Integrand size = 14, antiderivative size = 54 \[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\frac {3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{6},\frac {9+m}{6},\coth ^2(c+d x)\right )}{d (3+m)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557, 371} \[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\frac {3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {m+3}{6},\frac {m+9}{6},\coth ^2(c+d x)\right )}{d (m+3)} \]
[In]
[Out]
Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left (\coth ^{-\frac {m}{3}}(c+d x) \sqrt [3]{b \coth ^m(c+d x)}\right ) \int \coth ^{\frac {m}{3}}(c+d x) \, dx \\ & = -\frac {\left (\coth ^{-\frac {m}{3}}(c+d x) \sqrt [3]{b \coth ^m(c+d x)}\right ) \text {Subst}\left (\int \frac {x^{m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{6},\frac {9+m}{6},\coth ^2(c+d x)\right )}{d (3+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\frac {3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {3+m}{6},\frac {9+m}{6},\coth ^2(c+d x)\right )}{d (3+m)} \]
[In]
[Out]
\[\int \left (b \coth \left (d x +c \right )^{m}\right )^{\frac {1}{3}}d x\]
[In]
[Out]
Exception generated. \[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\int \sqrt [3]{b \coth ^{m}{\left (c + d x \right )}}\, dx \]
[In]
[Out]
\[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\int { \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {1}{3}} \,d x } \]
[In]
[Out]
\[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\int { \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {1}{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt [3]{b \coth ^m(c+d x)} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{1/3} \,d x \]
[In]
[Out]