3.1.76 \(\int \sinh ^3(e+f x) (a+b \sinh ^2(e+f x))^{3/2} \, dx\) [76]

3.1.76.1 Optimal result

Integrand size = 25, antiderivative size = 177 \[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=-\frac {(a-b)^2 (a+5 b) \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{16 b^{3/2} f}-\frac {(a-b) (a+5 b) \cosh (e+f x) \sqrt {a-b+b \cosh ^2(e+f x)}}{16 b f}-\frac {(a+5 b) \cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}{24 b f}+\frac {\cosh (e+f x) \left (a-b+b \cosh ^2(e+f x)\right )^{5/2}}{6 b f} \]

output

-1/16*(a-b)^2*(a+5*b)*arctanh(cosh(f*x+e)*b^(1/2)/(a-b+b*cosh(f*x+e)^2)^(1 
/2))/b^(3/2)/f-1/24*(a+5*b)*cosh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(3/2)/b/f+1/ 
6*cosh(f*x+e)*(a-b+b*cosh(f*x+e)^2)^(5/2)/b/f-1/16*(a-b)*(a+5*b)*cosh(f*x+ 
e)*(a-b+b*cosh(f*x+e)^2)^(1/2)/b/f
 

3.1.76.2 Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.85 \[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {2} \sqrt {b} \sqrt {2 a-b+b \cosh (2 (e+f x))} \left (\left (6 a^2-51 a b+37 b^2\right ) \cosh (e+f x)+b ((7 a-8 b) \cosh (3 (e+f x))+b \cosh (5 (e+f x)))\right )-12 (a-b)^2 (a+5 b) \log \left (\sqrt {2} \sqrt {b} \cosh (e+f x)+\sqrt {2 a-b+b \cosh (2 (e+f x))}\right )}{192 b^{3/2} f} \]

input

Integrate[Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

output

(Sqrt[2]*Sqrt[b]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*((6*a^2 - 51*a*b + 37 
*b^2)*Cosh[e + f*x] + b*((7*a - 8*b)*Cosh[3*(e + f*x)] + b*Cosh[5*(e + f*x 
)])) - 12*(a - b)^2*(a + 5*b)*Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a 
 - b + b*Cosh[2*(e + f*x)]]])/(192*b^(3/2)*f)
 

3.1.76.3 Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 26, 3665, 299, 211, 211, 224, 219}

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.

\(\displaystyle \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int i \sin (i e+i f x)^3 \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \sin (i e+i f x)^3 \left (a-b \sin (i e+i f x)^2\right )^{3/2}dx\)

\(\Big \downarrow \) 3665

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

\(\Big \downarrow \) 299

\(\displaystyle -\frac {\frac {(a+5 b) \int \left (b \cosh ^2(e+f x)+a-b\right )^{3/2}d\cosh (e+f x)}{6 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b}}{f}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {\frac {(a+5 b) \left (\frac {3}{4} (a-b) \int \sqrt {b \cosh ^2(e+f x)+a-b}d\cosh (e+f x)+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}\right )}{6 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b}}{f}\)

\(\Big \downarrow \) 211

\(\displaystyle -\frac {\frac {(a+5 b) \left (\frac {3}{4} (a-b) \left (\frac {1}{2} (a-b) \int \frac {1}{\sqrt {b \cosh ^2(e+f x)+a-b}}d\cosh (e+f x)+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}\right )}{6 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b}}{f}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {\frac {(a+5 b) \left (\frac {3}{4} (a-b) \left (\frac {1}{2} (a-b) \int \frac {1}{1-\frac {b \cosh ^2(e+f x)}{b \cosh ^2(e+f x)+a-b}}d\frac {\cosh (e+f x)}{\sqrt {b \cosh ^2(e+f x)+a-b}}+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}\right )}{6 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {(a+5 b) \left (\frac {3}{4} (a-b) \left (\frac {(a-b) \text {arctanh}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{2 \sqrt {b}}+\frac {1}{2} \cosh (e+f x) \sqrt {a+b \cosh ^2(e+f x)-b}\right )+\frac {1}{4} \cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}\right )}{6 b}-\frac {\cosh (e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{5/2}}{6 b}}{f}\)

input

Int[Sinh[e + f*x]^3*(a + b*Sinh[e + f*x]^2)^(3/2),x]
 

output

-((-1/6*(Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(5/2))/b + ((a + 5*b)*( 
(Cosh[e + f*x]*(a - b + b*Cosh[e + f*x]^2)^(3/2))/4 + (3*(a - b)*(((a - b) 
*ArcTanh[(Sqrt[b]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(2*Sqrt 
[b]) + (Cosh[e + f*x]*Sqrt[a - b + b*Cosh[e + f*x]^2])/2))/4))/(6*b))/f)
 

3.1.76.3.1 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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !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 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]
 

3.1.76.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs. \(2(157)=314\).

