\(\int (b \sinh (c+d x))^n \, dx\) [37]

Optimal result

Integrand size = 10, antiderivative size = 70 \[ \int (b \sinh (c+d x))^n \, dx=\frac {\cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{1+n}}{b d (1+n) \sqrt {\cosh ^2(c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2722} \[ \int (b \sinh (c+d x))^n \, dx=\frac {\cosh (c+d x) (b \sinh (c+d x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {n+1}{2},\frac {n+3}{2},-\sinh ^2(c+d x)\right )}{b d (n+1) \sqrt {\cosh ^2(c+d x)}} \]

[In]

Int[(b*Sinh[c + d*x])^n,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93 \[ \int (b \sinh (c+d x))^n \, dx=\frac {\sqrt {\cosh ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+n}{2},\frac {3+n}{2},-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^n \tanh (c+d x)}{d (1+n)} \]

[In]

Integrate[(b*Sinh[c + d*x])^n,x]

[Out]

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

Maple [F]

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

[In]

int((b*sinh(d*x+c))^n,x)

[Out]

int((b*sinh(d*x+c))^n,x)

Fricas [F]

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

[In]

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

[Out]

integral((b*sinh(d*x + c))^n, x)

Sympy [F]

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

[In]

integrate((b*sinh(d*x+c))**n,x)

[Out]

Integral((b*sinh(c + d*x))**n, x)

Maxima [F]

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

[In]

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

[Out]

integrate((b*sinh(d*x + c))^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*sinh(d*x + c))^n, x)

Mupad [F(-1)]

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

[In]

int((b*sinh(c + d*x))^n,x)

[Out]

int((b*sinh(c + d*x))^n, x)