Integrand size = 25, antiderivative size = 244 \[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=-\frac {a \cosh (e+f x) \sinh (e+f x)}{3 (a-b) b f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {2 \sqrt {a} (a-2 b) \cosh (e+f x) E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 (a-b)^2 b^{3/2} f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-3 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a (a-b)^2 b f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}} \]
-1/3*a*cosh(f*x+e)*sinh(f*x+e)/(a-b)/b/f/(a+b*sinh(f*x+e)^2)^(3/2)+2/3*(a- 2*b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)^2/a))^(1/2)*(1+b*sinh(f*x+e)^2/a)^(1/ 2)*EllipticE(sinh(f*x+e)*b^(1/2)/a^(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/2),(1-a/ b)^(1/2))*a^(1/2)/(a-b)^2/b^(3/2)/f/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2))^ (1/2)/(a+b*sinh(f*x+e)^2)^(1/2)-1/3*(a-3*b)*(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/(a-b)^2/b/f/(sech(f*x+e)^ 2*(a+b*sinh(f*x+e)^2)/a)^(1/2)
Result contains complex when optimal does not.
Time = 1.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.81 \[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 i a^2 (a-2 b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i a \left (2 a^2-5 a b+3 b^2\right ) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )-\sqrt {2} b \left (-a^2+4 a b-2 b^2-(a-2 b) b \cosh (2 (e+f x))\right ) \sinh (2 (e+f x))}{3 (a-b)^2 b^2 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \]
((2*I)*a^2*(a - 2*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticE[I *(e + f*x), b/a] - I*a*(2*a^2 - 5*a*b + 3*b^2)*((2*a - b + b*Cosh[2*(e + f *x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] - Sqrt[2]*b*(-a^2 + 4*a*b - 2*b ^2 - (a - 2*b)*b*Cosh[2*(e + f*x)])*Sinh[2*(e + f*x)])/(3*(a - b)^2*b^2*f* (2*a - b + b*Cosh[2*(e + f*x)])^(3/2))
Time = 0.43 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 3667, 372, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (i e+i f x)^4}{\left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\sinh ^4(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {(2 a-3 b) \sinh ^2(e+f x)+a}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 b (a-b)}-\frac {a \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 b (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 a (a-2 b) \int \frac {\sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}-\frac {(a-3 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}}{3 b (a-b)}-\frac {a \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 b (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 \sqrt {a} (a-2 b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {b} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-3 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}}{3 b (a-b)}-\frac {a \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 b (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 \sqrt {a} (a-2 b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {b} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-3 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a (a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{3 b (a-b)}-\frac {a \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 b (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-1/3*(a*Sinh[e + f*x]*Sqrt[1 + Sinh[ e + f*x]^2])/((a - b)*b*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((2*Sqrt[a]*(a - 2*b)*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b]*Sqrt[1 + Sinh[e + f*x]^2])/((a - b)*Sqrt[b]*Sqrt[(a*(1 + Sinh[e + f*x]^2))/(a + b*S inh[e + f*x]^2)]*Sqrt[a + b*Sinh[e + f*x]^2]) - ((a - 3*b)*EllipticF[ArcTa n[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))] ))/(3*(a - b)*b)))/f
3.2.21.3.1 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[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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff^(m + 1 )*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[x^m*((a + b*ff^2*x^2) ^p/Sqrt[1 - ff^2*x^2]), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(658\) vs. \(2(314)=628\).
Time = 1.21 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.70
| method | result | size |
| default | \(\frac {2 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{5}-4 \sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{5}+\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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 ) a b \sinh \left (f x +e \right )^{2}-\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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} \sinh \left (f x +e \right )^{2}-2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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 b \sinh \left (f x +e \right )^{2}+4 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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^{2} \sinh \left (f x +e \right )^{2}+\sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{3}-\sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{3}-4 \sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{3}+a^{2} \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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 )-a \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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 {a +b \sinh \left (f x +e \right )^{2}}{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^{2}+4 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{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 b +\sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )-3 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )}{3 \sqrt {-\frac {b}{a}}\, \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \left (a -b \right )^{2} b \cosh \left (f x +e \right ) f}\) | \(659\) |
1/3*(2*(-b/a)^(1/2)*a*b*sinh(f*x+e)^5-4*(-b/a)^(1/2)*b^2*sinh(f*x+e)^5+((a +b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b /a)^(1/2),(a/b)^(1/2))*a*b*sinh(f*x+e)^2-((a+b*sinh(f*x+e)^2)/a)^(1/2)*(co sh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2*sin h(f*x+e)^2-2*((a+b*sinh(f*x+e)^2)/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*sinh(f*x+e)^2+4*((a+b*sinh(f*x+ e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a /b)^(1/2))*b^2*sinh(f*x+e)^2+(-b/a)^(1/2)*a^2*sinh(f*x+e)^3-(-b/a)^(1/2)*a *b*sinh(f*x+e)^3-4*(-b/a)^(1/2)*b^2*sinh(f*x+e)^3+a^2*((a+b*sinh(f*x+e)^2) /a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^( 1/2))-a*((a+b*sinh(f*x+e)^2)/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*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh( f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a^2+4*((a+ b*sinh(f*x+e)^2)/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+(-b/a)^(1/2)*a^2*sinh(f*x+e)-3*(-b/a)^(1/2)*a*b* sinh(f*x+e))/(-b/a)^(1/2)/(a+b*sinh(f*x+e)^2)^(3/2)/(a-b)^2/b/cosh(f*x+e)/ f
Leaf count of result is larger than twice the leaf count of optimal. 4985 vs. \(2 (252) = 504\).
Time = 0.21 (sec) , antiderivative size = 4985, normalized size of antiderivative = 20.43 \[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
-2/3*(((2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^8 + 8*(2*a^2*b^2 - 5*a* b^3 + 2*b^4)*cosh(f*x + e)*sinh(f*x + e)^7 + (2*a^2*b^2 - 5*a*b^3 + 2*b^4) *sinh(f*x + e)^8 + 4*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e )^6 + 4*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4 + 7*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^6 + 8*(7*(2*a^2*b^2 - 5*a*b^3 + 2*b ^4)*cosh(f*x + e)^3 + 3*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e))*sinh(f*x + e)^5 + 2*(16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b ^4)*cosh(f*x + e)^4 + 2*(35*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^4 + 16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4 + 30*(4*a^3*b - 12*a^2 *b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 2*a^2*b^2 - 5*a *b^3 + 2*b^4 + 8*(7*(2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^5 + 10*(4* a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^3 + (16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4)*cosh(f*x + e))*sinh(f*x + e)^3 + 4*(4*a^ 3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^2 + 4*(7*(2*a^2*b^2 - 5* a*b^3 + 2*b^4)*cosh(f*x + e)^6 + 15*(4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^ 4)*cosh(f*x + e)^4 + 4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - 2*b^4 + 3*(16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 8*((2*a^2*b^2 - 5*a*b^3 + 2*b^4)*cosh(f*x + e)^7 + 3*(4*a^3*b - 12*a^2* b^2 + 9*a*b^3 - 2*b^4)*cosh(f*x + e)^5 + (16*a^4 - 56*a^3*b + 62*a^2*b^2 - 31*a*b^3 + 6*b^4)*cosh(f*x + e)^3 + (4*a^3*b - 12*a^2*b^2 + 9*a*b^3 - ...
Timed out. \[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\sinh \left (f x + e\right )^{4}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\sinh ^4(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {sinh}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]