Integrand size = 23, antiderivative size = 134 \[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {b \sinh (e+f x)}{3 a (a-b) f \left (a+b \sinh ^2(e+f x)\right )^{3/2}}-\frac {(5 a-2 b) b \sinh (e+f x)}{3 a^2 (a-b)^2 f \sqrt {a+b \sinh ^2(e+f x)}} \]
arctan(sinh(f*x+e)*(a-b)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2))/(a-b)^(5/2)/f-1/ 3*b*sinh(f*x+e)/a/(a-b)/f/(a+b*sinh(f*x+e)^2)^(3/2)-1/3*(5*a-2*b)*b*sinh(f *x+e)/a^2/(a-b)^2/f/(a+b*sinh(f*x+e)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.10 (sec) , antiderivative size = 1331, normalized size of antiderivative = 9.93 \[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx =\text {Too large to display} \]
(Sech[e + f*x]*Tanh[e + f*x]*(1575*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a ]] + (2100*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2)/a + (840*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4)/a^2 - (3150*(a - b)*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Tanh[e + f*x]^2) /a - (4200*(a - b)*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f* x]^2*Tanh[e + f*x]^2)/a^2 - (1680*(a - b)*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^4*Tanh[e + f*x]^2)/a^3 + (1575*(a - b)^2*ArcSi n[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Tanh[e + f*x]^4)/a^2 + (2100*(a - b)^ 2*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2*Tanh[e + f*x ]^4)/a^3 + (840*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Si nh[e + f*x]^4*Tanh[e + f*x]^4)/a^4 + 2100*Sqrt[(Sech[e + f*x]^2*(a + b*Sin h[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) + (2800*b*Sinh[e + f *x]^2*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))/a + (1120*b^2*Sinh[e + f*x]^4*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2))/a^2 + 96*Hyp ergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Sech[e + f*x] ^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7/2) + 24*Hy pergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Tanh[e + f*x]^2)/a]*Sqrt[(Se ch[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(7 /2) + (168*b*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Tanh[e + f*x]^2)/a]*...
Time = 0.35 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3669, 316, 402, 27, 291, 216}
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.
\(\displaystyle \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos (i e+i f x) \left (a-b \sin (i e+i f x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3669 |
\(\displaystyle \frac {\int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \left (b \sinh ^2(e+f x)+a\right )^{5/2}}d\sinh (e+f x)}{f}\) |
\(\Big \downarrow \) 316 |
\(\displaystyle \frac {\frac {\int \frac {-2 b \sinh ^2(e+f x)+3 a-2 b}{\left (\sinh ^2(e+f x)+1\right ) \left (b \sinh ^2(e+f x)+a\right )^{3/2}}d\sinh (e+f x)}{3 a (a-b)}-\frac {b \sinh (e+f x)}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{f}\) |
\(\Big \downarrow \) 402 |
\(\displaystyle \frac {\frac {\frac {\int \frac {3 a^2}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a (a-b)}-\frac {b (5 a-2 b) \sinh (e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {b \sinh (e+f x)}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\frac {3 a \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)}{a-b}-\frac {b (5 a-2 b) \sinh (e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {b \sinh (e+f x)}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {\frac {\frac {3 a \int \frac {1}{1-\frac {(b-a) \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}}{a-b}-\frac {b (5 a-2 b) \sinh (e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {b \sinh (e+f x)}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{f}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {\frac {3 a \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{(a-b)^{3/2}}-\frac {b (5 a-2 b) \sinh (e+f x)}{a (a-b) \sqrt {a+b \sinh ^2(e+f x)}}}{3 a (a-b)}-\frac {b \sinh (e+f x)}{3 a (a-b) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}}{f}\) |
(-1/3*(b*Sinh[e + f*x])/(a*(a - b)*(a + b*Sinh[e + f*x]^2)^(3/2)) + ((3*a* ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqrt[a + b*Sinh[e + f*x]^2]])/(a - b)^( 3/2) - ((5*a - 2*b)*b*Sinh[e + f*x])/(a*(a - b)*Sqrt[a + b*Sinh[e + f*x]^2 ]))/(3*a*(a - b)))/f
3.4.94.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
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[((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/f S ubst[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 - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.36 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {\operatorname {`\,int/indef0`\,}\left (\frac {-b^{2} \sinh \left (f x +e \right )^{4}-2 \sinh \left (f x +e \right )^{2} a b -a^{2}}{\left (-b^{4} \sinh \left (f x +e \right )^{10}+\left (-4 a \,b^{3}-b^{4}\right ) \sinh \left (f x +e \right )^{8}+\left (-6 a^{2} b^{2}-4 a \,b^{3}\right ) \sinh \left (f x +e \right )^{6}+\left (-4 a^{3} b -6 a^{2} b^{2}\right ) \sinh \left (f x +e \right )^{4}+\left (-a^{4}-4 a^{3} b \right ) \sinh \left (f x +e \right )^{2}-a^{4}\right ) \sqrt {a +b \sinh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\) | \(169\) |
risch | \(\text {Expression too large to display}\) | \(791529\) |
`int/indef0`((-b^2*sinh(f*x+e)^4-2*sinh(f*x+e)^2*a*b-a^2)/(-b^4*sinh(f*x+e )^10+(-4*a*b^3-b^4)*sinh(f*x+e)^8+(-6*a^2*b^2-4*a*b^3)*sinh(f*x+e)^6+(-4*a ^3*b-6*a^2*b^2)*sinh(f*x+e)^4+(-a^4-4*a^3*b)*sinh(f*x+e)^2-a^4)/(a+b*sinh( f*x+e)^2)^(1/2),sinh(f*x+e))/f
Leaf count of result is larger than twice the leaf count of optimal. 2640 vs. \(2 (120) = 240\).
