Integrand size = 12, antiderivative size = 90 \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\frac {5 \arcsin (\tanh (a+b x))}{16 b}+\frac {5 \sqrt {\text {sech}^2(a+b x)} \tanh (a+b x)}{16 b}+\frac {5 \text {sech}^2(a+b x)^{3/2} \tanh (a+b x)}{24 b}+\frac {\text {sech}^2(a+b x)^{5/2} \tanh (a+b x)}{6 b} \] Output:
5/16*arcsin(tanh(b*x+a))/b+5/16*(sech(b*x+a)^2)^(1/2)*tanh(b*x+a)/b+5/24*( sech(b*x+a)^2)^(3/2)*tanh(b*x+a)/b+1/6*(sech(b*x+a)^2)^(5/2)*tanh(b*x+a)/b
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.90 \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\frac {\text {sech}(a+b x) \left (15 \arctan (\sinh (a+b x))+15 \text {sech}(a+b x) \tanh (a+b x)+10 \text {sech}^3(a+b x) \tanh (a+b x)+8 \text {sech}^5(a+b x) \tanh (a+b x)\right )}{48 b \sqrt {\text {sech}^2(a+b x)}} \] Input:
Integrate[(Sech[a + b*x]^2)^(7/2),x]
Output:
(Sech[a + b*x]*(15*ArcTan[Sinh[a + b*x]] + 15*Sech[a + b*x]*Tanh[a + b*x] + 10*Sech[a + b*x]^3*Tanh[a + b*x] + 8*Sech[a + b*x]^5*Tanh[a + b*x]))/(48 *b*Sqrt[Sech[a + b*x]^2])
Time = 0.23 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4610, 211, 211, 211, 223}
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.
\(\displaystyle \int \text {sech}^2(a+b x)^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (\sec (i a+i b x)^2\right )^{7/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle \frac {\int \left (1-\tanh ^2(a+b x)\right )^{5/2}d\tanh (a+b x)}{b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {5}{6} \int \left (1-\tanh ^2(a+b x)\right )^{3/2}d\tanh (a+b x)+\frac {1}{6} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{5/2}}{b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-\tanh ^2(a+b x)}d\tanh (a+b x)+\frac {1}{4} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{3/2}\right )+\frac {1}{6} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{5/2}}{b}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-\tanh ^2(a+b x)}}d\tanh (a+b x)+\frac {1}{2} \sqrt {1-\tanh ^2(a+b x)} \tanh (a+b x)\right )+\frac {1}{4} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{3/2}\right )+\frac {1}{6} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{5/2}}{b}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \arcsin (\tanh (a+b x))+\frac {1}{2} \tanh (a+b x) \sqrt {1-\tanh ^2(a+b x)}\right )+\frac {1}{4} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{3/2}\right )+\frac {1}{6} \tanh (a+b x) \left (1-\tanh ^2(a+b x)\right )^{5/2}}{b}\) |
Input:
Int[(Sech[a + b*x]^2)^(7/2),x]
Output:
((Tanh[a + b*x]*(1 - Tanh[a + b*x]^2)^(5/2))/6 + (5*((Tanh[a + b*x]*(1 - T anh[a + b*x]^2)^(3/2))/4 + (3*(ArcSin[Tanh[a + b*x]]/2 + (Tanh[a + b*x]*Sq rt[1 - Tanh[a + b*x]^2])/2))/4))/6)/b
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 6.79 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.56
method | result | size |
risch | \(\frac {\sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (15 \,{\mathrm e}^{10 b x +10 a}+85 \,{\mathrm e}^{8 b x +8 a}+198 \,{\mathrm e}^{6 b x +6 a}-198 \,{\mathrm e}^{4 b x +4 a}-85 \,{\mathrm e}^{2 b x +2 a}-15\right )}{24 \left (1+{\mathrm e}^{2 b x +2 a}\right )^{5} b}+\frac {5 i \ln \left ({\mathrm e}^{b x}+i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{16 b}-\frac {5 i \ln \left ({\mathrm e}^{b x}-i {\mathrm e}^{-a}\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left (1+{\mathrm e}^{2 b x +2 a}\right )^{2}}}\, \left (1+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{-b x -a}}{16 b}\) | \(230\) |
Input:
int((sech(b*x+a)^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/24/(1+exp(2*b*x+2*a))^5*(1/(1+exp(2*b*x+2*a))^2*exp(2*b*x+2*a))^(1/2)*(1 5*exp(10*b*x+10*a)+85*exp(8*b*x+8*a)+198*exp(6*b*x+6*a)-198*exp(4*b*x+4*a) -85*exp(2*b*x+2*a)-15)/b+5/16*I*ln(exp(b*x)+I*exp(-a))/b*(1/(1+exp(2*b*x+2 *a))^2*exp(2*b*x+2*a))^(1/2)*(1+exp(2*b*x+2*a))*exp(-b*x-a)-5/16*I*ln(exp( b*x)-I*exp(-a))/b*(1/(1+exp(2*b*x+2*a))^2*exp(2*b*x+2*a))^(1/2)*(1+exp(2*b *x+2*a))*exp(-b*x-a)
Leaf count of result is larger than twice the leaf count of optimal. 1604 vs. \(2 (76) = 152\).
