Integrand size = 25, antiderivative size = 292 \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {b (3 a+b) \cosh (e+f x) \sinh (e+f x)}{3 a (a-b)^2 f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}+\frac {\sqrt {b} \left (3 a^2+7 a b-2 b^2\right ) \cosh (e+f x) E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{3 a^{3/2} (a-b)^3 f \sqrt {\frac {a \cosh ^2(e+f x)}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}-\frac {(9 a-b) b \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 (a-b)^3 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {\tanh (e+f x)}{(a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}} \]
1/3*b*(3*a+b)*cosh(f*x+e)*sinh(f*x+e)/a/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(3/2 )+1/3*(3*a^2+7*a*b-2*b^2)*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^(3/2)/(a-b)^3/f/(a*cosh(f*x+e)^2/ (a+b*sinh(f*x+e)^2))^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2)-1/3*(9*a-b)*b*(1/(1+s inh(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)^3/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+tanh(f*x+e)/(a-b)/f/ (a+b*sinh(f*x+e)^2)^(3/2)
Result contains complex when optimal does not.
Time = 2.82 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.89 \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 i a^2 \left (3 a^2+7 a b-2 b^2\right ) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )-2 i a^2 \left (3 a^2-2 a b-b^2\right ) \left (\frac {2 a-b+b \cosh (2 (e+f x))}{a}\right )^{3/2} \operatorname {EllipticF}\left (i (e+f x),\frac {b}{a}\right )+\frac {\left (24 a^4-24 a^3 b+41 a^2 b^2-19 a b^3+2 b^4+4 a b \left (6 a^2+5 a b-3 b^2\right ) \cosh (2 (e+f x))+b^2 \left (3 a^2+7 a b-2 b^2\right ) \cosh (4 (e+f x))\right ) \tanh (e+f x)}{\sqrt {2}}}{6 a^2 (a-b)^3 f (2 a-b+b \cosh (2 (e+f x)))^{3/2}} \]
((2*I)*a^2*(3*a^2 + 7*a*b - 2*b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/ 2)*EllipticE[I*(e + f*x), b/a] - (2*I)*a^2*(3*a^2 - 2*a*b - b^2)*((2*a - b + b*Cosh[2*(e + f*x)])/a)^(3/2)*EllipticF[I*(e + f*x), b/a] + ((24*a^4 - 24*a^3*b + 41*a^2*b^2 - 19*a*b^3 + 2*b^4 + 4*a*b*(6*a^2 + 5*a*b - 3*b^2)*C osh[2*(e + f*x)] + b^2*(3*a^2 + 7*a*b - 2*b^2)*Cosh[4*(e + f*x)])*Tanh[e + f*x])/Sqrt[2])/(6*a^2*(a - b)^3*f*(2*a - b + b*Cosh[2*(e + f*x)])^(3/2))
Time = 0.49 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.25, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3671, 316, 27, 402, 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.
| \(\displaystyle \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (i e+i f x)^2 \left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right )^{3/2} \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {\int \frac {b \left (1-3 \sinh ^2(e+f x)\right )}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b \int \frac {1-3 \sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b \left (\frac {\int \frac {2 (3 a-b)-(3 a+b) \sinh ^2(e+f x)}{\sqrt {\sinh ^2(e+f x)+1} \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 a (a-b)}-\frac {(3 a+b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b \left (\frac {\frac {(9 a-b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {\left (3 a^2+7 a b-2 b^2\right ) \int \frac {\sqrt {\sinh ^2(e+f x)+1}}{\left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{a-b}}{3 a (a-b)}-\frac {(3 a+b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b \left (\frac {\frac {(9 a-b) \int \frac {1}{\sqrt {\sinh ^2(e+f x)+1} \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {\left (3 a^2+7 a b-2 b^2\right ) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {(3 a+b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{a-b}\right )}{f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x) \left (\frac {\sinh (e+f x)}{(a-b) \sqrt {\sinh ^2(e+f x)+1} \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {b \left (\frac {\frac {(9 a-b) \sqrt {a+b \sinh ^2(e+f x)} \operatorname {EllipticF}\left (\arctan (\sinh (e+f x)),1-\frac {b}{a}\right )}{a (a-b) \sqrt {\sinh ^2(e+f x)+1} \sqrt {\frac {a+b \sinh ^2(e+f x)}{a \left (\sinh ^2(e+f x)+1\right )}}}-\frac {\left (3 a^2+7 a b-2 b^2\right ) \sqrt {\sinh ^2(e+f x)+1} E\left (\arctan \left (\frac {\sqrt {b} \sinh (e+f x)}{\sqrt {a}}\right )|1-\frac {a}{b}\right )}{\sqrt {a} \sqrt {b} (a-b) \sqrt {\frac {a \left (\sinh ^2(e+f x)+1\right )}{a+b \sinh ^2(e+f x)}} \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {(3 a+b) \sinh (e+f x) \sqrt {\sinh ^2(e+f x)+1}}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}\right )}{a-b}\right )}{f}\) |
(Sqrt[Cosh[e + f*x]^2]*Sech[e + f*x]*(Sinh[e + f*x]/((a - b)*Sqrt[1 + Sinh [e + f*x]^2]*(a + b*Sinh[e + f*x]^2)^(3/2)) - (b*(-1/3*((3*a + b)*Sinh[e + f*x]*Sqrt[1 + Sinh[e + f*x]^2])/(a*(a - b)*(a + b*Sinh[e + f*x]^2)^(3/2)) + (-(((3*a^2 + 7*a*b - 2*b^2)*EllipticE[ArcTan[(Sqrt[b]*Sinh[e + f*x])/Sq rt[a]], 1 - a/b]*Sqrt[1 + Sinh[e + f*x]^2])/(Sqrt[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])) + ((9*a - b)*EllipticF[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sqrt[a + b*S inh[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))]))/(3*a*(a - b))))/(a - b)))/f
3.4.99.3.1 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1001\) vs. \(2(360)=720\).
Time = 3.23 (sec) , antiderivative size = 1002, normalized size of antiderivative = 3.43
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1002\) |
| risch | \(\text {Expression too large to display}\) | \(88750\) |
-1/3*(-3*(-b/a)^(1/2)*a^2*b^2*sinh(f*x+e)^5-7*(-b/a)^(1/2)*a*b^3*sinh(f*x+ e)^5+2*(-b/a)^(1/2)*b^4*sinh(f*x+e)^5+3*EllipticE(sinh(f*x+e)*(-b/a)^(1/2) ,(a/b)^(1/2))*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*a^2*b^2* sinh(f*x+e)^2+7*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*((a+b*sinh (f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*a*b^3*sinh(f*x+e)^2-2*EllipticE( sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh( f*x+e)^2)^(1/2)*b^4*sinh(f*x+e)^2+6*((a+b*sinh(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))*a^2*b^2*sinh (f*x+e)^2-8*((a+b*sinh(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))*a*b^3*sinh(f*x+e)^2+2*((a+b*sinh(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^4*sinh(f*x+e)^2-6*(-b/a)^(1/2)*a^3*b*sinh(f*x+e)^3-8*(-b/a)^ (1/2)*a^2*b^2*sinh(f*x+e)^3-4*(-b/a)^(1/2)*a*b^3*sinh(f*x+e)^3+2*(-b/a)^(1 /2)*b^4*sinh(f*x+e)^3+3*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*(( a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*a^3*b+7*EllipticE(sinh(f *x+e)*(-b/a)^(1/2),(a/b)^(1/2))*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e) ^2)^(1/2)*a^2*b^2-2*EllipticE(sinh(f*x+e)*(-b/a)^(1/2),(a/b)^(1/2))*((a+b* sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*a*b^3+6*((a+b*sinh(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))*a^3*b-8*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*Ellip...
Leaf count of result is larger than twice the leaf count of optimal. 8928 vs. \(2 (298) = 596\).
Time = 0.38 (sec) , antiderivative size = 8928, normalized size of antiderivative = 30.58 \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\operatorname {sech}^{2}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\operatorname {sech}\left (f x + e\right )^{2}}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Exception generated. \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\text {sech}^2(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\mathrm {cosh}\left (e+f\,x\right )}^2\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]