Integrand size = 10, antiderivative size = 46 \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\frac {1}{2} a^{3/2} \arctan \left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a \text {sech}^2(x)}}\right )+\frac {1}{2} a \sqrt {a \text {sech}^2(x)} \tanh (x) \] Output:
1/2*a^(3/2)*arctan(a^(1/2)*tanh(x)/(a*sech(x)^2)^(1/2))+1/2*a*(a*sech(x)^2 )^(1/2)*tanh(x)
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.52 \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\frac {1}{2} a \sqrt {a \text {sech}^2(x)} (\arctan (\sinh (x)) \cosh (x)+\tanh (x)) \] Input:
Integrate[(a*Sech[x]^2)^(3/2),x]
Output:
(a*Sqrt[a*Sech[x]^2]*(ArcTan[Sinh[x]]*Cosh[x] + Tanh[x]))/2
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4610, 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.
\(\displaystyle \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle a \int \sqrt {a-a \tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a \left (\frac {1}{2} a \int \frac {1}{\sqrt {a-a \tanh ^2(x)}}d\tanh (x)+\frac {1}{2} \tanh (x) \sqrt {a-a \tanh ^2(x)}\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a \left (\frac {1}{2} a \int \frac {1}{\frac {a \tanh ^2(x)}{a-a \tanh ^2(x)}+1}d\frac {\tanh (x)}{\sqrt {a-a \tanh ^2(x)}}+\frac {1}{2} \tanh (x) \sqrt {a-a \tanh ^2(x)}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle a \left (\frac {1}{2} \sqrt {a} \arctan \left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a-a \tanh ^2(x)}}\right )+\frac {1}{2} \tanh (x) \sqrt {a-a \tanh ^2(x)}\right )\) |
Input:
Int[(a*Sech[x]^2)^(3/2),x]
Output:
a*((Sqrt[a]*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[a - a*Tanh[x]^2]])/2 + (Tanh[x]* Sqrt[a - a*Tanh[x]^2])/2)
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 = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.30
method | result | size |
risch | \(\frac {a \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}{{\mathrm e}^{2 x}+1}+\frac {i a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+i\right )}{2}-\frac {i a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left ({\mathrm e}^{2 x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-i\right )}{2}\) | \(106\) |
Input:
int((sech(x)^2*a)^(3/2),x,method=_RETURNVERBOSE)
Output:
a/(exp(2*x)+1)*(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)*(exp(2*x)-1)+1/2*I*a*exp( -x)*(exp(2*x)+1)*(exp(2*x)*a/(exp(2*x)+1)^2)^(1/2)*ln(exp(x)+I)-1/2*I*a*ex p(-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. 310 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 310, normalized size of antiderivative = 6.74 \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\frac {{\left (a \cosh \left (x\right )^{3} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{3} + 3 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{4} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} + {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - a \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + {\left (3 \, a \cosh \left (x\right )^{2} + {\left (3 \, a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}} \] Input:
integrate((a*sech(x)^2)^(3/2),x, algorithm="fricas")
Output:
(a*cosh(x)^3 + (a*e^(2*x) + a)*sinh(x)^3 + 3*(a*cosh(x)*e^(2*x) + a*cosh(x ))*sinh(x)^2 + (a*cosh(x)^4 + (a*e^(2*x) + a)*sinh(x)^4 + 4*(a*cosh(x)*e^( 2*x) + a*cosh(x))*sinh(x)^3 + 2*a*cosh(x)^2 + 2*(3*a*cosh(x)^2 + (3*a*cosh (x)^2 + a)*e^(2*x) + a)*sinh(x)^2 + (a*cosh(x)^4 + 2*a*cosh(x)^2 + a)*e^(2 *x) + 4*(a*cosh(x)^3 + a*cosh(x) + (a*cosh(x)^3 + a*cosh(x))*e^(2*x))*sinh (x) + a)*arctan(cosh(x) + sinh(x)) - a*cosh(x) + (a*cosh(x)^3 - a*cosh(x)) *e^(2*x) + (3*a*cosh(x)^2 + (3*a*cosh(x)^2 - a)*e^(2*x) - a)*sinh(x))*sqrt (a/(e^(4*x) + 2*e^(2*x) + 1))*e^x/(4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^x*sinh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^x*sinh(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^x)
\[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\int \left (a \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((a*sech(x)**2)**(3/2),x)
Output:
Integral((a*sech(x)**2)**(3/2), x)
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.85 \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=a^{\frac {3}{2}} \arctan \left (e^{x}\right ) + \frac {a^{\frac {3}{2}} e^{\left (3 \, x\right )} - a^{\frac {3}{2}} e^{x}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \] Input:
integrate((a*sech(x)^2)^(3/2),x, algorithm="maxima")
Output:
a^(3/2)*arctan(e^x) + (a^(3/2)*e^(3*x) - a^(3/2)*e^x)/(e^(4*x) + 2*e^(2*x) + 1)
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (\pi - \frac {4 \, {\left (e^{\left (-x\right )} - e^{x}\right )}}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4} + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} a^{\frac {3}{2}} \] Input:
integrate((a*sech(x)^2)^(3/2),x, algorithm="giac")
Output:
1/4*(pi - 4*(e^(-x) - e^x)/((e^(-x) - e^x)^2 + 4) + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*a^(3/2)
Timed out. \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2} \,d x \] Input:
int((a/cosh(x)^2)^(3/2),x)
Output:
int((a/cosh(x)^2)^(3/2), x)
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.22 \[ \int \left (a \text {sech}^2(x)\right )^{3/2} \, dx=\frac {\sqrt {a}\, a \left (e^{4 x} \mathit {atan} \left (e^{x}\right )+2 e^{2 x} \mathit {atan} \left (e^{x}\right )+\mathit {atan} \left (e^{x}\right )+e^{3 x}-e^{x}\right )}{e^{4 x}+2 e^{2 x}+1} \] Input:
int((a*sech(x)^2)^(3/2),x)
Output:
(sqrt(a)*a*(e**(4*x)*atan(e**x) + 2*e**(2*x)*atan(e**x) + atan(e**x) + e** (3*x) - e**x))/(e**(4*x) + 2*e**(2*x) + 1)