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

Optimal. Leaf size=25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \cosh (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])

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Rubi [A]  time = 0.0357397, antiderivative size = 25, 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 = {3190, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \cosh (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(a + b*Cosh[x]^2),x]

[Out]

ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])

Rule 3190

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e +
f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sinh (x)}{a+b \cosh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\cosh (x)\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} \cosh (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0218683, size = 25, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \cosh (x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])

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Maple [A]  time = 0.007, size = 17, normalized size = 0.7 \begin{align*}{\arctan \left ({b\cosh \left ( x \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a*b)^(1/2)*arctan(cosh(x)*b/(a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (x\right )}{b \cosh \left (x\right )^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.91175, size = 910, normalized size = 36.4 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} - 2 \,{\left (2 \, a - b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} - 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} -{\left (2 \, a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) - 4 \,{\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )\right )} \sqrt{-a b} + b}{b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \,{\left (2 \, a + b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a + b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} +{\left (2 \, a + b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b}\right )}{2 \, a b}, \frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{2 \, a}\right ) - \sqrt{a b} \arctan \left (\frac{{\left (b \cosh \left (x\right )^{3} + 3 \, b \cosh \left (x\right ) \sinh \left (x\right )^{2} + b \sinh \left (x\right )^{3} +{\left (4 \, a + b\right )} \cosh \left (x\right ) +{\left (3 \, b \cosh \left (x\right )^{2} + 4 \, a + b\right )} \sinh \left (x\right )\right )} \sqrt{a b}}{2 \, a b}\right )}{a b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*sqrt(-a*b)*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a - b)*cosh(x)^2 + 2*(3*b*cosh(
x)^2 - 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a - b)*cosh(x))*sinh(x) - 4*(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 +
sinh(x)^3 + (3*cosh(x)^2 + 1)*sinh(x) + cosh(x))*sqrt(-a*b) + b)/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sin
h(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh(x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*s
inh(x) + b))/(a*b), (sqrt(a*b)*arctan(1/2*sqrt(a*b)*(cosh(x) + sinh(x))/a) - sqrt(a*b)*arctan(1/2*(b*cosh(x)^3
+ 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + b)*cosh(x) + (3*b*cosh(x)^2 + 4*a + b)*sinh(x))*sqrt(a*b)/(a*b
)))/(a*b)]

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Sympy [A]  time = 1.59651, size = 87, normalized size = 3.48 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\cosh{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{b \cosh{\left (x \right )}} & \text{for}\: a = 0 \\\frac{\cosh{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{i \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \cosh{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} + \frac{i \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \cosh{\left (x \right )} \right )}}{2 \sqrt{a} b \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/cosh(x), Eq(a, 0) & Eq(b, 0)), (-1/(b*cosh(x)), Eq(a, 0)), (cosh(x)/a, Eq(b, 0)), (-I*log(-I*sq
rt(a)*sqrt(1/b) + cosh(x))/(2*sqrt(a)*b*sqrt(1/b)) + I*log(I*sqrt(a)*sqrt(1/b) + cosh(x))/(2*sqrt(a)*b*sqrt(1/
b)), True))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError