\(\int e^{a+b x} \sinh ^n(a+b x) \, dx\) [18]

Optimal result

Integrand size = 16, antiderivative size = 76 \[ \int e^{a+b x} \sinh ^n(a+b x) \, dx=\frac {e^{a+b x} \left (1-e^{2 a+2 b x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},-n,\frac {3-n}{2},e^{2 a+2 b x}\right ) \sinh ^n(a+b x)}{b (1-n)} \] Output:

exp(b*x+a)*hypergeom([-n, 1/2-1/2*n],[3/2-1/2*n],exp(2*b*x+2*a))*sinh(b*x+ 
a)^n/b/((1-exp(2*b*x+2*a))^n)/(1-n)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.13 \[ \int e^{a+b x} \sinh ^n(a+b x) \, dx=\frac {e^{a+b x} \left (2-2 e^{-2 (a+b x)}\right )^{-n} \left (-e^{-a-b x}+e^{a+b x}\right )^n \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-n),-n,\frac {1-n}{2},e^{-2 (a+b x)}\right )}{b (1+n)} \] Input:

Integrate[E^(a + b*x)*Sinh[a + b*x]^n,x]
 

Output:

(E^(a + b*x)*(-E^(-a - b*x) + E^(a + b*x))^n*Hypergeometric2F1[(-1 - n)/2, 
 -n, (1 - n)/2, E^(-2*(a + b*x))])/(b*(2 - 2/E^(2*(a + b*x)))^n*(1 + n))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2720, 27, 1917, 279, 278}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int e^{a+b x} \sinh ^n(a+b x) \, dx\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\int 2^{-n} \left (-e^{-a-b x}+e^{a+b x}\right )^nde^{a+b x}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2^{-n} \int \left (-e^{-a-b x}+e^{a+b x}\right )^nde^{a+b x}}{b}\)

\(\Big \downarrow \) 1917

\(\displaystyle \frac {2^{-n} \left (e^{a+b x}\right )^n \left (e^{a+b x}-e^{-a-b x}\right )^n \left (e^{2 a+2 b x}-1\right )^{-n} \int \left (e^{a+b x}\right )^{-n} \left (-1+e^{2 a+2 b x}\right )^nde^{a+b x}}{b}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {2^{-n} \left (e^{a+b x}\right )^n \left (e^{a+b x}-e^{-a-b x}\right )^n \left (1-e^{2 a+2 b x}\right )^{-n} \int \left (e^{a+b x}\right )^{-n} \left (1-e^{2 a+2 b x}\right )^nde^{a+b x}}{b}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {2^{-n} e^{a+b x} \left (e^{a+b x}-e^{-a-b x}\right )^n \left (1-e^{2 a+2 b x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (\frac {1-n}{2},-n,\frac {3-n}{2},e^{2 a+2 b x}\right )}{b (1-n)}\)

Input:

Int[E^(a + b*x)*Sinh[a + b*x]^n,x]
 

Output:

(E^(a + b*x)*(-E^(-a - b*x) + E^(a + b*x))^n*Hypergeometric2F1[(1 - n)/2, 
-n, (3 - n)/2, E^(2*a + 2*b*x)])/(2^n*b*(1 - E^(2*a + 2*b*x))^n*(1 - n))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 1917

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])   Int[ 
x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !Integ 
erQ[p] && NeQ[n, j] && PosQ[n - j]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

Maple [F]

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

Input:

int(exp(b*x+a)*sinh(b*x+a)^n,x)
 

Output:

int(exp(b*x+a)*sinh(b*x+a)^n,x)
 

Fricas [F]

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

integrate(exp(b*x+a)*sinh(b*x+a)^n,x, algorithm="fricas")
 

Output:

integral(sinh(b*x + a)^n*e^(b*x + a), x)
 

Sympy [F]

\[ \int e^{a+b x} \sinh ^n(a+b x) \, dx=e^{a} \int e^{b x} \sinh ^{n}{\left (a + b x \right )}\, dx \] Input:

integrate(exp(b*x+a)*sinh(b*x+a)**n,x)
 

Output:

exp(a)*Integral(exp(b*x)*sinh(a + b*x)**n, x)
 

Maxima [F]

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

integrate(exp(b*x+a)*sinh(b*x+a)^n,x, algorithm="maxima")
 

Output:

integrate(sinh(b*x + a)^n*e^(b*x + a), x)
 

Giac [F]

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

integrate(exp(b*x+a)*sinh(b*x+a)^n,x, algorithm="giac")
 

Output:

integrate(sinh(b*x + a)^n*e^(b*x + a), x)
 

Mupad [F(-1)]

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

int(exp(a + b*x)*sinh(a + b*x)^n,x)
 

Output:

int(exp(a + b*x)*sinh(a + b*x)^n, x)
 

Reduce [F]

\[ \int e^{a+b x} \sinh ^n(a+b x) \, dx=\frac {e^{a} \left (-e^{b x} \left (e^{2 b x +2 a}-1\right )^{n} n +e^{b n x +a n +b x} \sinh \left (b x +a \right )^{n} 2^{n} n +e^{b n x +a n +b x} \sinh \left (b x +a \right )^{n} 2^{n}-2 e^{b n x +a n} \left (\int \frac {e^{b x} \left (e^{2 b x +2 a}-1\right )^{n}}{e^{b n x +a n +2 b x +2 a} n +e^{b n x +a n +2 b x +2 a}-e^{b n x +a n} n -e^{b n x +a n}}d x \right ) b \,n^{2}-2 e^{b n x +a n} \left (\int \frac {e^{b x} \left (e^{2 b x +2 a}-1\right )^{n}}{e^{b n x +a n +2 b x +2 a} n +e^{b n x +a n +2 b x +2 a}-e^{b n x +a n} n -e^{b n x +a n}}d x \right ) b n \right )}{e^{b n x +a n} 2^{n} b \left (n +1\right )} \] Input:

int(exp(b*x+a)*sinh(b*x+a)^n,x)
 

Output:

(e**a*( - e**(b*x)*(e**(2*a + 2*b*x) - 1)**n*n + e**(a*n + b*n*x + b*x)*si 
nh(a + b*x)**n*2**n*n + e**(a*n + b*n*x + b*x)*sinh(a + b*x)**n*2**n - 2*e 
**(a*n + b*n*x)*int((e**(b*x)*(e**(2*a + 2*b*x) - 1)**n)/(e**(a*n + 2*a + 
b*n*x + 2*b*x)*n + e**(a*n + 2*a + b*n*x + 2*b*x) - e**(a*n + b*n*x)*n - e 
**(a*n + b*n*x)),x)*b*n**2 - 2*e**(a*n + b*n*x)*int((e**(b*x)*(e**(2*a + 2 
*b*x) - 1)**n)/(e**(a*n + 2*a + b*n*x + 2*b*x)*n + e**(a*n + 2*a + b*n*x + 
 2*b*x) - e**(a*n + b*n*x)*n - e**(a*n + b*n*x)),x)*b*n))/(e**(a*n + b*n*x 
)*2**n*b*(n + 1))