3.37 \(\int (b \sinh (c+d x))^n \, dx\)

Optimal. Leaf size=70 \[ \frac{\cosh (c+d x) (b \sinh (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{b d (n+1) \sqrt{\cosh ^2(c+d x)}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.0193254, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2643} \[ \frac{\cosh (c+d x) (b \sinh (c+d x))^{n+1} \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{b d (n+1) \sqrt{\cosh ^2(c+d x)}} \]

Antiderivative was successfully verified.

[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 2643

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1)*Hypergeomet
ric2F1[1/2, (n + 1)/2, (n + 3)/2, Sin[c + d*x]^2])/(b*d*(n + 1)*Sqrt[Cos[c + d*x]^2]), x] /; FreeQ[{b, c, d, n
}, x] &&  !IntegerQ[2*n]

Rubi steps

\begin{align*} \int (b \sinh (c+d x))^n \, dx &=\frac{\cosh (c+d x) \, _2F_1\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]  time = 0.0450741, size = 65, normalized size = 0.93 \[ \frac{\sqrt{\cosh ^2(c+d x)} \tanh (c+d x) (b \sinh (c+d x))^n \, _2F_1\left (\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};-\sinh ^2(c+d x)\right )}{d (n+1)} \]

Antiderivative was successfully verified.

[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]  time = 0.18, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sinh \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sinh(d*x + c))^n, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sinh \left (d x + c\right )\right )^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sinh(d*x + c))^n, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh{\left (c + d x \right )}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*sinh(c + d*x))**n, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*sinh(d*x + c))^n, x)