Integrand size = 10, antiderivative size = 65 \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\frac {3}{8} a^{5/2} \arctan \left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {3}{8} a^2 \sqrt {a \text {sech}^2(x)} \tanh (x)+\frac {1}{4} a \left (a \text {sech}^2(x)\right )^{3/2} \tanh (x) \] Output:
3/8*a^(5/2)*arctan(a^(1/2)*tanh(x)/(a*sech(x)^2)^(1/2))+3/8*a^2*(a*sech(x) ^2)^(1/2)*tanh(x)+1/4*a*(a*sech(x)^2)^(3/2)*tanh(x)
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58 \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\frac {1}{8} \cosh (x) \left (a \text {sech}^2(x)\right )^{5/2} \left (3 \arctan (\sinh (x)) \cosh ^4(x)+2 \sinh (x)+3 \cosh ^2(x) \sinh (x)\right ) \] Input:
Integrate[(a*Sech[x]^2)^(5/2),x]
Output:
(Cosh[x]*(a*Sech[x]^2)^(5/2)*(3*ArcTan[Sinh[x]]*Cosh[x]^4 + 2*Sinh[x] + 3* Cosh[x]^2*Sinh[x]))/8
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4610, 211, 211, 224, 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.
\(\displaystyle \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec (i x)^2\right )^{5/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle a \int \left (a-a \tanh ^2(x)\right )^{3/2}d\tanh (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a \left (\frac {3}{4} a \int \sqrt {a-a \tanh ^2(x)}d\tanh (x)+\frac {1}{4} \tanh (x) \left (a-a \tanh ^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {a-a \tanh ^2(x)}}d\tanh (x)+\frac {1}{2} \tanh (x) \sqrt {a-a \tanh ^2(x)}\right )+\frac {1}{4} \tanh (x) \left (a-a \tanh ^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\frac {a \tanh ^2(x)}{a-a \tanh ^2(x)}+1}d\frac {\tanh (x)}{\sqrt {a-a \tanh ^2(x)}}+\frac {1}{2} \tanh (x) \sqrt {a-a \tanh ^2(x)}\right )+\frac {1}{4} \tanh (x) \left (a-a \tanh ^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle a \left (\frac {3}{4} a \left (\frac {1}{2} \sqrt {a} \arctan \left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-a \tanh ^2(x)}}\right )+\frac {1}{2} \tanh (x) \sqrt {a-a \tanh ^2(x)}\right )+\frac {1}{4} \tanh (x) \left (a-a \tanh ^2(x)\right )^{3/2}\right )\) |
Input:
Int[(a*Sech[x]^2)^(5/2),x]
Output:
a*((Tanh[x]*(a - a*Tanh[x]^2)^(3/2))/4 + (3*a*((Sqrt[a]*ArcTan[(Sqrt[a]*Ta nh[x])/Sqrt[a - a*Tanh[x]^2]])/2 + (Tanh[x]*Sqrt[a - a*Tanh[x]^2])/2))/4)
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[((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], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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 = 6.44 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.95
method | result | size |
risch | \(\frac {a^{2} \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 x}+11 \,{\mathrm e}^{4 x}-11 \,{\mathrm e}^{2 x}-3\right )}{4 \left ({\mathrm e}^{2 x}+1\right )^{3}}+\frac {3 i a^{2} {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+i\right )}{8}-\frac {3 i a^{2} {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-i\right )}{8}\) | \(127\) |
Input:
int((sech(x)^2*a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/4*a^2/(exp(2*x)+1)^3*(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)*(3*exp(6*x)+11*ex p(4*x)-11*exp(2*x)-3)+3/8*I*a^2*exp(-x)*(exp(2*x)+1)*(exp(2*x)*a/(exp(2*x) +1)^2)^(1/2)*ln(exp(x)+I)-3/8*I*a^2*exp(-x)*(exp(2*x)+1)*(exp(2*x)*a/(exp( 2*x)+1)^2)^(1/2)*ln(exp(x)-I)
Leaf count of result is larger than twice the leaf count of optimal. 1082 vs. \(2 (49) = 98\).
