3.2.16 \(\int \frac {\sinh ^5(e+f x)}{(a+b \sinh ^2(e+f x))^{5/2}} \, dx\) [116]

Optimal. Leaf size=143 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{b^{5/2} f}-\frac {a (3 a-5 b) \cosh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {a \cosh (e+f x) \sinh ^2(e+f x)}{3 (a-b) b f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3265, 424, 393, 223, 212} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a+b \cosh ^2(e+f x)-b}}\right )}{b^{5/2} f}-\frac {a (3 a-5 b) \cosh (e+f x)}{3 b^2 f (a-b)^2 \sqrt {a+b \cosh ^2(e+f x)-b}}-\frac {a \sinh ^2(e+f x) \cosh (e+f x)}{3 b f (a-b) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

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 393

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

Rule 424

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

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-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]

Rubi steps

\begin {align*} \int \frac {\sinh ^5(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a-b+b x^2\right )^{5/2}} \, dx,x,\cosh (e+f x)\right )}{f}\\ &=-\frac {a \cosh (e+f x) \sinh ^2(e+f x)}{3 (a-b) b f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-a+3 b+3 (a-b) x^2}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\cosh (e+f x)\right )}{3 (a-b) b f}\\ &=-\frac {a (3 a-5 b) \cosh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {a \cosh (e+f x) \sinh ^2(e+f x)}{3 (a-b) b f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-b+b x^2}} \, dx,x,\cosh (e+f x)\right )}{b^2 f}\\ &=-\frac {a (3 a-5 b) \cosh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {a \cosh (e+f x) \sinh ^2(e+f x)}{3 (a-b) b f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{b^2 f}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \cosh (e+f x)}{\sqrt {a-b+b \cosh ^2(e+f x)}}\right )}{b^{5/2} f}-\frac {a (3 a-5 b) \cosh (e+f x)}{3 (a-b)^2 b^2 f \sqrt {a-b+b \cosh ^2(e+f x)}}-\frac {a \cosh (e+f x) \sinh ^2(e+f x)}{3 (a-b) b f \left (a-b+b \cosh ^2(e+f x)\right )^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.60, size = 130, normalized size = 0.91 \begin {gather*} \frac {-\frac {2 \sqrt {2} a \cosh (e+f x) \left (3 a^2-7 a b+3 b^2+(2 a-3 b) b \cosh (2 (e+f x))\right )}{3 (a-b)^2 b^2 (2 a-b+b \cosh (2 (e+f x)))^{3/2}}+\frac {\log \left (\sqrt {2} \sqrt {b} \cosh (e+f x)+\sqrt {2 a-b+b \cosh (2 (e+f x))}\right )}{b^{5/2}}}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*Sqrt[2]*a*Cosh[e + f*x]*(3*a^2 - 7*a*b + 3*b^2 + (2*a - 3*b)*b*Cosh[2*(e + f*x)]))/(3*(a - b)^2*b^2*(2*a
- b + b*Cosh[2*(e + f*x)])^(3/2)) + Log[Sqrt[2]*Sqrt[b]*Cosh[e + f*x] + Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]]/b
^(5/2))/f

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Maple [A]
time = 1.40, size = 230, normalized size = 1.61