Time = 0.09 (sec) , antiderivative size = 1604, normalized size of antiderivative = 17.82 \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\text {Too large to display} \] Input:
integrate((sech(b*x+a)^2)^(7/2),x, algorithm="fricas")
Output:
1/24*(15*cosh(b*x + a)^11 + 165*cosh(b*x + a)*sinh(b*x + a)^10 + 15*sinh(b *x + a)^11 + 5*(165*cosh(b*x + a)^2 + 17)*sinh(b*x + a)^9 + 85*cosh(b*x + a)^9 + 45*(55*cosh(b*x + a)^3 + 17*cosh(b*x + a))*sinh(b*x + a)^8 + 18*(27 5*cosh(b*x + a)^4 + 170*cosh(b*x + a)^2 + 11)*sinh(b*x + a)^7 + 198*cosh(b *x + a)^7 + 42*(165*cosh(b*x + a)^5 + 170*cosh(b*x + a)^3 + 33*cosh(b*x + a))*sinh(b*x + a)^6 + 18*(385*cosh(b*x + a)^6 + 595*cosh(b*x + a)^4 + 231* cosh(b*x + a)^2 - 11)*sinh(b*x + a)^5 - 198*cosh(b*x + a)^5 + 90*(55*cosh( b*x + a)^7 + 119*cosh(b*x + a)^5 + 77*cosh(b*x + a)^3 - 11*cosh(b*x + a))* sinh(b*x + a)^4 + 5*(495*cosh(b*x + a)^8 + 1428*cosh(b*x + a)^6 + 1386*cos h(b*x + a)^4 - 396*cosh(b*x + a)^2 - 17)*sinh(b*x + a)^3 - 85*cosh(b*x + a )^3 + 3*(275*cosh(b*x + a)^9 + 1020*cosh(b*x + a)^7 + 1386*cosh(b*x + a)^5 - 660*cosh(b*x + a)^3 - 85*cosh(b*x + a))*sinh(b*x + a)^2 + 15*(cosh(b*x + a)^12 + 12*cosh(b*x + a)*sinh(b*x + a)^11 + sinh(b*x + a)^12 + 6*(11*cos h(b*x + a)^2 + 1)*sinh(b*x + a)^10 + 6*cosh(b*x + a)^10 + 20*(11*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^9 + 15*(33*cosh(b*x + a)^4 + 18*co sh(b*x + a)^2 + 1)*sinh(b*x + a)^8 + 15*cosh(b*x + a)^8 + 24*(33*cosh(b*x + a)^5 + 30*cosh(b*x + a)^3 + 5*cosh(b*x + a))*sinh(b*x + a)^7 + 4*(231*co sh(b*x + a)^6 + 315*cosh(b*x + a)^4 + 105*cosh(b*x + a)^2 + 5)*sinh(b*x + a)^6 + 20*cosh(b*x + a)^6 + 24*(33*cosh(b*x + a)^7 + 63*cosh(b*x + a)^5 + 35*cosh(b*x + a)^3 + 5*cosh(b*x + a))*sinh(b*x + a)^5 + 15*(33*cosh(b*x...
Timed out. \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\text {Timed out} \] Input:
integrate((sech(b*x+a)**2)**(7/2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (76) = 152\).
