3.1.40 \(\int (a \text {sech}^3(x))^{3/2} \, dx\) [40]

Optimal. Leaf size=69 \[ -\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \]

[Out]

-10/21*I*a*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))*(a*sech(x)^3)^(1/2
)+10/21*a*sinh(x)*(a*sech(x)^3)^(1/2)+2/7*a*sech(x)*(a*sech(x)^3)^(1/2)*tanh(x)

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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2720} \begin {gather*} \frac {10}{21} a \sinh (x) \sqrt {a \text {sech}^3(x)}+\frac {2}{7} a \tanh (x) \text {sech}(x) \sqrt {a \text {sech}^3(x)}-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^3)^(3/2),x]

[Out]

((-10*I)/21)*a*Cosh[x]^(3/2)*EllipticF[(I/2)*x, 2]*Sqrt[a*Sech[x]^3] + (10*a*Sqrt[a*Sech[x]^3]*Sinh[x])/21 + (
2*a*Sech[x]*Sqrt[a*Sech[x]^3]*Tanh[x])/7

Rule 2720

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ
[{c, d}, x]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3856

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {9}{2}}(x) \, dx}{\text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (5 a \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {5}{2}}(x) \, dx}{7 \text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (5 a \sqrt {a \text {sech}^3(x)}\right ) \int \sqrt {\text {sech}(x)} \, dx}{21 \text {sech}^{\frac {3}{2}}(x)}\\ &=\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {1}{21} \left (5 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx\\ &=-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 47, normalized size = 0.68 \begin {gather*} \frac {2}{21} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \left (-5 i \cosh ^{\frac {5}{2}}(x) F\left (\left .\frac {i x}{2}\right |2\right )+5 \cosh (x) \sinh (x)+3 \tanh (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^3)^(3/2),x]

[Out]

(2*a*Sech[x]*Sqrt[a*Sech[x]^3]*((-5*I)*Cosh[x]^(5/2)*EllipticF[(I/2)*x, 2] + 5*Cosh[x]*Sinh[x] + 3*Tanh[x]))/2
1

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Maple [F]
time = 0.83, size = 0, normalized size = 0.00 \[\int \left (a \mathrm {sech}\left (x \right )^{3}\right )^{\frac {3}{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^3)^(3/2),x)

[Out]

int((a*sech(x)^3)^(3/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sech(x)^3)^(3/2), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 391, normalized size = 5.67 \begin {gather*} \frac {2 \, {\left (5 \, \sqrt {2} {\left (a \cosh \left (x\right )^{6} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + a \sinh \left (x\right )^{6} + 3 \, a \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a \cosh \left (x\right )^{4} + 6 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a \cosh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (5 \, a \cosh \left (x\right )^{6} + 30 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + 5 \, a \sinh \left (x\right )^{6} + 17 \, a \cosh \left (x\right )^{4} + {\left (75 \, a \cosh \left (x\right )^{2} + 17 \, a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (25 \, a \cosh \left (x\right )^{3} + 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )^{2} + {\left (75 \, a \cosh \left (x\right )^{4} + 102 \, a \cosh \left (x\right )^{2} - 17 \, a\right )} \sinh \left (x\right )^{2} + 2 \, {\left (15 \, a \cosh \left (x\right )^{5} + 34 \, a \cosh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}\right )}}{21 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(3/2),x, algorithm="fricas")

[Out]

2/21*(5*sqrt(2)*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*cosh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sin
h(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sin
h(x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(a)*weierstrassPInverse(-4, 0, cosh(x) +
sinh(x)) + sqrt(2)*(5*a*cosh(x)^6 + 30*a*cosh(x)*sinh(x)^5 + 5*a*sinh(x)^6 + 17*a*cosh(x)^4 + (75*a*cosh(x)^2
+ 17*a)*sinh(x)^4 + 4*(25*a*cosh(x)^3 + 17*a*cosh(x))*sinh(x)^3 - 17*a*cosh(x)^2 + (75*a*cosh(x)^4 + 102*a*co
sh(x)^2 - 17*a)*sinh(x)^2 + 2*(15*a*cosh(x)^5 + 34*a*cosh(x)^3 - 17*a*cosh(x))*sinh(x) - 5*a)*sqrt((a*cosh(x)
+ a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 +
3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(
x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**3)**(3/2),x)

[Out]

Integral((a*sech(x)**3)**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(3/2),x, algorithm="giac")

[Out]

integrate((a*sech(x)^3)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cosh(x)^3)^(3/2),x)

[Out]

int((a/cosh(x)^3)^(3/2), x)

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