Integrand size = 25, antiderivative size = 205 \[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {a^2 (5 a-6 b) \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{16 (a-b)^{3/2} f}+\frac {a (5 a-6 b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{16 (a-b) f}+\frac {(5 a-6 b) \text {sech}^3(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \tanh (e+f x)}{24 (a-b) f}+\frac {\text {sech}^5(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2} \tanh (e+f x)}{6 (a-b) f} \]
1/16*a^2*(5*a-6*b)*arctan(sinh(f*x+e)*(a-b)^(1/2)/(a+b*sinh(f*x+e)^2)^(1/2 ))/(a-b)^(3/2)/f+1/24*(5*a-6*b)*sech(f*x+e)^3*(a+b*sinh(f*x+e)^2)^(3/2)*ta nh(f*x+e)/(a-b)/f+1/6*sech(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(5/2)*tanh(f*x+e)/ (a-b)/f+1/16*a*(5*a-6*b)*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e) /(a-b)/f
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 6.68 (sec) , antiderivative size = 959, normalized size of antiderivative = 4.68 \[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\frac {a^2 \text {sech}^3(e+f x) \left (1+\frac {b \sinh ^2(e+f x)}{a}\right )^2 \tanh (e+f x) \left (45 a \arcsin \left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right )+30 b \arcsin \left (\sqrt {\frac {(a-b) \tanh ^2(e+f x)}{a}}\right ) \sinh ^2(e+f x)+210 a \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}+140 b \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}-120 a \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}+256 a \operatorname {Hypergeometric2F1}\left (2,5,\frac {7}{2},\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}-80 b \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}+256 b \operatorname {Hypergeometric2F1}\left (2,5,\frac {7}{2},\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{5/2}-512 a \operatorname {Hypergeometric2F1}\left (2,5,\frac {7}{2},\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}-512 b \operatorname {Hypergeometric2F1}\left (2,5,\frac {7}{2},\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{7/2}+256 a \operatorname {Hypergeometric2F1}\left (2,5,\frac {7}{2},\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{9/2}+256 b \operatorname {Hypergeometric2F1}\left (2,5,\frac {7}{2},\frac {(a-b) \tanh ^2(e+f x)}{a}\right ) \sinh ^2(e+f x) \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{9/2}-45 a \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}-30 b \sinh ^2(e+f x) \sqrt {\frac {(a-b) \text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right ) \tanh ^2(e+f x)}{a^2}}\right )}{240 f \left (a+b \sinh ^2(e+f x)\right )^{3/2} \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}} \left (\frac {(a-b) \tanh ^2(e+f x)}{a}\right )^{3/2}} \]
(a^2*Sech[e + f*x]^3*(1 + (b*Sinh[e + f*x]^2)/a)^2*Tanh[e + f*x]*(45*a*Arc Sin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]] + 30*b*ArcSin[Sqrt[((a - b)*Tanh[e + f*x]^2)/a]]*Sinh[e + f*x]^2 + 210*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(3/2) + 140*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) - 120*a*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]*(((a - b)*Tanh[e + f*x]^2)/a)^(5/2) + 256*a*Hypergeometric2F1[2, 5, 7/2, ((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)^(5/2) - 80*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)^(5/2) + 2 56*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*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)^(5/2) - 512*a*Hypergeometric2F1[2, 5, 7/2, ((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) - 512*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*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)^(7/2) + 256*a*Hypergeometric2F1[2, 5, 7 /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)^(9/2) + 256*b*Hypergeometric2F1[2, 5, 7/2, ((a - b)*Tanh[e + f*x]^2)/a]*Sinh[e + f*x]^2*Sqrt[(Sech[e + f*x...
Time = 0.36 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3042, 3669, 296, 292, 292, 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.