Time = 0.10 (sec) , antiderivative size = 1082, normalized size of antiderivative = 16.65 \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((a*sech(x)^2)^(5/2),x, algorithm="fricas")
Output:
1/4*(3*a^2*cosh(x)^7 + 3*(a^2*e^(2*x) + a^2)*sinh(x)^7 + 11*a^2*cosh(x)^5 + 21*(a^2*cosh(x)*e^(2*x) + a^2*cosh(x))*sinh(x)^6 + (63*a^2*cosh(x)^2 + 1 1*a^2 + (63*a^2*cosh(x)^2 + 11*a^2)*e^(2*x))*sinh(x)^5 - 11*a^2*cosh(x)^3 + 5*(21*a^2*cosh(x)^3 + 11*a^2*cosh(x) + (21*a^2*cosh(x)^3 + 11*a^2*cosh(x ))*e^(2*x))*sinh(x)^4 + (105*a^2*cosh(x)^4 + 110*a^2*cosh(x)^2 - 11*a^2 + (105*a^2*cosh(x)^4 + 110*a^2*cosh(x)^2 - 11*a^2)*e^(2*x))*sinh(x)^3 - 3*a^ 2*cosh(x) + (63*a^2*cosh(x)^5 + 110*a^2*cosh(x)^3 - 33*a^2*cosh(x) + (63*a ^2*cosh(x)^5 + 110*a^2*cosh(x)^3 - 33*a^2*cosh(x))*e^(2*x))*sinh(x)^2 + 3* (a^2*cosh(x)^8 + (a^2*e^(2*x) + a^2)*sinh(x)^8 + 4*a^2*cosh(x)^6 + 8*(a^2* cosh(x)*e^(2*x) + a^2*cosh(x))*sinh(x)^7 + 4*(7*a^2*cosh(x)^2 + a^2 + (7*a ^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^6 + 6*a^2*cosh(x)^4 + 8*(7*a^2*cosh(x )^3 + 3*a^2*cosh(x) + (7*a^2*cosh(x)^3 + 3*a^2*cosh(x))*e^(2*x))*sinh(x)^5 + 2*(35*a^2*cosh(x)^4 + 30*a^2*cosh(x)^2 + 3*a^2 + (35*a^2*cosh(x)^4 + 30 *a^2*cosh(x)^2 + 3*a^2)*e^(2*x))*sinh(x)^4 + 4*a^2*cosh(x)^2 + 8*(7*a^2*co sh(x)^5 + 10*a^2*cosh(x)^3 + 3*a^2*cosh(x) + (7*a^2*cosh(x)^5 + 10*a^2*cos h(x)^3 + 3*a^2*cosh(x))*e^(2*x))*sinh(x)^3 + 4*(7*a^2*cosh(x)^6 + 15*a^2*c osh(x)^4 + 9*a^2*cosh(x)^2 + a^2 + (7*a^2*cosh(x)^6 + 15*a^2*cosh(x)^4 + 9 *a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^2 + a^2 + (a^2*cosh(x)^8 + 4*a^2*co sh(x)^6 + 6*a^2*cosh(x)^4 + 4*a^2*cosh(x)^2 + a^2)*e^(2*x) + 8*(a^2*cosh(x )^7 + 3*a^2*cosh(x)^5 + 3*a^2*cosh(x)^3 + a^2*cosh(x) + (a^2*cosh(x)^7 ...
\[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\int \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((a*sech(x)**2)**(5/2),x)
Output:
Integral((a*sech(x)**2)**(5/2), x)
Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11 \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\frac {3}{4} \, a^{\frac {5}{2}} \arctan \left (e^{x}\right ) + \frac {3 \, a^{\frac {5}{2}} e^{\left (7 \, x\right )} + 11 \, a^{\frac {5}{2}} e^{\left (5 \, x\right )} - 11 \, a^{\frac {5}{2}} e^{\left (3 \, x\right )} - 3 \, a^{\frac {5}{2}} e^{x}}{4 \, {\left (e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1\right )}} \] Input:
integrate((a*sech(x)^2)^(5/2),x, algorithm="maxima")
Output:
3/4*a^(5/2)*arctan(e^x) + 1/4*(3*a^(5/2)*e^(7*x) + 11*a^(5/2)*e^(5*x) - 11 *a^(5/2)*e^(3*x) - 3*a^(5/2)*e^x)/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^( 2*x) + 1)
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\frac {1}{16} \, {\left (3 \, \pi - \frac {4 \, {\left (3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 20 \, e^{\left (-x\right )} - 20 \, e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}^{2}} + 6 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a^{\frac {5}{2}} \] Input:
integrate((a*sech(x)^2)^(5/2),x, algorithm="giac")
Output:
1/16*(3*pi - 4*(3*(e^(-x) - e^x)^3 + 20*e^(-x) - 20*e^x)/((e^(-x) - e^x)^2 + 4)^2 + 6*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*a^(5/2)
Timed out. \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{5/2} \,d x \] Input:
int((a/cosh(x)^2)^(5/2),x)
Output:
int((a/cosh(x)^2)^(5/2), x)
Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.77 \[ \int \left (a \text {sech}^2(x)\right )^{5/2} \, dx=\frac {\sqrt {a}\, a^{2} \left (3 e^{8 x} \mathit {atan} \left (e^{x}\right )+12 e^{6 x} \mathit {atan} \left (e^{x}\right )+18 e^{4 x} \mathit {atan} \left (e^{x}\right )+12 e^{2 x} \mathit {atan} \left (e^{x}\right )+3 \mathit {atan} \left (e^{x}\right )+3 e^{7 x}+11 e^{5 x}-11 e^{3 x}-3 e^{x}\right )}{4 e^{8 x}+16 e^{6 x}+24 e^{4 x}+16 e^{2 x}+4} \] Input:
int((a*sech(x)^2)^(5/2),x)
Output:
(sqrt(a)*a**2*(3*e**(8*x)*atan(e**x) + 12*e**(6*x)*atan(e**x) + 18*e**(4*x )*atan(e**x) + 12*e**(2*x)*atan(e**x) + 3*atan(e**x) + 3*e**(7*x) + 11*e** (5*x) - 11*e**(3*x) - 3*e**x))/(4*(e**(8*x) + 4*e**(6*x) + 6*e**(4*x) + 4* e**(2*x) + 1))