3.126 \(\int \frac{1}{(a \cosh ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{\tanh (x)}{2 a \sqrt{a \cosh ^2(x)}}+\frac{\cosh (x) \tan ^{-1}(\sinh (x))}{2 a \sqrt{a \cosh ^2(x)}} \]

[Out]

(ArcTan[Sinh[x]]*Cosh[x])/(2*a*Sqrt[a*Cosh[x]^2]) + Tanh[x]/(2*a*Sqrt[a*Cosh[x]^2])

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Rubi [A]  time = 0.0248997, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3204, 3207, 3770} \[ \frac{\tanh (x)}{2 a \sqrt{a \cosh ^2(x)}}+\frac{\cosh (x) \tan ^{-1}(\sinh (x))}{2 a \sqrt{a \cosh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[Sinh[x]]*Cosh[x])/(2*a*Sqrt[a*Cosh[x]^2]) + Tanh[x]/(2*a*Sqrt[a*Cosh[x]^2])

Rule 3204

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1))/(b*f*(
2*p + 1)), x] + Dist[(2*(p + 1))/(b*(2*p + 1)), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[p, -1]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \cosh ^2(x)\right )^{3/2}} \, dx &=\frac{\tanh (x)}{2 a \sqrt{a \cosh ^2(x)}}+\frac{\int \frac{1}{\sqrt{a \cosh ^2(x)}} \, dx}{2 a}\\ &=\frac{\tanh (x)}{2 a \sqrt{a \cosh ^2(x)}}+\frac{\cosh (x) \int \text{sech}(x) \, dx}{2 a \sqrt{a \cosh ^2(x)}}\\ &=\frac{\tan ^{-1}(\sinh (x)) \cosh (x)}{2 a \sqrt{a \cosh ^2(x)}}+\frac{\tanh (x)}{2 a \sqrt{a \cosh ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0144693, size = 31, normalized size = 0.74 \[ \frac{\tanh (x)+2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{2 a \sqrt{a \cosh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTan[Tanh[x/2]]*Cosh[x] + Tanh[x])/(2*a*Sqrt[a*Cosh[x]^2])

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Maple [B]  time = 0.059, size = 82, normalized size = 2. \begin{align*}{\frac{1}{2\,{a}^{2}\cosh \left ( x \right ) \sinh \left ( x \right ) }\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}} \left ( -\ln \left ( 2\,{\frac{\sqrt{-a}\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}-a}{\cosh \left ( x \right ) }} \right ) a \left ( \cosh \left ( x \right ) \right ) ^{2}+\sqrt{-a}\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{-a}}}{\frac{1}{\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*cosh(x)^2)^(3/2),x)

[Out]

1/2/a^2/cosh(x)*(a*sinh(x)^2)^(1/2)*(-ln(2*((-a)^(1/2)*(a*sinh(x)^2)^(1/2)-a)/cosh(x))*a*cosh(x)^2+(-a)^(1/2)*
(a*sinh(x)^2)^(1/2))/(-a)^(1/2)/sinh(x)/(a*cosh(x)^2)^(1/2)

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Maxima [A]  time = 1.77283, size = 55, normalized size = 1.31 \begin{align*} \frac{e^{\left (3 \, x\right )} - e^{x}}{a^{\frac{3}{2}} e^{\left (4 \, x\right )} + 2 \, a^{\frac{3}{2}} e^{\left (2 \, x\right )} + a^{\frac{3}{2}}} + \frac{\arctan \left (e^{x}\right )}{a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

(e^(3*x) - e^x)/(a^(3/2)*e^(4*x) + 2*a^(3/2)*e^(2*x) + a^(3/2)) + arctan(e^x)/a^(3/2)

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Fricas [B]  time = 2.18795, size = 878, normalized size = 20.9 \begin{align*} \frac{{\left (3 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + e^{x} \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{x} \sinh \left (x\right ) +{\left (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}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) +{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{x}\right )} \sqrt{a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{a^{2} \cosh \left (x\right )^{4} +{\left (a^{2} e^{\left (2 \, x\right )} + a^{2}\right )} \sinh \left (x\right )^{4} + 2 \, a^{2} \cosh \left (x\right )^{2} + 4 \,{\left (a^{2} \cosh \left (x\right ) e^{\left (2 \, x\right )} + a^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \,{\left (3 \, a^{2} \cosh \left (x\right )^{2} + a^{2} +{\left (3 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )^{2} + a^{2} +{\left (a^{2} \cosh \left (x\right )^{4} + 2 \, a^{2} \cosh \left (x\right )^{2} + a^{2}\right )} e^{\left (2 \, x\right )} + 4 \,{\left (a^{2} \cosh \left (x\right )^{3} + a^{2} \cosh \left (x\right ) +{\left (a^{2} \cosh \left (x\right )^{3} + a^{2} \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

(3*cosh(x)*e^x*sinh(x)^2 + e^x*sinh(x)^3 + (3*cosh(x)^2 - 1)*e^x*sinh(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)*arctan(cosh(x) + sinh(x)) + (cosh(x)^3 - cosh(x))*e^x)*sqrt(a*e^(4*x) + 2*a*e^(2*x) + a)*e^(-x)/(a^2*co
sh(x)^4 + (a^2*e^(2*x) + a^2)*sinh(x)^4 + 2*a^2*cosh(x)^2 + 4*(a^2*cosh(x)*e^(2*x) + a^2*cosh(x))*sinh(x)^3 +
2*(3*a^2*cosh(x)^2 + a^2 + (3*a^2*cosh(x)^2 + a^2)*e^(2*x))*sinh(x)^2 + a^2 + (a^2*cosh(x)^4 + 2*a^2*cosh(x)^2
+ a^2)*e^(2*x) + 4*(a^2*cosh(x)^3 + a^2*cosh(x) + (a^2*cosh(x)^3 + a^2*cosh(x))*e^(2*x))*sinh(x))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)**2)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.15458, size = 76, normalized size = 1.81 \begin{align*} \frac{\frac{\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )}{\sqrt{a}} - \frac{4 \,{\left (e^{\left (-x\right )} - e^{x}\right )}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )} \sqrt{a}}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*cosh(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/4*((pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))/sqrt(a) - 4*(e^(-x) - e^x)/(((e^(-x) - e^x)^2 + 4)*sqrt(a)))/a