Time = 0.55 (sec) , antiderivative size = 5396, normalized size of antiderivative = 40.27 \[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\operatorname {sech}{\left (e + f x \right )}}{\left (a + b \sinh ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\operatorname {sech}\left (f x + e\right )}{{\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (120) = 240\).
Time = 0.60 (sec) , antiderivative size = 1177, normalized size of antiderivative = 8.78 \[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
-1/3*(((((5*a^9*b^2*e^(12*e) - 42*a^8*b^3*e^(12*e) + 156*a^7*b^4*e^(12*e) - 336*a^6*b^5*e^(12*e) + 462*a^5*b^6*e^(12*e) - 420*a^4*b^7*e^(12*e) + 252 *a^3*b^8*e^(12*e) - 96*a^2*b^9*e^(12*e) + 21*a*b^10*e^(12*e) - 2*b^11*e^(1 2*e))*e^(2*f*x)/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10*b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210*a^8*b^4*e^(12*e) - 252*a^7*b^5*e^(12*e) + 21 0*a^6*b^6*e^(12*e) - 120*a^5*b^7*e^(12*e) + 45*a^4*b^8*e^(12*e) - 10*a^3*b ^9*e^(12*e) + a^2*b^10*e^(12*e)) + 3*(8*a^10*b*e^(10*e) - 73*a^9*b^2*e^(10 *e) + 298*a^8*b^3*e^(10*e) - 716*a^7*b^4*e^(10*e) + 1120*a^6*b^5*e^(10*e) - 1190*a^5*b^6*e^(10*e) + 868*a^4*b^7*e^(10*e) - 428*a^3*b^8*e^(10*e) + 13 6*a^2*b^9*e^(10*e) - 25*a*b^10*e^(10*e) + 2*b^11*e^(10*e))/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10*b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210*a ^8*b^4*e^(12*e) - 252*a^7*b^5*e^(12*e) + 210*a^6*b^6*e^(12*e) - 120*a^5*b^ 7*e^(12*e) + 45*a^4*b^8*e^(12*e) - 10*a^3*b^9*e^(12*e) + a^2*b^10*e^(12*e) ))*e^(2*f*x) - 3*(8*a^10*b*e^(8*e) - 73*a^9*b^2*e^(8*e) + 298*a^8*b^3*e^(8 *e) - 716*a^7*b^4*e^(8*e) + 1120*a^6*b^5*e^(8*e) - 1190*a^5*b^6*e^(8*e) + 868*a^4*b^7*e^(8*e) - 428*a^3*b^8*e^(8*e) + 136*a^2*b^9*e^(8*e) - 25*a*b^1 0*e^(8*e) + 2*b^11*e^(8*e))/(a^12*e^(12*e) - 10*a^11*b*e^(12*e) + 45*a^10* b^2*e^(12*e) - 120*a^9*b^3*e^(12*e) + 210*a^8*b^4*e^(12*e) - 252*a^7*b^5*e ^(12*e) + 210*a^6*b^6*e^(12*e) - 120*a^5*b^7*e^(12*e) + 45*a^4*b^8*e^(12*e ) - 10*a^3*b^9*e^(12*e) + a^2*b^10*e^(12*e)))*e^(2*f*x) - (5*a^9*b^2*e^...
Timed out. \[ \int \frac {\text {sech}(e+f x)}{\left (a+b \sinh ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\mathrm {cosh}\left (e+f\,x\right )\,{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]