Integrand size = 25, antiderivative size = 244 \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {(a-b) \cosh (e+f x) \sinh (e+f x)}{a b f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(2 a-b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a b^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{a b^2 f} \]
-(a-b)*cosh(f*x+e)*sinh(f*x+e)/a/b/f/(a+b*sinh(f*x+e)^2)^(1/2)-(2*a-b)*(1/ (1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)/(1+ sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/ a/b^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+(1/(1+sinh(f*x+e)^2))^ (1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2 ),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a/b/f/(sech(f*x+e)^ 2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+(2*a-b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+ e)/a/b^2/f
Result contains complex when optimal does not.
Time = 0.72 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.64 \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {-2 i a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+(a-b) \left (4 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )-\sqrt {2} b \sinh (2 (e+f x))\right )}{2 a b^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
((-2*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (a - b)*((4*I)*a*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*E llipticF[I*(e + f*x), b/a] - Sqrt[2]*b*Sinh[2*(e + f*x)]))/(2*a*b^2*f*Sqrt [2*a - b + b*Cosh[2*(e + f*x)]])
Time = 0.42 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3671, 315, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i e+i f x)^4}{\left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\left (\sinh ^2(e+f x)+1\right )^{3/2}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {(2 a-b) \sinh ^2(e+f x)+a}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{a b \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {a \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+(2 a-b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{a b \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {(2 a-b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{a b \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {(2 a-b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{a b \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+(2 a-b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a b}-\frac {(a-b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{a b \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-(((a - b)*Sinh[e + f*x]*Sqrt[1 + Si nh[e + f*x]^2])/(a*b*Sqrt[a + b*Sinh[e + f*x]^2])) + ((EllipticF[ArcTan[Si nh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(Sqrt[1 + Sinh[e + f*x ]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]) + (2*a - b)* ((Sinh[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x]^2]) - (EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2]) /(b*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])))/(a*b)))/f
3.4.87.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 2.26 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.37
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right ) a -\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right ) b +a \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b -2 \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a +\sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b}{b \sqrt {-\frac {b}{a}}\, a \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(334\) |
-((-b/a)^(1/2)*cosh(f*x+e)^2*sinh(f*x+e)*a-(-b/a)^(1/2)*cosh(f*x+e)^2*sinh (f*x+e)*b+a*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ellipt icF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))-(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2 )*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b- 2*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f *x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh( f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b)/b/(-b/a )^(1/2)/a/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
\[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cosh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(b*sinh(f*x + e)^2 + a)*cosh(f*x + e)^4/(b^2*sinh(f*x + e)^4 + 2*a*b*sinh(f*x + e)^2 + a^2), x)
Timed out. \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cosh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (260) = 520\).
Time = 2.44 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.63 \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {{\left (\frac {{\left (a^{4} b^{2} e^{\left (7 \, e\right )} - 3 \, a^{3} b^{3} e^{\left (7 \, e\right )} + 3 \, a^{2} b^{4} e^{\left (7 \, e\right )} - a b^{5} e^{\left (7 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{a^{4} b^{3} e^{\left (4 \, e\right )} - 2 \, a^{3} b^{4} e^{\left (4 \, e\right )} + a^{2} b^{5} e^{\left (4 \, e\right )}} + \frac {2 \, a^{5} b e^{\left (5 \, e\right )} - 7 \, a^{4} b^{2} e^{\left (5 \, e\right )} + 9 \, a^{3} b^{3} e^{\left (5 \, e\right )} - 5 \, a^{2} b^{4} e^{\left (5 \, e\right )} + a b^{5} e^{\left (5 \, e\right )}}{a^{4} b^{3} e^{\left (4 \, e\right )} - 2 \, a^{3} b^{4} e^{\left (4 \, e\right )} + a^{2} b^{5} e^{\left (4 \, e\right )}}\right )} e^{\left (f x\right )}}{\sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b} f} - \frac {{\left (\frac {{\left (3 \, a e^{\left (2 \, e\right )} - 2 \, b e^{\left (2 \, e\right )}\right )} \log \left ({\left | {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} \sqrt {b} + 2 \, a - b \right |}\right )}{\sqrt {b}} + \frac {2 \, {\left (2 \, a^{2} e^{\left (2 \, e\right )} - a b e^{\left (2 \, e\right )} - 2 \, b^{2} e^{\left (2 \, e\right )}\right )} \arctan \left (-\frac {\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {2 \, {\left (2 \, {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} a^{2} e^{\left (2 \, e\right )} - {\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )} a b e^{\left (2 \, e\right )} + a b^{\frac {3}{2}} e^{\left (2 \, e\right )}\right )}}{{\left ({\left (\sqrt {b} e^{\left (2 \, f x + 2 \, e\right )} - \sqrt {b e^{\left (4 \, f x + 4 \, e\right )} + 4 \, a e^{\left (2 \, f x + 2 \, e\right )} - 2 \, b e^{\left (2 \, f x + 2 \, e\right )} + b}\right )}^{2} - b\right )} b}\right )} e^{\left (-e\right )}}{4 \, a b f^{2}} \]
-((a^4*b^2*e^(7*e) - 3*a^3*b^3*e^(7*e) + 3*a^2*b^4*e^(7*e) - a*b^5*e^(7*e) )*e^(2*f*x)/(a^4*b^3*e^(4*e) - 2*a^3*b^4*e^(4*e) + a^2*b^5*e^(4*e)) + (2*a ^5*b*e^(5*e) - 7*a^4*b^2*e^(5*e) + 9*a^3*b^3*e^(5*e) - 5*a^2*b^4*e^(5*e) + a*b^5*e^(5*e))/(a^4*b^3*e^(4*e) - 2*a^3*b^4*e^(4*e) + a^2*b^5*e^(4*e)))*e ^(f*x)/(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)*f) - 1/4*((3*a*e^(2*e) - 2*b*e^(2*e))*log(abs((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) + 2*a - b))/sqrt(b) + 2*(2*a^2*e^(2*e) - a*b*e^(2*e) - 2*b^2*e ^(2*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) - 2*(2*(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))*a^2*e^(2*e) - (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))*a*b*e^(2*e ) + a*b^(3/2)*e^(2*e))/(((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))^2 - b)*b))*e^(-e)/(a*b* f^2)
Timed out. \[ \int \frac {\cosh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cosh}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]