) \sqrt {b \coth ^m(c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^{m/2}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {2+m}{4};\frac {6+m}{4};\coth ^2(c+d x)\right )}{d (2+m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 54, normalized size = 1.00 \[ \frac {2 \coth (c+d x) \sqrt {b \coth ^m(c+d x)} \, _2F_1\left (1,\frac {m+2}{4};\frac {m+6}{4};\coth ^2(c+d x)\right )}{d (m+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \coth \left (d x + c\right )^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \sqrt {b \left (\coth ^{m}\left (d x +c \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \coth \left (d x + c\right )^{m}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,{\mathrm {coth}\left (c+d\,x\right )}^m} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \coth ^{m}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________