Integrand size = 25, antiderivative size = 285 \[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=-\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {2 (2 a-b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^2 f} \]
-2/3*(2*a-b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/f-1/3*coth(f*x+e)*c sch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/2)/a/f-2/3*(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^2/f/(sech(f*x+ e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+1/3*(3*a-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*(2*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 = 2.77 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.73 \[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\frac {-\frac {\left (2 \left (4 a^2-5 a b+2 b^2\right ) \cosh (2 (e+f x))-(2 a-b) (2 a-3 b-b \cosh (4 (e+f x)))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}-4 i a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+2 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{6 a^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
(-(((2*(4*a^2 - 5*a*b + 2*b^2)*Cosh[2*(e + f*x)] - (2*a - b)*(2*a - 3*b - b*Cosh[4*(e + f*x)]))*Coth[e + f*x]*Csch[e + f*x]^2)/Sqrt[2]) - (4*I)*a*(2 *a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a ] + (2*I)*a*(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 = 349, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3675, 376, 445, 25, 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.
| \(\displaystyle \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (i e+i f x)^4 \sqrt {a-b \sin (i e+i f x)^2}}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^4(e+f x) \left (\sinh ^2(e+f x)+1\right )^{3/2}}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 376 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {\text {csch}^2(e+f x) \left ((3 a-b) \sinh ^2(e+f x)+2 (2 a-b)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {\int -\frac {2 (2 a-b) b \sinh ^2(e+f x)+a (3 a-b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 (2 a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\int \frac {2 (2 a-b) b \sinh ^2(e+f x)+a (3 a-b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 (2 a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {a (3 a-b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+2 b (2 a-b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 (2 a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 b (2 a-b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(3 a-b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {2 (2 a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 b (2 a-b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(3 a-b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {2 (2 a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {(3 a-b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+2 b (2 a-b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a}-\frac {2 (2 a-b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{3 a}-\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}^3(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a}\right )}{f}\) |
(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*(2*a - b)*Csch[e + f*x]* Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a + (((3*a - b)*Ell ipticF[ArcTan[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*(2*a - b)*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
3.5.89.3.1 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[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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^ (m_), 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/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b , e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 1.78 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.83
| method | result | size |
| default | \(\frac {-4 \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}+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 ) \sinh \left (f x +e \right )^{3}-5 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}+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 \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}-4 \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}-5 \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 \sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{3} \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(522\) |
1/3*(-4*(-b/a)^(1/2)*a*b*sinh(f*x+e)^6+2*(-b/a)^(1/2)*b^2*sinh(f*x+e)^6+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))*sinh(f*x+e)^3-5*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)*Elli pticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2*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))*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)^3-4*(-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)^4-5*(-b/a)^(1/2)*a^2*sinh(f*x+e)^2+(-b/a)^(1/ 2)*a*b*sinh(f*x+e)^2-(-b/a)^(1/2)*a^2)/(-b/a)^(1/2)/a^2/sinh(f*x+e)^3/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. 2584 vs. \(2 (289) = 578\).
Time = 0.12 (sec) , antiderivative size = 2584, normalized size of antiderivative = 9.07 \[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
2/3*(((4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^6 + 6*(4*a^2*b - 4*a*b^2 + b ^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (4*a^2*b - 4*a*b^2 + b^3)*sinh(f*x + e )^6 - 3*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^4 - 3*(4*a^2*b - 4*a*b^2 + b^3 - 5*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5 *(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^3 - 3*(4*a^2*b - 4*a*b^2 + b^3)*c osh(f*x + e))*sinh(f*x + e)^3 - 4*a^2*b + 4*a*b^2 - b^3 + 3*(4*a^2*b - 4*a *b^2 + b^3)*cosh(f*x + e)^2 + 3*(5*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e) ^4 + 4*a^2*b - 4*a*b^2 + b^3 - 6*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2 )*sinh(f*x + e)^2 + 6*((4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^5 - 2*(4*a^ 2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^3 + (4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e) - 2*((2*a*b^2 - b^3)*cosh(f*x + e)^6 + 6*(2*a*b^2 - b^ 3)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a*b^2 - b^3)*sinh(f*x + e)^6 - 3*(2* a*b^2 - b^3)*cosh(f*x + e)^4 - 3*(2*a*b^2 - b^3 - 5*(2*a*b^2 - b^3)*cosh(f *x + e)^2)*sinh(f*x + e)^4 + 4*(5*(2*a*b^2 - b^3)*cosh(f*x + e)^3 - 3*(2*a *b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - 2*a*b^2 + b^3 + 3*(2*a*b^2 - b^3)*cosh(f*x + e)^2 + 3*(5*(2*a*b^2 - b^3)*cosh(f*x + e)^4 + 2*a*b^2 - b^ 3 - 6*(2*a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 6*((2*a*b^2 - b^3 )*cosh(f*x + e)^5 - 2*(2*a*b^2 - b^3)*cosh(f*x + e)^3 + (2*a*b^2 - b^3)*co sh(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 - ...
\[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {\coth ^{4}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int { \frac {\coth \left (f x + e\right )^{4}}{\sqrt {b \sinh \left (f x + e\right )^{2} + a}} \,d x } \]
Exception generated. \[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx=\int \frac {{\mathrm {coth}\left (e+f\,x\right )}^4}{\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]