3.2.31 \(\int \sinh ^5(e+f x) (a+b \sinh ^2(e+f x))^p \, dx\) [131]

3.2.31.1 Optimal result

Integrand size = 23, antiderivative size = 226 \[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=-\frac {(3 a+2 b (2+p)) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{1+p}}{b^2 f (3+2 p) (5+2 p)}+\frac {\left (3 a^2+4 a b (1+p)+4 b^2 \left (2+3 p+p^2\right )\right ) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^p \left (1+\frac {b \cosh ^2(e+f x)}{a-b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \cosh ^2(e+f x)}{a-b}\right )}{b^2 f (3+2 p) (5+2 p)}+\frac {\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{1+p} \sinh ^2(e+f x)}{b f (5+2 p)} \]

output

-(3*a+2*b*(2+p))*cosh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(p+1)/b^2/f/(4*p^2+16*p 
+15)+(3*a^2+4*a*b*(p+1)+4*b^2*(p^2+3*p+2))*cosh(f*x+e)*(a-b+b*cosh(f*x+e)^ 
2)^p*hypergeom([1/2, -p],[3/2],-b*cosh(f*x+e)^2/(a-b))/b^2/f/(4*p^2+16*p+1 
5)/((1+b*cosh(f*x+e)^2/(a-b))^p)+cosh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(p+1)*s 
inh(f*x+e)^2/b/f/(5+2*p)
 

3.2.31.2 Mathematica [F]

\[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx \]

input

Integrate[Sinh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p,x]
 

output

Integrate[Sinh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p, x]
 

3.2.31.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 3665, 318, 299, 238, 237}

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.

\(\displaystyle \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -i \sin (i e+i f x)^5 \left (a-b \sin (i e+i f x)^2\right )^pdx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \sin (i e+i f x)^5 \left (a-b \sin (i e+i f x)^2\right )^pdx\)

\(\Big \downarrow \) 3665

\(\displaystyle \frac {\int \left (1-\cosh ^2(e+f x)\right )^2 \left (b \cosh ^2(e+f x)+a-b\right )^pd\cosh (e+f x)}{f}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {\frac {\int \left (b \cosh ^2(e+f x)+a-b\right )^p \left (-\left ((3 a+2 b (p+2)) \cosh ^2(e+f x)\right )+a+2 b (p+2)\right )d\cosh (e+f x)}{b (2 p+5)}-\frac {\cosh (e+f x) \left (1-\cosh ^2(e+f x)\right ) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+5)}}{f}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {\left (3 a^2+4 a b (p+1)+4 b^2 \left (p^2+3 p+2\right )\right ) \int \left (b \cosh ^2(e+f x)+a-b\right )^pd\cosh (e+f x)}{b (2 p+3)}-\frac {(3 a+2 b (p+2)) \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+3)}}{b (2 p+5)}-\frac {\cosh (e+f x) \left (1-\cosh ^2(e+f x)\right ) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+5)}}{f}\)

\(\Big \downarrow \) 238

\(\displaystyle \frac {\frac {\frac {\left (3 a^2+4 a b (p+1)+4 b^2 \left (p^2+3 p+2\right )\right ) \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} \int \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^pd\cosh (e+f x)}{b (2 p+3)}-\frac {(3 a+2 b (p+2)) \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+3)}}{b (2 p+5)}-\frac {\cosh (e+f x) \left (1-\cosh ^2(e+f x)\right ) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+5)}}{f}\)

\(\Big \downarrow \) 237

\(\displaystyle \frac {\frac {\frac {\left (3 a^2+4 a b (p+1)+4 b^2 \left (p^2+3 p+2\right )\right ) \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^p \left (\frac {b \cosh ^2(e+f x)}{a-b}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b \cosh ^2(e+f x)}{a-b}\right )}{b (2 p+3)}-\frac {(3 a+2 b (p+2)) \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+3)}}{b (2 p+5)}-\frac {\cosh (e+f x) \left (1-\cosh ^2(e+f x)\right ) \left (a+b \cosh ^2(e+f x)-b\right )^{p+1}}{b (2 p+5)}}{f}\)

input

Int[Sinh[e + f*x]^5*(a + b*Sinh[e + f*x]^2)^p,x]
 

output

(-((Cosh[e + f*x]*(1 - Cosh[e + f*x]^2)*(a - b + b*Cosh[e + f*x]^2)^(1 + p 
))/(b*(5 + 2*p))) + (-(((3*a + 2*b*(2 + p))*Cosh[e + f*x]*(a - b + b*Cosh[ 
e + f*x]^2)^(1 + p))/(b*(3 + 2*p))) + ((3*a^2 + 4*a*b*(1 + p) + 4*b^2*(2 + 
 3*p + p^2))*Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^p*Hypergeometric2F1 
[1/2, -p, 3/2, -((b*Cosh[e + f*x]^2)/(a - b))])/(b*(3 + 2*p)*(1 + (b*Cosh[ 
e + f*x]^2)/(a - b))^p))/(b*(5 + 2*p)))/f
 

3.2.31.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 237

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

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3665

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f 
Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + 
 f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
 

3.2.31.4 Maple [F]

\[\int \sinh \left (f x +e \right )^{5} \left (a +b \sinh \left (f x +e \right )^{2}\right )^{p}d x\]

input

int(sinh(f*x+e)^5*(a+b*sinh(f*x+e)^2)^p,x)
 

output

int(sinh(f*x+e)^5*(a+b*sinh(f*x+e)^2)^p,x)
 

3.2.31.5 Fricas [F]

\[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \sinh \left (f x + e\right )^{5} \,d x } \]

input

integrate(sinh(f*x+e)^5*(a+b*sinh(f*x+e)^2)^p,x, algorithm="fricas")
 

output

integral((b*sinh(f*x + e)^2 + a)^p*sinh(f*x + e)^5, x)
 

3.2.31.6 Sympy [F(-1)]

Timed out. \[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\text {Timed out} \]

input

integrate(sinh(f*x+e)**5*(a+b*sinh(f*x+e)**2)**p,x)
 

output

Timed out
 

3.2.31.7 Maxima [F]

\[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \sinh \left (f x + e\right )^{5} \,d x } \]

input

integrate(sinh(f*x+e)^5*(a+b*sinh(f*x+e)^2)^p,x, algorithm="maxima")
 

output

integrate((b*sinh(f*x + e)^2 + a)^p*sinh(f*x + e)^5, x)
 

3.2.31.8 Giac [F]

\[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{p} \sinh \left (f x + e\right )^{5} \,d x } \]

input

integrate(sinh(f*x+e)^5*(a+b*sinh(f*x+e)^2)^p,x, algorithm="giac")
 

output

integrate((b*sinh(f*x + e)^2 + a)^p*sinh(f*x + e)^5, x)
 

3.2.31.9 Mupad [F(-1)]

Timed out. \[ \int \sinh ^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^p \, dx=\int {\mathrm {sinh}\left (e+f\,x\right )}^5\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]

input

int(sinh(e + f*x)^5*(a + b*sinh(e + f*x)^2)^p,x)
 

output

int(sinh(e + f*x)^5*(a + b*sinh(e + f*x)^2)^p, x)