Integrand size = 25, antiderivative size = 199 \[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=-\frac {\coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f}-\frac {E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{f} \]
-coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/f-(1/(1+sinh(f*x+e)^2))^(1/2)*(1+si nh(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)+b*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*Elli pticF(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/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+(a+b* sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Result contains complex when optimal does not.
Time = 0.74 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.76 \[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {\sqrt {2} (-2 a+b-b \cosh (2 (e+f x))) \coth (e+f x)-2 i a \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+2 i (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )}{2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \]
(Sqrt[2]*(-2*a + b - b*Cosh[2*(e + f*x)])*Coth[e + f*x] - (2*I)*a*Sqrt[(2* a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b/a] + (2*I)*(a - b )*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(2* f*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]])
Time = 0.42 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.35, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 25, 3667, 377, 27, 324, 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.
| \(\displaystyle \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sqrt {a-b \sin (i e+i f x)^2}}{\sin (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sqrt {a-b \sin (i e+i f x)^2}}{\sin (i e+i f x)^2}dx\) |
| \(\Big \downarrow \) 3667 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {\text {csch}^2(e+f x) \sqrt {b \sinh ^2(e+f x)+a}}{\sqrt {\sinh ^2(e+f x)+1}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\int \frac {b \sqrt {\sinh ^2(e+f x)+1}}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (b \int \frac {\sqrt {\sinh ^2(e+f x)+1}}{\sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 324 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (b \left (\int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)\right )-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (b \left (\int \frac {\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}\right )-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (b \left (-\frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^{3/2}}d\sinh (e+f x)}{b}+\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}\right )-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (b \left (\frac {\sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {\sqrt {a+b \sinh ^2(e+f x)} E\left (\arctan (\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{b \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{b \sqrt {\sinh ^2(e+f x)+1}}\right )-\sqrt {\sinh ^2(e+f x)+1} \text {csch}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(-(Csch[e + f*x]*Sqrt[1 + Sinh[e + f* x]^2]*Sqrt[a + b*Sinh[e + f*x]^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))]) + (EllipticF[ArcTan[Si nh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*Sqrt[1 + Sinh[e + f *x]^2]*Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))]))))/f
3.1.74.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
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*( 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 - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, 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[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 = 0.52 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84
| method | result | size |
| default | \(-\frac {\sqrt {-\frac {b}{a}}\, b \cosh \left (f x +e \right )^{4}-b \sqrt {\frac {b \cosh \left (f x +e \right )^{2}}{a}+\frac {a -b}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sinh \left (f x +e \right ) \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right )+\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{2} a -\sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right )^{2} b}{\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(167\) |
-((-b/a)^(1/2)*b*cosh(f*x+e)^4-b*(b/a*cosh(f*x+e)^2+(a-b)/a)^(1/2)*(cosh(f *x+e)^2)^(1/2)*sinh(f*x+e)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2)) +(-b/a)^(1/2)*cosh(f*x+e)^2*a-(-b/a)^(1/2)*cosh(f*x+e)^2*b)/sinh(f*x+e)/(- b/a)^(1/2)/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. 542 vs. \(2 (215) = 430\).
Time = 0.12 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.72 \[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\frac {{\left ({\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (2 \, a - b\right )} \sinh \left (f x + e\right )^{2} - 2 \, {\left (b \cosh \left (f x + e\right )^{2} + 2 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + b \sinh \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - 2 \, {\left ({\left (2 \, a - b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (2 \, a - b\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (2 \, a - b\right )} \sinh \left (f x + e\right )^{2} - 2 \, a + b\right )} \sqrt {b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} - a b}{b^{2}}} - 2 \, a + b}{b}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (2 \, a b - b^{2}\right )} \sqrt {\frac {a^{2} - a b}{b^{2}}}}{b^{2}}) - \sqrt {2} {\left (b \cosh \left (f x + e\right ) + b \sinh \left (f x + e\right )\right )} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{b f \cosh \left (f x + e\right )^{2} + 2 \, b f \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + b f \sinh \left (f x + e\right )^{2} - b f} \]
(((2*a - b)*cosh(f*x + e)^2 + 2*(2*a - b)*cosh(f*x + e)*sinh(f*x + e) + (2 *a - b)*sinh(f*x + e)^2 - 2*(b*cosh(f*x + e)^2 + 2*b*cosh(f*x + e)*sinh(f* x + e) + b*sinh(f*x + e)^2 - b)*sqrt((a^2 - a*b)/b^2) - 2*a + b)*sqrt(b)*s qrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*s qrt((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*((2*a - b)*cosh(f*x + e)^2 + 2*(2*a - b)*cosh(f*x + e)*sinh(f*x + e) + (2*a - b)*s inh(f*x + e)^2 - 2*a + b)*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cos h(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) - sqrt(2)*(b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt( (b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cos h(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(b*f*cosh(f*x + e)^2 + 2*b*f *cosh(f*x + e)*sinh(f*x + e) + b*f*sinh(f*x + e)^2 - b*f)
\[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \operatorname {csch}^{2}{\left (e + f x \right )}\, dx \]
\[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int { \sqrt {b \sinh \left (f x + e\right )^{2} + a} \operatorname {csch}\left (f x + e\right )^{2} \,d x } \]
Exception generated. \[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx=\int \frac {\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}}{{\mathrm {sinh}\left (e+f\,x\right )}^2} \,d x \]