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

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Rubi [A]  time = 0.04, antiderivative size = 25, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

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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fricas [B]  time = 0.52, size = 300, normalized size = 12.00 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} - 2 \, {\left (2 \, a - b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} - 2 \, a + b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} - {\left (2 \, a - b\right )} \cosh \relax (x)\right )} \sinh \relax (x) - 4 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3} + {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)\right )} \sqrt {-a b} + b}{b \cosh \relax (x)^{4} + 4 \, b \cosh \relax (x) \sinh \relax (x)^{3} + b \sinh \relax (x)^{4} + 2 \, {\left (2 \, a + b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} + 2 \, a + b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} + {\left (2 \, a + b\right )} \cosh \relax (x)\right )} \sinh \relax (x) + b}\right )}{2 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{2 \, a}\right ) - \sqrt {a b} \arctan \left (\frac {{\left (b \cosh \relax (x)^{3} + 3 \, b \cosh \relax (x) \sinh \relax (x)^{2} + b \sinh \relax (x)^{3} + {\left (4 \, a + b\right )} \cosh \relax (x) + {\left (3 \, b \cosh \relax (x)^{2} + 4 \, a + b\right )} \sinh \relax (x)\right )} \sqrt {a b}}{2 \, a b}\right )}{a b}\right ] \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[-58,31]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-85,-18]Undef/Unsigned Inf encountered in limitLimit: Max
order reached or unable to make series expansion Error: Bad Argument Value

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maple [A]  time = 0.05, size = 17, normalized size = 0.68 \[ \frac {\arctan \left (\frac {\cosh \relax (x ) b}{\sqrt {a b}}\right )}{\sqrt {a b}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \relax (x)}{b \cosh \relax (x)^{2} + a}\,{d x} \]

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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mupad [B]  time = 0.97, size = 16, normalized size = 0.64 \[ \frac {\mathrm {atan}\left (\frac {b\,\mathrm {cosh}\relax (x)}{\sqrt {a\,b}}\right )}{\sqrt {a\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(a + b*cosh(x)^2),x)

[Out]

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

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sympy [A]  time = 1.06, size = 87, normalized size = 3.48 \[ \begin {cases} \frac {\tilde {\infty }}{\cosh {\relax (x )}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{b \cosh {\relax (x )}} & \text {for}\: a = 0 \\\frac {\cosh {\relax (x )}}{a} & \text {for}\: b = 0 \\- \frac {i \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \cosh {\relax (x )} \right )}}{2 \sqrt {a} b \sqrt {\frac {1}{b}}} + \frac {i \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \cosh {\relax (x )} \right )}}{2 \sqrt {a} b \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]

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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