| \(\displaystyle \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-b \sin (i e+i f x)^2\right )^{3/2}}{\cos (i e+i f x)^7}dx\) |
| \(\Big \downarrow \) 3669 |
| \(\displaystyle \frac {\int \frac {\left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^4}d\sinh (e+f x)}{f}\) |
| \(\Big \downarrow \) 296 |
| \(\displaystyle \frac {\frac {(5 a-6 b) \int \frac {\left (b \sinh ^2(e+f x)+a\right )^{3/2}}{\left (\sinh ^2(e+f x)+1\right )^3}d\sinh (e+f x)}{6 (a-b)}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 (a-b) \left (\sinh ^2(e+f x)+1\right )^3}}{f}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {\frac {(5 a-6 b) \left (\frac {3}{4} a \int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\left (\sinh ^2(e+f x)+1\right )^2}d\sinh (e+f x)+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}\right )}{6 (a-b)}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 (a-b) \left (\sinh ^2(e+f x)+1\right )^3}}{f}\) |
| \(\Big \downarrow \) 292 |
| \(\displaystyle \frac {\frac {(5 a-6 b) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh (e+f x)+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}\right )+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}\right )}{6 (a-b)}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 (a-b) \left (\sinh ^2(e+f x)+1\right )^3}}{f}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {(5 a-6 b) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {(b-a) \sinh ^2(e+f x)}{b \sinh ^2(e+f x)+a}}d\frac {\sinh (e+f x)}{\sqrt {b \sinh ^2(e+f x)+a}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}\right )+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}\right )}{6 (a-b)}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 (a-b) \left (\sinh ^2(e+f x)+1\right )^3}}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {(5 a-6 b) \left (\frac {3}{4} a \left (\frac {a \arctan \left (\frac {\sqrt {a-b} \sinh (e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}}\right )}{2 \sqrt {a-b}}+\frac {\sinh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{2 \left (\sinh ^2(e+f x)+1\right )}\right )+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2}}{4 \left (\sinh ^2(e+f x)+1\right )^2}\right )}{6 (a-b)}+\frac {\sinh (e+f x) \left (a+b \sinh ^2(e+f x)\right )^{5/2}}{6 (a-b) \left (\sinh ^2(e+f x)+1\right )^3}}{f}\) |
((Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(5/2))/(6*(a - b)*(1 + Sinh[e + f* x]^2)^3) + ((5*a - 6*b)*((Sinh[e + f*x]*(a + b*Sinh[e + f*x]^2)^(3/2))/(4* (1 + Sinh[e + f*x]^2)^2) + (3*a*((a*ArcTan[(Sqrt[a - b]*Sinh[e + f*x])/Sqr t[a + b*Sinh[e + f*x]^2]])/(2*Sqrt[a - b]) + (Sinh[e + f*x]*Sqrt[a + b*Sin h[e + f*x]^2])/(2*(1 + Sinh[e + f*x]^2))))/4))/(6*(a - b)))/f
3.4.67.3.1 Defintions of rubi rules used
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] :> Si mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( a*(p + 1))) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ {a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt Q[q, 0] && NeQ[p, -1]
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[(b*c + 2*(p + 1)*(b*c - a*d))/(2*a*(p + 1)*(b*c - a*d)) Int[ (a + b*x^2)^(p + 1)*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, q}, x] && N eQ[b*c - a*d, 0] && EqQ[2*(p + q + 2) + 1, 0] && (LtQ[p, -1] || !LtQ[q, -1 ]) && NeQ[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 = 0.94 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.31
\[\frac {\operatorname {`\,int/indef0`\,}\left (\frac {b^{2} \sinh \left (f x +e \right )^{4}+2 \sinh \left (f x +e \right )^{2} a b +a^{2}}{\cosh \left (f x +e \right )^{8} \sqrt {a +b \sinh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\]
`int/indef0`((b^2*sinh(f*x+e)^4+2*sinh(f*x+e)^2*a*b+a^2)/cosh(f*x+e)^8/(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. 3758 vs. \(2 (185) = 370\).
Time = 0.82 (sec) , antiderivative size = 7633, normalized size of antiderivative = 37.23 \[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sinh \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \operatorname {sech}\left (f x + e\right )^{7} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 4555 vs. \(2 (185) = 370\).
Time = 1.79 (sec) , antiderivative size = 4555, normalized size of antiderivative = 22.22 \[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\text {Too large to display} \]
1/24*(3*(5*a^3 - 6*a^2*b)*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^ (4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b) + sqrt(b))/ sqrt(a - b))/(a - b)^(3/2) - 2*(15*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4* f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^11*a^3 - 18*( sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2 *b*e^(2*f*x + 2*e) + b))^11*a^2*b + 165*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b* e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^10*a^3*s qrt(b) - 198*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2* f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^10*a^2*b^(3/2) - 192*(sqrt(b)*e^(2* f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^10*a*b^(5/2) + 192*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^10*b^(7/2) + 340* (sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^9*a^4 + 77*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^( 4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^9*a^3*b - 2 886*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e ) - 2*b*e^(2*f*x + 2*e) + b))^9*a^2*b^2 + 2944*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^9 *a*b^3 - 640*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2* f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^9*b^4 + 3060*(sqrt(b)*e^(2*f*x +...
Timed out. \[ \int \text {sech}^7(e+f x) \left (a+b \sinh ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\mathrm {cosh}\left (e+f\,x\right )}^7} \,d x \]