Time = 0.13 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.73 \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=-\frac {5 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{8 \, b} + \frac {15 \, e^{\left (-b x - a\right )} + 85 \, e^{\left (-3 \, b x - 3 \, a\right )} + 198 \, e^{\left (-5 \, b x - 5 \, a\right )} - 198 \, e^{\left (-7 \, b x - 7 \, a\right )} - 85 \, e^{\left (-9 \, b x - 9 \, a\right )} - 15 \, e^{\left (-11 \, b x - 11 \, a\right )}}{24 \, b {\left (6 \, e^{\left (-2 \, b x - 2 \, a\right )} + 15 \, e^{\left (-4 \, b x - 4 \, a\right )} + 20 \, e^{\left (-6 \, b x - 6 \, a\right )} + 15 \, e^{\left (-8 \, b x - 8 \, a\right )} + 6 \, e^{\left (-10 \, b x - 10 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )} + 1\right )}} \] Input:
integrate((sech(b*x+a)^2)^(7/2),x, algorithm="maxima")
Output:
-5/8*arctan(e^(-b*x - a))/b + 1/24*(15*e^(-b*x - a) + 85*e^(-3*b*x - 3*a) + 198*e^(-5*b*x - 5*a) - 198*e^(-7*b*x - 7*a) - 85*e^(-9*b*x - 9*a) - 15*e ^(-11*b*x - 11*a))/(b*(6*e^(-2*b*x - 2*a) + 15*e^(-4*b*x - 4*a) + 20*e^(-6 *b*x - 6*a) + 15*e^(-8*b*x - 8*a) + 6*e^(-10*b*x - 10*a) + e^(-12*b*x - 12 *a) + 1))
Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.38 \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\frac {15 \, \pi + \frac {4 \, {\left (15 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{5} + 160 \, {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{3} + 528 \, e^{\left (b x + a\right )} - 528 \, e^{\left (-b x - a\right )}\right )}}{{\left ({\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2} + 4\right )}^{3}} + 30 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{96 \, b} \] Input:
integrate((sech(b*x+a)^2)^(7/2),x, algorithm="giac")
Output:
1/96*(15*pi + 4*(15*(e^(b*x + a) - e^(-b*x - a))^5 + 160*(e^(b*x + a) - e^ (-b*x - a))^3 + 528*e^(b*x + a) - 528*e^(-b*x - a))/((e^(b*x + a) - e^(-b* x - a))^2 + 4)^3 + 30*arctan(1/2*(e^(2*b*x + 2*a) - 1)*e^(-b*x - a)))/b
Timed out. \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\int {\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{7/2} \,d x \] Input:
int((1/cosh(a + b*x)^2)^(7/2),x)
Output:
int((1/cosh(a + b*x)^2)^(7/2), x)
Time = 0.25 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.10 \[ \int \text {sech}^2(a+b x)^{7/2} \, dx=\frac {15 e^{12 b x +12 a} \mathit {atan} \left (e^{b x +a}\right )+90 e^{10 b x +10 a} \mathit {atan} \left (e^{b x +a}\right )+225 e^{8 b x +8 a} \mathit {atan} \left (e^{b x +a}\right )+300 e^{6 b x +6 a} \mathit {atan} \left (e^{b x +a}\right )+225 e^{4 b x +4 a} \mathit {atan} \left (e^{b x +a}\right )+90 e^{2 b x +2 a} \mathit {atan} \left (e^{b x +a}\right )+15 \mathit {atan} \left (e^{b x +a}\right )+15 e^{11 b x +11 a}+85 e^{9 b x +9 a}+198 e^{7 b x +7 a}-198 e^{5 b x +5 a}-85 e^{3 b x +3 a}-15 e^{b x +a}}{24 b \left (e^{12 b x +12 a}+6 e^{10 b x +10 a}+15 e^{8 b x +8 a}+20 e^{6 b x +6 a}+15 e^{4 b x +4 a}+6 e^{2 b x +2 a}+1\right )} \] Input:
int((sech(b*x+a)^2)^(7/2),x)
Output:
(15*e**(12*a + 12*b*x)*atan(e**(a + b*x)) + 90*e**(10*a + 10*b*x)*atan(e** (a + b*x)) + 225*e**(8*a + 8*b*x)*atan(e**(a + b*x)) + 300*e**(6*a + 6*b*x )*atan(e**(a + b*x)) + 225*e**(4*a + 4*b*x)*atan(e**(a + b*x)) + 90*e**(2* a + 2*b*x)*atan(e**(a + b*x)) + 15*atan(e**(a + b*x)) + 15*e**(11*a + 11*b *x) + 85*e**(9*a + 9*b*x) + 198*e**(7*a + 7*b*x) - 198*e**(5*a + 5*b*x) - 85*e**(3*a + 3*b*x) - 15*e**(a + b*x))/(24*b*(e**(12*a + 12*b*x) + 6*e**(1 0*a + 10*b*x) + 15*e**(8*a + 8*b*x) + 20*e**(6*a + 6*b*x) + 15*e**(4*a + 4 *b*x) + 6*e**(2*a + 2*b*x) + 1))