3.280 \(\int \frac{\cosh (x)}{a+a \sinh ^2(x)} \, dx\)

Optimal. Leaf size=7 \[ \frac{\tan ^{-1}(\sinh (x))}{a} \]

[Out]

ArcTan[Sinh[x]]/a

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Rubi [A]  time = 0.0276392, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3175, 3770} \[ \frac{\tan ^{-1}(\sinh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + a*Sinh[x]^2),x]

[Out]

ArcTan[Sinh[x]]/a

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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{\cosh (x)}{a+a \sinh ^2(x)} \, dx &=\frac{\int \text{sech}(x) \, dx}{a}\\ &=\frac{\tan ^{-1}(\sinh (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0027435, size = 12, normalized size = 1.71 \[ \frac{2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]/(a + a*Sinh[x]^2),x]

[Out]

(2*ArcTan[Tanh[x/2]])/a

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Maple [A]  time = 0.008, size = 8, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \sinh \left ( x \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(a+a*sinh(x)^2),x)

[Out]

arctan(sinh(x))/a

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Maxima [A]  time = 1.57621, size = 14, normalized size = 2. \begin{align*} -\frac{2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*sinh(x)^2),x, algorithm="maxima")

[Out]

-2*arctan(e^(-x))/a

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Fricas [A]  time = 1.44555, size = 42, normalized size = 6. \begin{align*} \frac{2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*sinh(x)^2),x, algorithm="fricas")

[Out]

2*arctan(cosh(x) + sinh(x))/a

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Sympy [A]  time = 0.250555, size = 5, normalized size = 0.71 \begin{align*} \frac{\operatorname{atan}{\left (\sinh{\left (x \right )} \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*sinh(x)**2),x)

[Out]

atan(sinh(x))/a

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Giac [A]  time = 1.10845, size = 11, normalized size = 1.57 \begin{align*} \frac{2 \, \arctan \left (e^{x}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)/(a+a*sinh(x)^2),x, algorithm="giac")

[Out]

2*arctan(e^x)/a