Integrand size = 25, antiderivative size = 267 \[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {2 (a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {a \coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}+\frac {2 (a-2 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {(a-3 b) b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \]
2/3*(a-2*b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f-1/3*a*coth(f*x+e)*csch (f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/f+2/3*(a-2*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)/f/(sech(f*x+e)^2*(a+b *sinh(f*x+e)^2)/a)^(1/2)-1/3*(a-3*b)*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/f/(sech(f*x+e)^2*(a+b*sinh(f*x +e)^2)/a)^(1/2)-2/3*(a-2*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Result contains complex when optimal does not.
Time = 3.03 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.80 \[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {\frac {\left (-8 a^2+13 a b-6 b^2+2 \left (2 a^2-7 a b+4 b^2\right ) \cosh (2 (e+f x))+(a-2 b) b \cosh (4 (e+f x))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}+4 i a (a-2 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i \left (2 a^2-5 a b+3 b^2\right ) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{6 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
(((-8*a^2 + 13*a*b - 6*b^2 + 2*(2*a^2 - 7*a*b + 4*b^2)*Cosh[2*(e + f*x)] + (a - 2*b)*b*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + ( 4*I)*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] - (2*I)*(2*a^2 - 5*a*b + 3*b^2)*Sqrt[(2*a - b + b*Cosh[2*(e + f *x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*f*Sqrt[2*a - b + b*Cosh[2*(e + f* x)]])
Time = 0.52 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.26, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3667, 376, 25, 445, 25, 27, 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.
| \(\displaystyle \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\sin (i e+i f x)^4}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^4(e+f x) \left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\sqrt {\sinh ^2(e+f x)+1}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 376 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \int -\frac {\text {csch}^2(e+f x) \left ((a-3 b) b \sinh ^2(e+f x)+2 a (a-2 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {1}{3} \int \frac {\text {csch}^2(e+f x) \left ((a-3 b) b \sinh ^2(e+f x)+2 a (a-2 b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (\frac {\int -\frac {a b \left (2 (a-2 b) \sinh ^2(e+f x)+a-3 b\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}+2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}-\frac {\int \frac {a b \left (2 (a-2 b) \sinh ^2(e+f x)+a-3 b\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}-b \int \frac {2 (a-2 b) \sinh ^2(e+f x)+a-3 b}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}-b \left ((a-3 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+2 (a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}-b \left (2 (a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(a-3 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}-b \left (2 (a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(a-3 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (2 (a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}-b \left (\frac {(a-3 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+2 (a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )\right )-\frac {1}{3} a \sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-1/3*(a*Csch[e + f*x]^3*Sqrt[1 + Sin h[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2]) + (2*(a - 2*b)*Csch[e + f*x]*Sq rt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2] - b*(((a - 3*b)*Ellipt icF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2)) ]) + 2*(a - 2*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))]))))/3))/f
3.1.86.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[c*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1 )/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^ 2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b*c - a*d)*(m + 1) + 2*c*(b*c*(p + 1) + a* d*(q - 1)) + d*((b*c - a*d)*(m + 1) + 2*b*c*(p + q))*x^2, x], x], x] /; Fre eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && LtQ[m, -1] & & IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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]
Time = 1.16 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.70
| method | result | size |
| default | \(\frac {2 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{6}-4 \sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{6}+b \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \sinh \left (f x +e \right )^{3}-\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \sinh \left (f x +e \right )^{3}-2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \sinh \left (f x +e \right )^{3}+4 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \sinh \left (f x +e \right )^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{4}-3 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{4}-4 \sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{4}+\sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{2}-5 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{2}-\sqrt {-\frac {b}{a}}\, a^{2}}{3 \sinh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(454\) |
1/3*(2*(-b/a)^(1/2)*a*b*sinh(f*x+e)^6-4*(-b/a)^(1/2)*b^2*sinh(f*x+e)^6+b*( (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*sinh(f*x+e)^3-((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)^3-2*((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)^3+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)^3+2*(-b/a)^(1/2)*a^2*sinh(f*x+e)^4-3*(-b/a)^(1/ 2)*a*b*sinh(f*x+e)^4-4*(-b/a)^(1/2)*b^2*sinh(f*x+e)^4+(-b/a)^(1/2)*a^2*sin h(f*x+e)^2-5*(-b/a)^(1/2)*a*b*sinh(f*x+e)^2-(-b/a)^(1/2)*a^2)/sinh(f*x+e)^ 3/(-b/a)^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 2222 vs. \(2 (271) = 542\).
Time = 0.12 (sec) , antiderivative size = 2222, normalized size of antiderivative = 8.32 \[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
-2/3*(((2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^6 + 6*(2*a^2 - 5*a*b + 2*b^2) *cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^6 - 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 - 5*a*b + 2*b^2)* cosh(f*x + e)^2 - 2*a^2 + 5*a*b - 2*b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 - 5 *a*b + 2*b^2)*cosh(f*x + e)^3 - 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e))*s inh(f*x + e)^3 + 3*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^4 - 6*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^ 2 + 2*a^2 - 5*a*b + 2*b^2)*sinh(f*x + e)^2 - 2*a^2 + 5*a*b - 2*b^2 + 6*((2 *a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^5 - 2*(2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e)^3 + (2*a^2 - 5*a*b + 2*b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b - 2*b^2)*cosh(f*x + e)^6 + 6*(a*b - 2*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b - 2*b^2)*sinh(f*x + e)^6 - 3*(a*b - 2*b^2)*cosh(f*x + e)^4 + 3*(5*(a* b - 2*b^2)*cosh(f*x + e)^2 - a*b + 2*b^2)*sinh(f*x + e)^4 + 4*(5*(a*b - 2* b^2)*cosh(f*x + e)^3 - 3*(a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3* (a*b - 2*b^2)*cosh(f*x + e)^2 + 3*(5*(a*b - 2*b^2)*cosh(f*x + e)^4 - 6*(a* b - 2*b^2)*cosh(f*x + e)^2 + a*b - 2*b^2)*sinh(f*x + e)^2 - a*b + 2*b^2 + 6*((a*b - 2*b^2)*cosh(f*x + e)^5 - 2*(a*b - 2*b^2)*cosh(f*x + e)^3 + (a*b - 2*b^2)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt ((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt ((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^...
Timed out. \[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {csch}\left (f x + e\right )^{4} \,d x } \]
Exception generated. \[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[8,6,0]%%%}+%%%{%%{[8,0]:[1,0,%%%{-1,[1]%%%}]%%},[7,6,0] %%%}+%%%{
Timed out. \[ \int \text {csch}^4(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {sinh}\left (e+f\,x\right )}^4} \,d x \]