3.3.80 \(\int \frac {\cosh (x)}{a+a \sinh ^2(x)} \, dx\) [280]

Optimal. Leaf size=7 \[ \frac {\text {ArcTan}(\sinh (x))}{a} \]

[Out]

arctan(sinh(x))/a

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Rubi [A]
time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, 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 = {3254, 3855} \begin {gather*} \frac {\text {ArcTan}(\sinh (x))}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[x]/(a + a*Sinh[x]^2),x]

[Out]

ArcTan[Sinh[x]]/a

Rule 3254

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 3855

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*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.71 \begin {gather*} \frac {2 \text {ArcTan}\left (\tanh \left (\frac {x}{2}\right )\right )}{a} \end {gather*}

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 = 0.32, size = 8, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {\arctan \left (\sinh \left (x \right )\right )}{a}\)	\(8\)
default	\(\frac {\arctan \left (\sinh \left (x \right )\right )}{a}\)	\(8\)
risch	\(\frac {i \ln \left ({\mathrm e}^{x}+i\right )}{a}-\frac {i \ln \left ({\mathrm e}^{x}-i\right )}{a}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sinh(x))/a

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Maxima [A]
time = 0.47, size = 10, normalized size = 1.43 \begin {gather*} -\frac {2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \end {gather*}

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 = 0.39, size = 11, normalized size = 1.57 \begin {gather*} \frac {2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a} \end {gather*}

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 = 0.08, size = 5, normalized size = 0.71 \begin {gather*} \frac {\operatorname {atan}{\left (\sinh {\left (x \right )} \right )}}{a} \end {gather*}

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 = 0.40, size = 8, normalized size = 1.14 \begin {gather*} \frac {2 \, \arctan \left (e^{x}\right )}{a} \end {gather*}

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

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Mupad [B]
time = 0.07, size = 7, normalized size = 1.00 \begin {gather*} \frac {\mathrm {atan}\left (\mathrm {sinh}\left (x\right )\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/(a + a*sinh(x)^2),x)

[Out]

atan(sinh(x))/a

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