Integrand size = 25, antiderivative size = 256 \[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {4 b \cosh (e+f x) \sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f}-\frac {\coth (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{f}-\frac {(7 a+b) E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a+5 b) \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(7 a+b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 f} \]
-coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(3/2)/f+4/3*b*cosh(f*x+e)*sinh(f*x+e)*(a+ b*sinh(f*x+e)^2)^(1/2)/f-1/3*(7*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)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^ 2)/a)^(1/2)+1/3*(3*a+5*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)/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2) +1/3*(7*a+b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Result contains complex when optimal does not.
Time = 2.05 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.72 \[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {2} \left (-24 a^2+8 a b+3 b^2-4 b (2 a+b) \cosh (2 (e+f x))+b^2 \cosh (4 (e+f x))\right ) \coth (e+f x)-8 i a (7 a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+32 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{24 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
(Sqrt[2]*(-24*a^2 + 8*a*b + 3*b^2 - 4*b*(2*a + b)*Cosh[2*(e + f*x)] + b^2* Cosh[4*(e + f*x)])*Coth[e + f*x] - (8*I)*a*(7*a + b)*Sqrt[(2*a - b + b*Cos h[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (32*I)*a*(a - b)*Sqrt[(2* a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(24*f*Sqrt[2* a - b + b*Cosh[2*(e + f*x)]])
Time = 0.46 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 25, 3675, 375, 27, 403, 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.
| \(\displaystyle \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\tan (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\tan (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 3675 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \text {csch}^2(e+f x) \sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 375 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (2 \int \frac {\sqrt {b \sinh ^2(e+f x)+a} \left (4 b \sinh ^2(e+f x)+a+3 b\right )}{2 \sqrt {\sinh ^2(e+f x)+1}}d\sinh (e+f x)-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\int \frac {\sqrt {b \sinh ^2(e+f x)+a} \left (4 b \sinh ^2(e+f x)+a+3 b\right )}{\sqrt {\sinh ^2(e+f x)+1}}d\sinh (e+f x)-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \int \frac {b (7 a+b) \sinh ^2(e+f x)+a (3 a+5 b)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {4}{3} b \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}+\sqrt {\sinh ^2(e+f x)+1} (-\text {csch}(e+f x)) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (a (3 a+5 b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+b (7 a+b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )+\frac {4}{3} b \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}+\sqrt {\sinh ^2(e+f x)+1} (-\text {csch}(e+f x)) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (b (7 a+b) \int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {(3 a+5 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {4}{3} b \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}+\sqrt {\sinh ^2(e+f x)+1} (-\text {csch}(e+f x)) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (b (7 a+b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}\right )+\frac {(3 a+5 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )+\frac {4}{3} b \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}+\sqrt {\sinh ^2(e+f x)+1} (-\text {csch}(e+f x)) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {1}{3} \left (\frac {(3 a+5 b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+b (7 a+b) \left (\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )\right )+\frac {4}{3} b \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}+\sqrt {\sinh ^2(e+f x)+1} (-\text {csch}(e+f x)) \left (a+b \sinh ^2(e+f x)\right )^{3/2}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*((4*b*Sinh[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2])/3 - Csch[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^(3/2) + (((3*a + 5*b)*EllipticF[ArcTan[Sin h[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))]) + b*(7*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))])))/3))/f
3.5.77.3.1 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*((c + d*x^2)^q/(e*(m + 1))) , x] - Simp[2/(e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^(p - 1)*(c + d* x^2)^(q - 1)*Simp[b*c*p + a*d*q + b*d*(p + q)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0 ] && 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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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[((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 = 2.62 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(\frac {\sqrt {-\frac {b}{a}}\, b^{2} \cosh \left (f x +e \right )^{6}+\left (-2 \sqrt {-\frac {b}{a}}\, a b -2 \sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{4}+\left (-3 \sqrt {-\frac {b}{a}}\, a^{2}+2 \sqrt {-\frac {b}{a}}\, a b +\sqrt {-\frac {b}{a}}\, b^{2}\right ) \cosh \left (f x +e \right )^{2}+\sinh \left (f x +e \right ) \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \left (3 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a^{2}-2 \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -\operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+7 \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b +\operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}\right )}{3 \sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(327\) |
1/3*((-b/a)^(1/2)*b^2*cosh(f*x+e)^6+(-2*(-b/a)^(1/2)*a*b-2*(-b/a)^(1/2)*b^ 2)*cosh(f*x+e)^4+(-3*(-b/a)^(1/2)*a^2+2*(-b/a)^(1/2)*a*b+(-b/a)^(1/2)*b^2) *cosh(f*x+e)^2+sinh(f*x+e)*(cosh(f*x+e)^2)^(1/2)*(b/a*cosh(f*x+e)^2+(a-b)/ a)^(1/2)*(3*EllipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a^2-2*Elliptic F(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a*b-EllipticF(sinh(f*x+e)*(-b/a)^( 1/2),(a/b)^(1/2))*b^2+7*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*a* b+EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2))/sinh(f*x+e)/(-b/a) ^(1/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f
\[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \coth \left (f x + e\right )^{2} \,d x } \]
integral((b*coth(f*x + e)^2*sinh(f*x + e)^2 + a*coth(f*x + e)^2)*sqrt(b*si nh(f*x + e)^2 + a), x)
Timed out. \[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \coth \left (f x + e\right )^{2} \,d x } \]
Exception generated. \[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[4,0,4]%%%}+%%%{%%{[4,0]:[1,0,%%%{-1,[1]%%%}]%%},[3,0,4] %%%}+%%%{
Timed out. \[ \int \coth ^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int {\mathrm {coth}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]