Integrand size = 25, antiderivative size = 267 \[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {2 (a+b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {2 (a+b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (a+b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^2 f} \]
2/3*(a+b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/f-1/3*coth(f*x+e)*csch (f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/a/f+2/3*(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^2/f/(sech(f*x+e)^2* (a+b*sinh(f*x+e)^2)/a)^(1/2)-1/3*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^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e )^2)/a)^(1/2)-2/3*(a+b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/a^2/f
Result contains complex when optimal does not.
Time = 2.65 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.75 \[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {\frac {\left (-8 a^2+a b+3 b^2+\left (4 a^2-2 a b-4 b^2\right ) \cosh (2 (e+f x))+b (a+b) \cosh (4 (e+f x))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}+4 i a (a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a (2 a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{6 a^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
(((-8*a^2 + a*b + 3*b^2 + (4*a^2 - 2*a*b - 4*b^2)*Cosh[2*(e + f*x)] + b*(a + b)*Cosh[4*(e + f*x)])*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2] + (4*I)*a* (a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a ] - (2*I)*a*(2*a + b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I* (e + f*x), b/a])/(6*a^2*f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
Time = 0.51 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.25, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 3667, 382, 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.
| \(\displaystyle \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sin (i e+i f x)^4 \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^4(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 382 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int -\frac {\text {csch}^2(e+f x) \left (b \sinh ^2(e+f x)+2 (a+b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\int \frac {\text {csch}^2(e+f x) \left (b \sinh ^2(e+f x)+2 (a+b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {-\frac {\int -\frac {b \left (2 (a+b) \sinh ^2(e+f x)+a\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {\int \frac {b \left (2 (a+b) \sinh ^2(e+f x)+a\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {b \int \frac {2 (a+b) \sinh ^2(e+f x)+a}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {b \left (a \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+2 (a+b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {b \left (2 (a+b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {b \left (2 (a+b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (-\frac {\frac {b \left (\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+2 (a+b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )}{a}-\frac {2 (a+b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-1/3*(Csch[e + f*x]^3*Sqrt[1 + Sinh[ e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a - ((-2*(a + b)*Csch[e + f*x]*Sq rt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a + (b*((EllipticF[Ar cTan[Sinh[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)/(3*a)))/f
3.2.6.3.1 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[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/ (a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b* x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m + 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ b*c - a*d, 0] && 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 = 0.90 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.71
| method | result | size |
| default | \(\frac {2 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{6}+2 \sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{6}+b \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \sinh \left (f x +e \right )^{3}+2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \sinh \left (f x +e \right )^{3}-2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \sinh \left (f x +e \right )^{3}-2 \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \sinh \left (f x +e \right )^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{4}+3 \sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{4}+2 \sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{4}+\sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{2}+\sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{2}-\sqrt {-\frac {b}{a}}\, a^{2}}{3 a^{2} \sinh \left (f x +e \right )^{3} \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(456\) |
| risch | \(\text {Expression too large to display}\) | \(122054\) |
1/3*(2*(-b/a)^(1/2)*a*b*sinh(f*x+e)^6+2*(-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+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))*b^2*s inh(f*x+e)^3-2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ellipti cE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b*sinh(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))*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+2*(-b/a)^(1/2)*b^2*sinh(f*x+e)^4+(-b/a)^(1/2)*a^2*s inh(f*x+e)^2+(-b/a)^(1/2)*a*b*sinh(f*x+e)^2-(-b/a)^(1/2)*a^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. 2131 vs. \(2 (271) = 542\).
Time = 0.11 (sec) , antiderivative size = 2131, normalized size of antiderivative = 7.98 \[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
-2/3*(((2*a^2 + a*b - b^2)*cosh(f*x + e)^6 + 6*(2*a^2 + a*b - b^2)*cosh(f* x + e)*sinh(f*x + e)^5 + (2*a^2 + a*b - b^2)*sinh(f*x + e)^6 - 3*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 - 2* a^2 - a*b + b^2)*sinh(f*x + e)^4 + 4*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^ 3 - 3*(2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(2*a^2 + a*b - b^2)*cosh(f*x + e)^2 + 3*(5*(2*a^2 + a*b - b^2)*cosh(f*x + e)^4 - 6*(2*a ^2 + a*b - b^2)*cosh(f*x + e)^2 + 2*a^2 + a*b - b^2)*sinh(f*x + e)^2 - 2*a ^2 - a*b + b^2 + 6*((2*a^2 + a*b - b^2)*cosh(f*x + e)^5 - 2*(2*a^2 + a*b - b^2)*cosh(f*x + e)^3 + (2*a^2 + a*b - b^2)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b + b^2)*cosh(f*x + e)^6 + 6*(a*b + b^2)*cosh(f*x + e)*sinh(f*x + e )^5 + (a*b + b^2)*sinh(f*x + e)^6 - 3*(a*b + b^2)*cosh(f*x + e)^4 + 3*(5*( a*b + b^2)*cosh(f*x + e)^2 - a*b - b^2)*sinh(f*x + e)^4 + 4*(5*(a*b + b^2) *cosh(f*x + e)^3 - 3*(a*b + b^2)*cosh(f*x + e))*sinh(f*x + e)^3 + 3*(a*b + b^2)*cosh(f*x + e)^2 + 3*(5*(a*b + b^2)*cosh(f*x + e)^4 - 6*(a*b + b^2)*c osh(f*x + e)^2 + a*b + b^2)*sinh(f*x + e)^2 - a*b - b^2 + 6*((a*b + b^2)*c osh(f*x + e)^5 - 2*(a*b + b^2)*cosh(f*x + e)^3 + (a*b + 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^2 - 8*a*b + b^2 + 4*(2*a* b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - ((2*a^2 - a*b)*cosh(f*x + e)^6 +...
\[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {\operatorname {csch}^{4}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int { \frac {\operatorname {csch}\left (f x + e\right )^{4}}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1118 vs. \(2 (271) = 542\).
Time = 1.45 (sec) , antiderivative size = 1118, normalized size of antiderivative = 4.19 \[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
1/3*(3*(a + b)*arctan(-1/2*(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(-a))*e ^(-4*e)/(sqrt(-a)*a) - 2*(3*(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*(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 - 15*(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*sqrt(b) - 15*(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*b^(3/2) - 32*(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))^3*a^2 - 66*(sqr t(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))^3*a*b + 30*(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))^3*b^2 - 96*(sqr t(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*a^2*sqrt(b) + 162*(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*a*b^ (3/2) - 30*(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^(5/2) - 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))*a^3 + 240*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e...
Timed out. \[ \int \frac {\text {csch}^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {1}{{\mathrm {sinh}\left (e+f\,x\right )}^4\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]