Integrand size = 25, antiderivative size = 290 \[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {b \coth (e+f x)}{a (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(a-2 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f}-\frac {(a-2 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a^2 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(a-2 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{a^2 (a-b) f} \]
-b*coth(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)-(a-2*b)*coth(f*x+e)*(a+ b*sinh(f*x+e)^2)^(1/2)/a^2/(a-b)/f-(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)/a^2/(a-b)/f/(sech(f*x+e)^2*( a+b*sinh(f*x+e)^2)/a)^(1/2)-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/(a-b)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+ e)^2)/a)^(1/2)+(a-2*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/a^2/(a-b)/f
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.64 \[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {-\left (\left (2 a^2-3 a b+2 b^2+(a-2 b) b \cosh (2 (e+f x))\right ) \coth (e+f x)\right )-i \sqrt {2} a (a-2 b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+i \sqrt {2} a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{a^2 (a-b) f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \]
(-((2*a^2 - 3*a*b + 2*b^2 + (a - 2*b)*b*Cosh[2*(e + f*x)])*Coth[e + f*x]) - I*Sqrt[2]*a*(a - 2*b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[ I*(e + f*x), b/a] + I*Sqrt[2]*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x) ])/a]*EllipticF[I*(e + f*x), b/a])/(a^2*(a - b)*f*Sqrt[4*a - 2*b + 2*b*Cos h[2*(e + f*x)]])
Time = 0.52 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 25, 3667, 374, 445, 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.
| \(\displaystyle \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {\text {csch}^2(e+f x) \left (-b \sinh ^2(e+f x)+a-2 b\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {\int \frac {b \left (a-(a-2 b) \sinh ^2(e+f x)\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {(a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {b \int \frac {a-(a-2 b) \sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {(a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {b \left (a \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-(a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )}{a}-\frac {(a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {b \left (\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-(a-2 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )}{a}-\frac {(a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {b \left (\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-(a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )\right )}{a}-\frac {(a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {-\frac {b \left (\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-(a-2 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )}{a}-\frac {(a-2 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}-\frac {b \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-((b*Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2])/(a*(a - b)*Sqrt[a + b*Sinh[e + f*x]^2])) + (-(((a - 2*b)*Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/a) - (b*((El lipticF[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 ))]) - (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))]))))/a)/(a*(a - b))))/f
3.2.15.3.1 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[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -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 = 1.28 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(-\frac {\left (\sqrt {-\frac {b}{a}}\, a b -2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4}+\left (\sqrt {-\frac {b}{a}}\, a^{2}-2 \sqrt {-\frac {b}{a}}\, a b +2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2}+\sinh \left (f x +e \right ) \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (2 a \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-2 b \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )-\operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a +2 b \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )\right )}{a^{2} \sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}\, \left (a -b \right ) \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(284\) |
| risch | \(\text {Expression too large to display}\) | \(49811\) |
-(((-b/a)^(1/2)*a*b-2*(-b/a)^(1/2)*b^2)*cosh(f*x+e)^4+((-b/a)^(1/2)*a^2-2* (-b/a)^(1/2)*a*b+2*(-b/a)^(1/2)*b^2)*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)*b*(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))-EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a+2*b*EllipticE(sinh(f *x+e)*(-b/a)^(1/2),(a/b)^(1/2))))/a^2/sinh(f*x+e)/(-b/a)^(1/2)/(a-b)/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. 2829 vs. \(2 (304) = 608\).
Time = 0.14 (sec) , antiderivative size = 2829, normalized size of antiderivative = 9.76 \[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
(((2*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^6 + 6*(2*a^2*b - 5*a*b^2 + 2*b ^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a^2*b - 5*a*b^2 + 2*b^3)*sinh(f*x + e)^6 + (8*a^3 - 26*a^2*b + 23*a*b^2 - 6*b^3)*cosh(f*x + e)^4 + (8*a^3 - 2 6*a^2*b + 23*a*b^2 - 6*b^3 + 15*(2*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^ 2)*sinh(f*x + e)^4 + 4*(5*(2*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^3 + (8 *a^3 - 26*a^2*b + 23*a*b^2 - 6*b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - 2*a^2 *b + 5*a*b^2 - 2*b^3 - (8*a^3 - 26*a^2*b + 23*a*b^2 - 6*b^3)*cosh(f*x + e) ^2 + (15*(2*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^4 - 8*a^3 + 26*a^2*b - 23*a*b^2 + 6*b^3 + 6*(8*a^3 - 26*a^2*b + 23*a*b^2 - 6*b^3)*cosh(f*x + e)^2 )*sinh(f*x + e)^2 + 2*(3*(2*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^5 + 2*( 8*a^3 - 26*a^2*b + 23*a*b^2 - 6*b^3)*cosh(f*x + e)^3 - (8*a^3 - 26*a^2*b + 23*a*b^2 - 6*b^3)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b^2 - 2*b^3)*cosh( f*x + e)^6 + 6*(a*b^2 - 2*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b^2 - 2* b^3)*sinh(f*x + e)^6 + (4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e)^4 + (4*a ^2*b - 11*a*b^2 + 6*b^3 + 15*(a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f*x + e )^4 + 4*(5*(a*b^2 - 2*b^3)*cosh(f*x + e)^3 + (4*a^2*b - 11*a*b^2 + 6*b^3)* cosh(f*x + e))*sinh(f*x + e)^3 - a*b^2 + 2*b^3 - (4*a^2*b - 11*a*b^2 + 6*b ^3)*cosh(f*x + e)^2 + (15*(a*b^2 - 2*b^3)*cosh(f*x + e)^4 - 4*a^2*b + 11*a *b^2 - 6*b^3 + 6*(4*a^2*b - 11*a*b^2 + 6*b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a*b^2 - 2*b^3)*cosh(f*x + e)^5 + 2*(4*a^2*b - 11*a*b^2 + 6...
\[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\operatorname {csch}^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\operatorname {csch}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\text {csch}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\mathrm {sinh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]