Time = 0.08 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.73

method	result	size
default	\(-\frac {\sqrt {\left (a +b \sinh \left (f x +e \right )^{2}\right ) \cosh \left (f x +e \right )^{2}}\, \left (-16 b^{\frac {7}{2}} \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \cosh \left (f x +e \right )^{4}-4 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, b^{\frac {5}{2}} \left (-13 b +7 a \right ) \cosh \left (f x +e \right )^{2}-66 b^{\frac {7}{2}} \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}+72 a \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, b^{\frac {5}{2}}-6 a^{2} \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, b^{\frac {3}{2}}+3 \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a^{3} b +9 \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a^{2} b^{2}-27 b^{3} \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right ) a +15 b^{4} \ln \left (\frac {2 b \cosh \left (f x +e \right )^{2}+2 \sqrt {b \cosh \left (f x +e \right )^{4}+\left (a -b \right ) \cosh \left (f x +e \right )^{2}}\, \sqrt {b}+a -b}{2 \sqrt {b}}\right )\right )}{96 b^{\frac {5}{2}} \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\)	\(483\)

input

int(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x,method=_RETURNVERBOSE)
 

output

-1/96*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(-16*b^(7/2)*(b*cosh(f*x+e 
)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*cosh(f*x+e)^4-4*(b*cosh(f*x+e)^4+(a-b)*cosh 
(f*x+e)^2)^(1/2)*b^(5/2)*(-13*b+7*a)*cosh(f*x+e)^2-66*b^(7/2)*(b*cosh(f*x+ 
e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+72*a*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2) 
^(1/2)*b^(5/2)-6*a^2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(3/2)+3 
*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b 
^(1/2)+a-b)/b^(1/2))*a^3*b+9*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+ 
(a-b)*cosh(f*x+e)^2)^(1/2)*b^(1/2)+a-b)/b^(1/2))*a^2*b^2-27*b^3*ln(1/2*(2* 
b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)*b^(1/2)+a-b) 
/b^(1/2))*a+15*b^4*ln(1/2*(2*b*cosh(f*x+e)^2+2*(b*cosh(f*x+e)^4+(a-b)*cosh 
(f*x+e)^2)^(1/2)*b^(1/2)+a-b)/b^(1/2)))/b^(5/2)/cosh(f*x+e)/(a+b*sinh(f*x+ 
e)^2)^(1/2)/f
 

3.1.76.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1965 vs. \(2 (157) = 314\).

Time = 0.41 (sec) , antiderivative size = 4608, normalized size of antiderivative = 26.03 \[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]

input

integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="fricas")
 