method	result	size
default	\(\frac {\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (\frac {\ln \left (\frac {\frac {a}{2}+\frac {b}{2}+b \left (\sinh ^{2}\left (f x +e \right )\right )}{\sqrt {b}}+\sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\right )}{2 b^{\frac {5}{2}}}-\frac {2 a \left (\cosh ^{2}\left (f x +e \right )\right )}{b^{2} \left (a -b \right ) \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}}+\frac {a^{2} \left (2 b \left (\sinh ^{2}\left (f x +e \right )\right )+3 a -b \right ) \left (\cosh ^{2}\left (f x +e \right )\right )}{3 b^{2} \sqrt {\left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (\cosh ^{2}\left (f x +e \right )\right )}\, \left (a +b \left (\sinh ^{2}\left (f x +e \right )\right )\right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(f*x+e)^5/(a+b*sinh(f*x+e)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(1/2/b^(5/2)*ln((1/2*a+1/2*b+b*sinh(f*x+e)^2)/b^(1/2)+((a+b*sinh(f*x
+e)^2)*cosh(f*x+e)^2)^(1/2))-2*a/b^2*cosh(f*x+e)^2/(a-b)/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)+1/3*a^2/b^2
*(2*b*sinh(f*x+e)^2+3*a-b)*cosh(f*x+e)^2/((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)/(a+b*sinh(f*x+e)^2)/(a^2-2*
a*b+b^2))/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3631 vs. \(2 (129) = 258\).
time = 0.73, size = 7938, normalized size = 55.51 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)*sinh(f*x + e)^
7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^6 + 4*(2
*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*
b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e))*sinh(f*x + e)^5
+ 2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(
f*x + e)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(
f*x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^5 + 10*(2
*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^3 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(
f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^2 + 4*(7*(a^2*b^2 - 2*a*b^3
+ b^4)*cosh(f*x + e)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*
b^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b^
2 - 2*a*b^3 + b^4)*cosh(f*x + e)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^5 + (8*a^4 - 24*a^3
*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + (2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e))*sin
h(f*x + e))*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 + e)^4 + 18*a^2*cosh(f*x + e)^2 + 4*a*b - b^2)*sinh(f*x + e)^2 + b^2 + 2*
(3*a^2*cosh(f*x + e)^5 + 6*a^2*cosh(f*x + e)^3 + (4*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*sqrt((b*c
osh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)
^2)) + 4*(2*a^2*b*cosh(f*x + e)^7 + 3*(a^3 + a^2*b)*cosh(f*x + e)^5 + (9*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^
3 + (3*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e))/(cosh(f*x + e)^6 + 6*cosh(f*x + e)^5*sinh(f*x + e) + 15*cosh
(f*x + e)^4*sinh(f*x + e)^2 + 20*cosh(f*x + e)^3*sinh(f*x + e)^3 + 15*cosh(f*x + e)^2*sinh(f*x + e)^4 + 6*cosh
(f*x + e)*sinh(f*x + e)^5 + sinh(f*x + e)^6)) + 3*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^8 + 8*(a^2*b^2 - 2*
a*b^3 + b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (a^2*b^2 - 2*a*b^3 + b^4)*sinh(f*x + e)^8 + 4*(2*a^3*b - 5*a^2*b^
2 + 4*a*b^3 - b^4)*cosh(f*x + e)^6 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh
(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^3 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b
^3 - b^4)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^
4 + 2*(35*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^4 + 8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4 + 30*(2
*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + a^2*b^2 - 2*a*b^3 + b^4 + 8*(7*(a^2*b^2
- 2*a*b^3 + b^4)*cosh(f*x + e)^5 + 10*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e)^3 + (8*a^4 - 24*a^3
*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*c
osh(f*x + e)^2 + 4*(7*(a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^6 + 15*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4)*cos
h(f*x + e)^4 + 2*a^3*b - 5*a^2*b^2 + 4*a*b^3 - b^4 + 3*(8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh
(f*x + e)^2)*sinh(f*x + e)^2 + 8*((a^2*b^2 - 2*a*b^3 + b^4)*cosh(f*x + e)^7 + 3*(2*a^3*b - 5*a^2*b^2 + 4*a*b^3
- b^4)*cosh(f*x + e)^5 + (8*a^4 - 24*a^3*b + 27*a^2*b^2 - 14*a*b^3 + 3*b^4)*cosh(f*x + e)^3 + (2*a^3*b - 5*a^
2*b^2 + 4*a*b^3 - b^4)*cosh(f*x + e))*sinh(f*x + e))*sqrt(b)*log(-(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(
f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(a - b)*cosh(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + a - b)*sinh(f*x + e)^2 +
sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 - 1)*sqrt(b)*sqrt((b*cosh(f*x + e)
^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*(b*
cosh(f*x + e)^3 + (a - b)*cosh(f*x + e))*sinh(f...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.98Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (e+f\,x\right )}^5}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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