\(\int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx\) [35]

Optimal result

Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\frac {3 \cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right ) \sqrt [3]{b \sinh (c+d x)}}{b d \sqrt {\cosh ^2(c+d x)}} \]

[Out]

3*cosh(d*x+c)*hypergeom([1/6, 1/2],[7/6],-sinh(d*x+c)^2)*(b*sinh(d*x+c))^(1/3)/b/d/(cosh(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\frac {3 \cosh (c+d x) \sqrt [3]{b \sinh (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right )}{b d \sqrt {\cosh ^2(c+d x)}} \]

[In]

Int[(b*Sinh[c + d*x])^(-2/3),x]

[Out]

(3*Cosh[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(1/3))/(b*d*Sqrt[Cosh[c
+ d*x]^2])

Rule 2722

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
)*Sqrt[Cos[c + d*x]^2]))*Hypergeometric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2], x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {3 \cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right ) \sqrt [3]{b \sinh (c+d x)}}{b d \sqrt {\cosh ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\frac {3 \sqrt {\cosh ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right ) \tanh (c+d x)}{d (b \sinh (c+d x))^{2/3}} \]

[In]

Integrate[(b*Sinh[c + d*x])^(-2/3),x]

[Out]

(3*Sqrt[Cosh[c + d*x]^2]*Hypergeometric2F1[1/6, 1/2, 7/6, -Sinh[c + d*x]^2]*Tanh[c + d*x])/(d*(b*Sinh[c + d*x]
)^(2/3))

Maple [F]

\[\int \frac {1}{\left (b \sinh \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]

[In]

int(1/(b*sinh(d*x+c))^(2/3),x)

[Out]

int(1/(b*sinh(d*x+c))^(2/3),x)

Fricas [F]

\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(b*sinh(d*x+c))^(2/3),x, algorithm="fricas")

[Out]

integral((b*sinh(d*x + c))^(1/3)/(b*sinh(d*x + c)), x)

Sympy [F]

\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (b \sinh {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]

[In]

integrate(1/(b*sinh(d*x+c))**(2/3),x)

[Out]

Integral((b*sinh(c + d*x))**(-2/3), x)

Maxima [F]

\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(b*sinh(d*x+c))^(2/3),x, algorithm="maxima")

[Out]

integrate((b*sinh(d*x + c))^(-2/3), x)

Giac [F]

\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]

[In]

integrate(1/(b*sinh(d*x+c))^(2/3),x, algorithm="giac")

[Out]

integrate((b*sinh(d*x + c))^(-2/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int \frac {1}{{\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{2/3}} \,d x \]

[In]

int(1/(b*sinh(c + d*x))^(2/3),x)

[Out]

int(1/(b*sinh(c + d*x))^(2/3), x)