Integrand size = 25, antiderivative size = 217 \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sqrt {b} (a+b) \cosh (e+f x) E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} (a-b)^2 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {2 b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{a (a-b)^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{(a-b) f \sqrt {a+b \sinh ^2(e+f x)}} \]
(a+b)*cosh(f*x+e)*(1/(1+b*sinh(f*x+e)^2/a))^(1/2)*(1+b*sinh(f*x+e)^2/a)^(1 /2)*EllipticE(sinh(f*x+e)*b^(1/2)/a^(1/2)/(1+b*sinh(f*x+e)^2/a)^(1/2),(1-a /b)^(1/2))*b^(1/2)/(a-b)^2/f/a^(1/2)/(a*cosh(f*x+e)^2/(a+b*sinh(f*x+e)^2)) ^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)-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/(a-b)^2/f/(sech(f*x+e)^2*(a+b*si nh(f*x+e)^2)/a)^(1/2)+tanh(f*x+e)/(a-b)/f/(a+b*sinh(f*x+e)^2)^(1/2)
Result contains complex when optimal does not.
Time = 1.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.82 \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {i \sqrt {2} a (a+b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-i \sqrt {2} a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\left (2 a^2-a b+b^2+b (a+b) \cosh (2 (e+f x))\right ) \tanh (e+f x)}{a (a-b)^2 f \sqrt {4 a-2 b+2 b \cosh (2 (e+f x))}} \]
(I*Sqrt[2]*a*(a + 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] + (2*a^2 - a*b + b^2 + b*(a + b)*Cosh[2*(e + f*x)])*Tanh[e + f*x])/(a*(a - b)^2*f*Sqrt[4*a - 2*b + 2*b*Cosh[2*(e + f* x)]])
Time = 0.41 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.28, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3671, 316, 27, 400, 313, 320}
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.
| \(\displaystyle \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {\int \frac {b \left (1-\sinh ^2(e+f x)\right )}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {b \int \frac {1-\sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {b \left (\frac {2 \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {(a+b) \int \frac {\sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}\right )}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {b \left (\frac {2 \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {(a+b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}\right )}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {b \left (\frac {2 \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a (a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {(a+b) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}\right )}{a-b}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(Sinh[e + f*x]/((a - b)*Sqrt[1 + Sinh [e + f*x]^2]*Sqrt[a + b*Sinh[e + f*x]^2]) - (b*(-(((a + b)*EllipticE[ArcTa n[(Sqrt[b]*Sinh[e + f*x])/Sqrt[a]], 1 - a/b]*Sqrt[1 + Sinh[e + f*x]^2])/(S qrt[a]*(a - b)*Sqrt[b]*Sqrt[(a*(1 + Sinh[e + f*x]^2))/(a + b*Sinh[e + f*x] ^2)]*Sqrt[a + b*Sinh[e + f*x]^2])) + (2*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*Sinh[e + f*x]^2])/(a*(a - b)*Sqrt[1 + Sinh[e + f*x]^2]* Sqrt[(a + b*Sinh[e + f*x]^2)/(a*(1 + Sinh[e + f*x]^2))])))/(a - b)))/f
3.4.90.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
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_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 2.16 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.58
| method | result | size |
| default | \(\frac {\sqrt {-\frac {b}{a}}\, a b \sinh \left (f x +e \right )^{3}+\sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )^{3}-a \sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b +\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticF}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}-\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b -\sqrt {\frac {a +b \sinh \left (f x +e \right )^{2}}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \operatorname {EllipticE}\left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2}+\sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )+\sqrt {-\frac {b}{a}}\, b^{2} \sinh \left (f x +e \right )}{\left (a -b \right )^{2} a \sqrt {-\frac {b}{a}}\, \cosh \left (f x +e \right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}\, f}\) | \(342\) |
| risch | \(\text {Expression too large to display}\) | \(54607\) |
((-b/a)^(1/2)*a*b*sinh(f*x+e)^3+(-b/a)^(1/2)*b^2*sinh(f*x+e)^3-a*((a+b*sin h(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-b/a)^(1 /2),(a/b)^(1/2))*b+((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ell ipticF(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2-((a+b*sinh(f*x+e)^2)/a)^( 1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2)) *a*b-((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f* x+e)*(-b/a)^(1/2),(a/b)^(1/2))*b^2+(-b/a)^(1/2)*a^2*sinh(f*x+e)+(-b/a)^(1/ 2)*b^2*sinh(f*x+e))/(a-b)^2/a/(-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. 2786 vs. \(2 (231) = 462\).
Time = 0.15 (sec) , antiderivative size = 2786, normalized size of antiderivative = 12.84 \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-(((2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)^6 + 6*(2*a^2*b + a*b^2 - b^3)*cos h(f*x + e)*sinh(f*x + e)^5 + (2*a^2*b + a*b^2 - b^3)*sinh(f*x + e)^6 + (8* a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^4 + (8*a^3 + 2*a^2*b - 5*a*b^ 2 + b^3 + 15*(2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4* (5*(2*a^2*b + a*b^2 - b^3)*cosh(f*x + e)^3 + (8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + 2*a^2*b + a*b^2 - b^3 + (8*a^3 + 2*a ^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^2 + (15*(2*a^2*b + a*b^2 - b^3)*cosh(f *x + e)^4 + 8*a^3 + 2*a^2*b - 5*a*b^2 + b^3 + 6*(8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(2*a^2*b + a*b^2 - b^3)*co sh(f*x + e)^5 + 2*(8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e)^3 + (8*a ^3 + 2*a^2*b - 5*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e) - 2*((a*b^2 + b ^3)*cosh(f*x + e)^6 + 6*(a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (a*b ^2 + b^3)*sinh(f*x + e)^6 + (4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^4 + (4 *a^2*b + 3*a*b^2 - b^3 + 15*(a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(a*b^2 + b^3)*cosh(f*x + e)^3 + (4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e)^3 + a*b^2 + b^3 + (4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^2 + (15*(a*b^2 + b^3)*cosh(f*x + e)^4 + 4*a^2*b + 3*a*b^2 - b^3 + 6*( 4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 2*(3*(a*b^2 + b^3)*cosh(f*x + e)^5 + 2*(4*a^2*b + 3*a*b^2 - b^3)*cosh(f*x + e)^3 + (4*a^ 2*b + 3*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2...
\[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\operatorname {sech}^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\operatorname {sech}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\mathrm {cosh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]