Integrand size = 25, antiderivative size = 351 \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {\coth (e+f x)}{3 a f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {(3 a-4 b) \coth (e+f x)}{3 a^2 (a-b) f \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(7 a-8 b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f}-\frac {(7 a-8 b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-4 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^3 (a-b) f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a-8 b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^3 (a-b) f} \]
1/3*coth(f*x+e)/a/f/(a+b*sinh(f*x+e)^2)^(3/2)+1/3*(3*a-4*b)*coth(f*x+e)/a^ 2/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)-1/3*(7*a-8*b)*coth(f*x+e)*(a+b*sinh(f* x+e)^2)^(1/2)/a^3/(a-b)/f-1/3*(7*a-8*b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sin h(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^3/(a-b)/f/(sech(f*x+e)^2*(a+b *sinh(f*x+e)^2)/a)^(1/2)+1/3*(3*a-4*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^3/(a-b)/f/(sech(f*x+e)^2*(a+b* sinh(f*x+e)^2)/a)^(1/2)+1/3*(7*a-8*b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e )/a^3/(a-b)/f
Result contains complex when optimal does not.
Time = 2.38 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.64 \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {-\frac {\left (24 a^3-68 a^2 b+69 a b^2-24 b^3+4 b \left (11 a^2-19 a b+8 b^2\right ) \cosh (2 (e+f x))+(7 a-8 b) b^2 \cosh (4 (e+f x))\right ) \coth (e+f x)}{\sqrt {2}}-2 i a^2 (7 a-8 b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+8 i a^2 (a-b) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{6 a^3 (a-b) f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \]
(-(((24*a^3 - 68*a^2*b + 69*a*b^2 - 24*b^3 + 4*b*(11*a^2 - 19*a*b + 8*b^2) *Cosh[2*(e + f*x)] + (7*a - 8*b)*b^2*Cosh[4*(e + f*x)])*Coth[e + f*x])/Sqr t[2]) - (2*I)*a^2*(7*a - 8*b)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*El lipticE[I*(e + f*x), b/a] + (8*I)*a^2*(a - b)*((2*a - b + b*Cosh[2*(e + f* x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a])/(6*a^3*(a - b)*f*(2*a - b + b*C osh[2*(e + f*x)])^(3/2))
Time = 0.58 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 25, 3675, 371, 25, 441, 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.
| \(\displaystyle \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (a-b \sin (i e+i f x)^2\right )^{5/2} \tan (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^2(e+f x) \sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 371 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\int -\frac {\text {csch}^2(e+f x) \left (3 \sinh ^2(e+f x)+4\right )}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\int \frac {\text {csch}^2(e+f x) \left (3 \sinh ^2(e+f x)+4\right )}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 441 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\int \frac {\text {csch}^2(e+f x) \left ((3 a-4 b) \sinh ^2(e+f x)+7 a-8 b\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {-\frac {\int -\frac {(7 a-8 b) b \sinh ^2(e+f x)+a (3 a-4 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {(7 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {\int \frac {(7 a-8 b) b \sinh ^2(e+f x)+a (3 a-4 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {(7 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {a (3 a-4 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+b (7 a-8 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a}-\frac {(7 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {b (7 a-8 b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {(7 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {b (7 a-8 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}}{a}-\frac {(7 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\frac {\frac {\frac {(3 a-4 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+b (7 a-8 b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )}{a}-\frac {(7 a-8 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a}}{a (a-b)}+\frac {(3 a-4 b) \sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a}+\frac {\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x)}{3 a \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x ]^2])/(3*a*(a + b*Sinh[e + f*x]^2)^(3/2)) + (((3*a - 4*b)*Csch[e + f*x]*Sq rt[1 + Sinh[e + f*x]^2])/(a*(a - b)*Sqrt[a + b*Sinh[e + f*x]^2]) + (-(((7* a - 8*b)*Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^ 2])/a) + (((3*a - 4*b)*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))]) + (7*a - 8*b)*b*((Sinh[e + f*x]*Sqrt[a + b*S inh[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) ))/(3*a)))/f
3.6.10.3.1 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[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si mp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 )^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && LtQ[p, -1]
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 = 3.04 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {\left (-7 \sqrt {-\frac {b}{a}}\, a \,b^{2}+8 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (f x +e \right )^{6}+\left (-11 \sqrt {-\frac {b}{a}}\, a^{2} b +26 \sqrt {-\frac {b}{a}}\, a \,b^{2}-16 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (f x +e \right )^{4}+\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 (3 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-11 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +8 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+7 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -8 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right ) \cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )+\left (-3 \sqrt {-\frac {b}{a}}\, a^{3}+14 \sqrt {-\frac {b}{a}}\, a^{2} b -19 \sqrt {-\frac {b}{a}}\, a \,b^{2}+8 \sqrt {-\frac {b}{a}}\, b^{3}\right ) \cosh \left (f x +e \right )^{2}+\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 (3 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{3}-14 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b +19 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}-8 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}+7 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2} b -15 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \,b^{2}+8 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{3}\right ) \sinh \left (f x +e \right )}{3 \sqrt {-\frac {b}{a}}\, \sinh \left (f x +e \right ) a^{3} \left (a -b \right ) \left (a +b \sinh \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \cosh \left (f x +e \right ) f}\) | \(640\) |
| risch | \(\text {Expression too large to display}\) | \(76488\) |
1/3*((-7*(-b/a)^(1/2)*a*b^2+8*(-b/a)^(1/2)*b^3)*cosh(f*x+e)^6+(-11*(-b/a)^ (1/2)*a^2*b+26*(-b/a)^(1/2)*a*b^2-16*(-b/a)^(1/2)*b^3)*cosh(f*x+e)^4+(b/a* cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*b*(3*EllipticF(sinh(f*x +e)*(-b/a)^(1/2),(a/b)^(1/2))*a^2-11*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a /b)^(1/2))*a*b+8*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2+7*Ell ipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b-8*EllipticE(sinh(f*x+e)*( -b/a)^(1/2),(a/b)^(1/2))*b^2)*cosh(f*x+e)^2*sinh(f*x+e)+(-3*(-b/a)^(1/2)*a ^3+14*(-b/a)^(1/2)*a^2*b-19*(-b/a)^(1/2)*a*b^2+8*(-b/a)^(1/2)*b^3)*cosh(f* x+e)^2+(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*(3*Elliptic F(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a^3-14*EllipticF(sinh(f*x+e)*(-b/a )^(1/2),(a/b)^(1/2))*a^2*b+19*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/ 2))*a*b^2-8*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^3+7*Elliptic E(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a^2*b-15*EllipticE(sinh(f*x+e)*(-b /a)^(1/2),(a/b)^(1/2))*a*b^2+8*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1 /2))*b^3)*sinh(f*x+e))/(-b/a)^(1/2)/sinh(f*x+e)/a^3/(a-b)/(a+b*sinh(f*x+e) ^2)^(3/2)/cosh(f*x+e)/f
Leaf count of result is larger than twice the leaf count of optimal. 7847 vs. \(2 (351) = 702\).
Time = 0.29 (sec) , antiderivative size = 7847, normalized size of antiderivative = 22.36 \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\coth ^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\coth ^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {coth}\left (e+f\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]