3.24 \(\int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx\)

Optimal. Leaf size=38 \[ \frac {\sinh (x)}{b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}} \]

[Out]

sinh(x)/b-a*arctan(sinh(x)*b^(1/2)/(a+b)^(1/2))/b^(3/2)/(a+b)^(1/2)

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Rubi [A]  time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3186, 388, 205} \[ \frac {\sinh (x)}{b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^3/(a + b*Cosh[x]^2),x]

[Out]

-((a*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sinh[x]/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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 3186

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 38, normalized size = 1.00 \[ \frac {\sinh (x)}{b}-\frac {a \tan ^{-1}\left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sinh[x]/b

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fricas [B]  time = 0.45, size = 498, normalized size = 13.11 \[ \left [\frac {{\left (a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a b + b^{2}\right )} \sinh \relax (x)^{2} - \sqrt {-a b - b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )} \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 + 3 \, b\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b \cosh \relax (x)^{2} - 2 \, a - 3 \, b\right )} \sinh \relax (x)^{2} + 4 \, {\left (b \cosh \relax (x)^{3} - {\left (2 \, a + 3 \, 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^{2}} + 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 ) - a b - b^{2}}{2 \, {\left ({\left (a b^{2} + b^{3}\right )} \cosh \relax (x) + {\left (a b^{2} + b^{3}\right )} \sinh \relax (x)\right )}}, \frac {{\left (a b + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (a b + b^{2}\right )} \cosh \relax (x) \sinh \relax (x) + {\left (a b + b^{2}\right )} \sinh \relax (x)^{2} - 2 \, \sqrt {a b + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )} \arctan \left (\frac {b \cosh \relax (x)^{3} + 3 \, b \cosh \relax (x) \sinh \relax (x)^{2} + b \sinh \relax (x)^{3} + {\left (4 \, a + 3 \, b\right )} \cosh \relax (x) + {\left (3 \, b \cosh \relax (x)^{2} + 4 \, a + 3 \, b\right )} \sinh \relax (x)}{2 \, \sqrt {a b + b^{2}}}\right ) - 2 \, \sqrt {a b + b^{2}} {\left (a \cosh \relax (x) + a \sinh \relax (x)\right )} \arctan \left (\frac {\sqrt {a b + b^{2}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{2 \, {\left (a + b\right )}}\right ) - a b - b^{2}}{2 \, {\left ({\left (a b^{2} + b^{3}\right )} \cosh \relax (x) + {\left (a b^{2} + b^{3}\right )} \sinh \relax (x)\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - sqrt(-a*b - b^2)*(a*cosh
(x) + a*sinh(x))*log((b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a + 3*b)*cosh(x)^2 + 2*(3*b*cos
h(x)^2 - 2*a - 3*b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a + 3*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^2) + b)/(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) + b)) - a*b - b^2)/((a*b^2 + b^3)*cosh(x) + (a*b^2 + b^3)*sinh(x)), 1/2*((a*b + b^2)*cosh(x
)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - 2*sqrt(a*b + b^2)*(a*cosh(x) + a*sinh(x))*arctan
(1/2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + 3*b)*cosh(x) + (3*b*cosh(x)^2 + 4*a + 3*b)*si
nh(x))/sqrt(a*b + b^2)) - 2*sqrt(a*b + b^2)*(a*cosh(x) + a*sinh(x))*arctan(1/2*sqrt(a*b + b^2)*(cosh(x) + sinh
(x))/(a + b)) - a*b - b^2)/((a*b^2 + b^3)*cosh(x) + (a*b^2 + b^3)*sinh(x))]

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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(cosh(x)^3/(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]=[81,-22]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]=[55,-12]Undef/Unsigned Inf encountered in limitLimit: Max
order reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.09, size = 96, normalized size = 2.53 \[ -\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {3}{2}} \sqrt {a +b}}-\frac {a \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{b^{\frac {3}{2}} \sqrt {a +b}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b/(tanh(1/2*x)+1)-a/b^(3/2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)-2*a^(1/2))/b^(1/2))-a/b^(3/2)
/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)+2*a^(1/2))/b^(1/2))-1/b/(tanh(1/2*x)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}}{2 \, b} - \frac {1}{8} \, \int \frac {16 \, {\left (a e^{\left (3 \, x\right )} + a e^{x}\right )}}{b^{2} e^{\left (4 \, x\right )} + b^{2} + 2 \, {\left (2 \, a b + b^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(e^(2*x) - 1)*e^(-x)/b - 1/8*integrate(16*(a*e^(3*x) + a*e^x)/(b^2*e^(4*x) + b^2 + 2*(2*a*b + b^2)*e^(2*x)
), x)

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mupad [B]  time = 1.16, size = 204, normalized size = 5.37 \[ \frac {{\mathrm {e}}^x}{2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,b}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^x\,\sqrt {b^3\,\left (a+b\right )}}{2\,b\,\left (a+b\right )\,{\left (a^2\right )}^{3/2}}\right )-2\,\mathrm {atan}\left (\left (\frac {b^5\,\sqrt {b^4+a\,b^3}}{4}+\frac {a\,b^4\,\sqrt {b^4+a\,b^3}}{4}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^3}{b^5\,{\left (a+b\right )}^2\,{\left (a^2\right )}^{3/2}}-\frac {4\,\left (2\,b^2\,{\left (a^2\right )}^{3/2}+2\,a\,b\,{\left (a^2\right )}^{3/2}\right )}{a^3\,b^4\,\left (a+b\right )\,\sqrt {b^3\,\left (a+b\right )}\,\sqrt {b^4+a\,b^3}}\right )-\frac {2\,a^3\,{\mathrm {e}}^{3\,x}}{b^5\,{\left (a+b\right )}^2\,{\left (a^2\right )}^{3/2}}\right )\right )\right )\,\sqrt {a^2}}{2\,\sqrt {b^4+a\,b^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^3/(a + b*cosh(x)^2),x)

[Out]

exp(x)/(2*b) - exp(-x)/(2*b) - ((2*atan((a^3*exp(x)*(b^3*(a + b))^(1/2))/(2*b*(a + b)*(a^2)^(3/2))) - 2*atan((
(b^5*(a*b^3 + b^4)^(1/2))/4 + (a*b^4*(a*b^3 + b^4)^(1/2))/4)*(exp(x)*((2*a^3)/(b^5*(a + b)^2*(a^2)^(3/2)) - (4
*(2*b^2*(a^2)^(3/2) + 2*a*b*(a^2)^(3/2)))/(a^3*b^4*(a + b)*(b^3*(a + b))^(1/2)*(a*b^3 + b^4)^(1/2))) - (2*a^3*
exp(3*x))/(b^5*(a + b)^2*(a^2)^(3/2)))))*(a^2)^(1/2))/(2*(a*b^3 + b^4)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**3/(a+b*cosh(x)**2),x)

[Out]

Timed out

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