output

[1/384*(6*((a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^6 + 6*(a^3 + 3* 
a^2*b - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^5*sinh(f*x + e) + 15*(a^3 + 3*a^2*b 
 - 9*a*b^2 + 5*b^3)*cosh(f*x + e)^4*sinh(f*x + e)^2 + 20*(a^3 + 3*a^2*b - 
9*a*b^2 + 5*b^3)*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*(a^3 + 3*a^2*b - 9*a 
*b^2 + 5*b^3)*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*(a^3 + 3*a^2*b - 9*a*b^2 
 + 5*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3 
)*sinh(f*x + e)^6)*sqrt(b)*log((a^2*b*cosh(f*x + e)^8 + 8*a^2*b*cosh(f*x + 
 e)*sinh(f*x + e)^7 + a^2*b*sinh(f*x + e)^8 + 2*(a^3 + a^2*b)*cosh(f*x + e 
)^6 + 2*(14*a^2*b*cosh(f*x + e)^2 + a^3 + a^2*b)*sinh(f*x + e)^6 + 4*(14*a 
^2*b*cosh(f*x + e)^3 + 3*(a^3 + a^2*b)*cosh(f*x + e))*sinh(f*x + e)^5 + (9 
*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^4 + (70*a^2*b*cosh(f*x + e)^4 + 9*a^ 
2*b - 4*a*b^2 + b^3 + 30*(a^3 + a^2*b)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 
4*(14*a^2*b*cosh(f*x + e)^5 + 10*(a^3 + a^2*b)*cosh(f*x + e)^3 + (9*a^2*b 
- 4*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + b^3 + 2*(3*a*b^2 - b^3)* 
cosh(f*x + e)^2 + 2*(14*a^2*b*cosh(f*x + e)^6 + 15*(a^3 + a^2*b)*cosh(f*x 
+ e)^4 + 3*a*b^2 - b^3 + 3*(9*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh 
(f*x + e)^2 - sqrt(2)*(a^2*cosh(f*x + e)^6 + 6*a^2*cosh(f*x + e)*sinh(f*x 
+ e)^5 + a^2*sinh(f*x + e)^6 + 3*a^2*cosh(f*x + e)^4 + 3*(5*a^2*cosh(f*x + 
 e)^2 + a^2)*sinh(f*x + e)^4 + 4*(5*a^2*cosh(f*x + e)^3 + 3*a^2*cosh(f*x + 
 e))*sinh(f*x + e)^3 + (4*a*b - b^2)*cosh(f*x + e)^2 + (15*a^2*cosh(f*x...
 

3.1.76.6 Sympy [F(-1)]

Timed out. \[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]

input

integrate(sinh(f*x+e)**3*(a+b*sinh(f*x+e)**2)**(3/2),x)
 

output

Timed out
 

3.1.76.7 Maxima [F]

\[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sinh \left (f x + e\right )^{3} \,d x } \]

input

integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="maxima")
 

output

integrate((b*sinh(f*x + e)^2 + a)^(3/2)*sinh(f*x + e)^3, x)
 

3.1.76.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1578 vs. \(2 (157) = 314\).

Time = 1.06 (sec) , antiderivative size = 1578, normalized size of antiderivative = 8.92 \[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]

input

integrate(sinh(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2),x, algorithm="giac")
 

output

1/384*(((b*e^(2*f*x + 10*e) + (7*a*b^2*e^(14*e) - 8*b^3*e^(14*e))*e^(-6*e) 
/b^2)*e^(2*f*x) + (6*a^2*b*e^(12*e) - 51*a*b^2*e^(12*e) + 37*b^3*e^(12*e)) 
*e^(-6*e)/b^2)*sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x 
 + 2*e) + b) + 24*(a^3*e^(6*e) + 3*a^2*b*e^(6*e) - 9*a*b^2*e^(6*e) + 5*b^3 
*e^(6*e))*arctan(-(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a* 
e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))/sqrt(-b))/(sqrt(-b)*b) + 12*(a 
^3*e^(6*e) + 3*a^2*b*e^(6*e) - 9*a*b^2*e^(6*e) + 5*b^3*e^(6*e))*log(abs((s 
qrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2* 
b*e^(2*f*x + 2*e) + b))*sqrt(b) + 2*a - b))/b^(3/2) - 2*(12*(sqrt(b)*e^(2* 
f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 
 2*e) + b))^5*a^3*e^(6*e) - 108*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x 
 + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b*e^(6*e) 
+ 132*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2 
*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a*b^2*e^(6*e) - 45*(sqrt(b)*e^(2*f*x + 2 
*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + 
 b))^5*b^3*e^(6*e) + 48*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) 
+ 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^2*b^(3/2)*e^(6*e) - 
120*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e 
) - 2*b*e^(2*f*x + 2*e) + b))^4*a*b^(5/2)*e^(6*e) + 63*(sqrt(b)*e^(2*f*x + 
 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2...
 

3.1.76.9 Mupad [F(-1)]

Timed out. \[ \int \sinh ^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {sinh}\left (e+f\,x\right )}^3\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]

input

int(sinh(e + f*x)^3*(a + b*sinh(e + f*x)^2)^(3/2),x)
 

output

int(sinh(e + f*x)^3*(a + b*sinh(e + f*x)^2)^(3/2), x)