Integrand size = 25, antiderivative size = 237 \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {\coth (e+f x)}{a f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f}-\frac {2 E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{a^2 f} \]
coth(f*x+e)/a/f/(a+b*sinh(f*x+e)^2)^(1/2)-2*coth(f*x+e)*(a+b*sinh(f*x+e)^2 )^(1/2)/a^2/f-2*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*Ellipt icE(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*si nh(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/(1 +sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+si nh(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*(a+b*sinh(f*x+e)^2)^(1/2 )*tanh(f*x+e)/a^2/f
Result contains complex when optimal does not.
Time = 0.97 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.65 \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {-2 (a-b+b \cosh (2 (e+f x))) \coth (e+f x)-2 i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+i \sqrt {2} a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{a^2 f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \]
(-2*(a - b + b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*Sqrt[2]*a*Sqrt[(2* a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + I*Sqrt[2]*a* Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(a^2* f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f*x)]])
Time = 0.47 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.35, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 25, 3675, 371, 25, 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.
| \(\displaystyle \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (a-b \sin (i e+i f x)^2\right )^{3/2} \tan (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^2(e+f x) \sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 371 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int -\frac {\text {csch}^2(e+f x) \left (\sinh ^2(e+f x)+2\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {\text {csch}^2(e+f x) \left (\sinh ^2(e+f x)+2\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {\int -\frac {2 b \sinh ^2(e+f x)+a}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\int \frac {2 b \sinh ^2(e+f x)+a}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {a \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+2 b \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {2 \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 b \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {2 \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {2 b \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {2 \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+2 b \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a}-\frac {2 \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x ]^2])/(a*Sqrt[a + b*Sinh[e + f*x]^2]) + ((-2*Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a + ((EllipticF[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*b*((Sinh[e + f*x] *Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Sinh[e + f*x]^2]) - (EllipticE[A rcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(b*Sqrt[1 + Si nh[e + f*x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])))/ a)/a))/f
3.5.99.3.1 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[(-(e*x)^(m + 1))*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a *e*2*(p + 1))), x] + Simp[1/(a*2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1) *(c + d*x^2)^(q - 1)*Simp[c*(m + 2*(p + 1) + 1) + d*(m + 2*(p + q + 1) + 1) *x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && Lt Q[p, -1] && LtQ[0, q, 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 = 2.18 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(-\frac {2 \sqrt {-\frac {b}{a}}\, b \cosh \left (f x +e \right )^{4}+\left (\sqrt {-\frac {b}{a}}\, a -2 \sqrt {-\frac {b}{a}}\, b \right ) \cosh \left (f x +e \right )^{2}-\sinh \left (f x +e \right ) \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+2 b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sinh \left (f x +e \right ) a^{2} \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(219\) |
| risch | \(\text {Expression too large to display}\) | \(76390\) |
-(2*(-b/a)^(1/2)*b*cosh(f*x+e)^4+((-b/a)^(1/2)*a-2*(-b/a)^(1/2)*b)*cosh(f* x+e)^2-sinh(f*x+e)*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2) *(a*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))-2*b*EllipticF(sinh(f*x +e)*(-b/a)^(1/2),(a/b)^(1/2))+2*b*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b) ^(1/2))))/(-b/a)^(1/2)/sinh(f*x+e)/a^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. 2436 vs. \(2 (251) = 502\).
Time = 0.12 (sec) , antiderivative size = 2436, normalized size of antiderivative = 10.28 \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
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 + (8*a^2*b - 10*a*b^2 + 3*b^ 3)*cosh(f*x + e)^4 + (8*a^2*b - 10*a*b^2 + 3*b^3 + 15*(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 + (8*a ^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - 2*a*b^2 + b^3 - (8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^2 + (15*(2*a*b^2 - b^3)*cosh(f* x + e)^4 - 8*a^2*b + 10*a*b^2 - 3*b^3 + 6*(8*a^2*b - 10*a*b^2 + 3*b^3)*cos h(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(2*a*b^2 - b^3)*cosh(f*x + e)^5 + 2*( 8*a^2*b - 10*a*b^2 + 3*b^3)*cosh(f*x + e)^3 - (8*a^2*b - 10*a*b^2 + 3*b^3) *cosh(f*x + e))*sinh(f*x + e) - 2*(b^3*cosh(f*x + e)^6 + 6*b^3*cosh(f*x + e)*sinh(f*x + e)^5 + b^3*sinh(f*x + e)^6 + (4*a*b^2 - 3*b^3)*cosh(f*x + e) ^4 + (15*b^3*cosh(f*x + e)^2 + 4*a*b^2 - 3*b^3)*sinh(f*x + e)^4 + 4*(5*b^3 *cosh(f*x + e)^3 + (4*a*b^2 - 3*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - b^3 - (4*a*b^2 - 3*b^3)*cosh(f*x + e)^2 + (15*b^3*cosh(f*x + e)^4 - 4*a*b^2 + 3*b^3 + 6*(4*a*b^2 - 3*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*b^3*co sh(f*x + e)^5 + 2*(4*a*b^2 - 3*b^3)*cosh(f*x + e)^3 - (4*a*b^2 - 3*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 - 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*b - a*b^2)*co...
\[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\coth ^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\coth \left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {coth}\left (e+f\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]