\(\int (b \sinh (c+d x))^{4/3} \, dx\) [31]

Optimal result

Integrand size = 12, antiderivative size = 60 \[ \int (b \sinh (c+d x))^{4/3} \, dx=\frac {3 \cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{7/3}}{7 b d \sqrt {\cosh ^2(c+d x)}} \]

[Out]

3/7*cosh(d*x+c)*hypergeom([1/2, 7/6],[13/6],-sinh(d*x+c)^2)*(b*sinh(d*x+c))^(7/3)/b/d/(cosh(d*x+c)^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, 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} \[ \int (b \sinh (c+d x))^{4/3} \, dx=\frac {3 \cosh (c+d x) (b \sinh (c+d x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},-\sinh ^2(c+d x)\right )}{7 b d \sqrt {\cosh ^2(c+d x)}} \]

[In]

Int[(b*Sinh[c + d*x])^(4/3),x]

[Out]

(3*Cosh[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(7/3))/(7*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}{2},\frac {7}{6},\frac {13}{6},-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{7/3}}{7 b d \sqrt {\cosh ^2(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95 \[ \int (b \sinh (c+d x))^{4/3} \, dx=\frac {3 \sqrt {\cosh ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{6},\frac {13}{6},-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{4/3} \tanh (c+d x)}{7 d} \]

[In]

Integrate[(b*Sinh[c + d*x])^(4/3),x]

[Out]

(3*Sqrt[Cosh[c + d*x]^2]*Hypergeometric2F1[1/2, 7/6, 13/6, -Sinh[c + d*x]^2]*(b*Sinh[c + d*x])^(4/3)*Tanh[c +
d*x])/(7*d)

Maple [F]

\[\int \left (b \sinh \left (d x +c \right )\right )^{\frac {4}{3}}d x\]

[In]

int((b*sinh(d*x+c))^(4/3),x)

[Out]

int((b*sinh(d*x+c))^(4/3),x)

Fricas [F]

\[ \int (b \sinh (c+d x))^{4/3} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((b*sinh(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int (b \sinh (c+d x))^{4/3} \, dx=\int \left (b \sinh {\left (c + d x \right )}\right )^{\frac {4}{3}}\, dx \]

[In]

integrate((b*sinh(d*x+c))**(4/3),x)

[Out]

Integral((b*sinh(c + d*x))**(4/3), x)

Maxima [F]

\[ \int (b \sinh (c+d x))^{4/3} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((b*sinh(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

integrate((b*sinh(d*x + c))^(4/3), x)

Giac [F]

\[ \int (b \sinh (c+d x))^{4/3} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate((b*sinh(d*x+c))^(4/3),x, algorithm="giac")

[Out]

integrate((b*sinh(d*x + c))^(4/3), x)

Mupad [F(-1)]

Timed out. \[ \int (b \sinh (c+d x))^{4/3} \, dx=\int {\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{4/3} \,d x \]

[In]

int((b*sinh(c + d*x))^(4/3),x)

[Out]

int((b*sinh(c + d*x))^(4/3), x)