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

Optimal. Leaf size=36 \[ \frac {\cosh (x)}{b}-\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 36, 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 = {3190, 388, 205} \[ \frac {\cosh (x)}{b}-\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \cosh (x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(a + b*Cosh[x]^2),x]

[Out]

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

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Mathematica [C]  time = 0.19, size = 83, normalized size = 2.31 \[ \frac {\cosh (x)}{b}-\frac {(a+b) \left (\tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )+\tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a+b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a}}\right )\right )}{\sqrt {a} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((a + b)*(ArcTan[(Sqrt[b] - I*Sqrt[a + b]*Tanh[x/2])/Sqrt[a]] + ArcTan[(Sqrt[b] + I*Sqrt[a + b]*Tanh[x/2])/S
qrt[a]]))/(Sqrt[a]*b^(3/2))) + Cosh[x]/b

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fricas [B]  time = 0.49, size = 416, normalized size = 11.56 \[ \left [\frac {a b \cosh \relax (x)^{2} + 2 \, a b \cosh \relax (x) \sinh \relax (x) + a b \sinh \relax (x)^{2} - \sqrt {-a b} {\left ({\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \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 - 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 ) + a b}{2 \, {\left (a b^{2} \cosh \relax (x) + a b^{2} \sinh \relax (x)\right )}}, \frac {a b \cosh \relax (x)^{2} + 2 \, a b \cosh \relax (x) \sinh \relax (x) + a b \sinh \relax (x)^{2} - 2 \, \sqrt {a b} {\left ({\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)\right )} \arctan \left (\frac {\sqrt {a b} {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{2 \, a}\right ) + 2 \, \sqrt {a b} {\left ({\left (a + b\right )} \cosh \relax (x) + {\left (a + b\right )} \sinh \relax (x)\right )} \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}{2 \, {\left (a b^{2} \cosh \relax (x) + a b^{2} \sinh \relax (x)\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(a*b*cosh(x)^2 + 2*a*b*cosh(x)*sinh(x) + a*b*sinh(x)^2 - sqrt(-a*b)*((a + b)*cosh(x) + (a + b)*sinh(x))*l
og((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)*si
nh(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*co
sh(x)^2 + 1)*sinh(x) + cosh(x))*sqrt(-a*b) + 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
)/(a*b^2*cosh(x) + a*b^2*sinh(x)), 1/2*(a*b*cosh(x)^2 + 2*a*b*cosh(x)*sinh(x) + a*b*sinh(x)^2 - 2*sqrt(a*b)*((
a + b)*cosh(x) + (a + b)*sinh(x))*arctan(1/2*sqrt(a*b)*(cosh(x) + sinh(x))/a) + 2*sqrt(a*b)*((a + b)*cosh(x) +
(a + b)*sinh(x))*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*cos
h(x)^2 + 4*a + b)*sinh(x))*sqrt(a*b)/(a*b)) + a*b)/(a*b^2*cosh(x) + a*b^2*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(sinh(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 = 0.08, size = 97, normalized size = 2.69 \[ \frac {1}{b \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {\arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right ) a}{b \sqrt {a b}}-\frac {\arctan \left (\frac {2 \left (a +b \right ) \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 a +2 b}{4 \sqrt {a b}}\right )}{\sqrt {a b}}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b/(tanh(1/2*x)+1)-1/b/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*x)^2-2*a+2*b)/(a*b)^(1/2))*a-1/(a*b)^(1/2)*ar
ctan(1/4*(2*(a+b)*tanh(1/2*x)^2-2*a+2*b)/(a*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 \[ \frac {{\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )}}{2 \, b} - \frac {1}{8} \, \int \frac {16 \, {\left ({\left (a + b\right )} e^{\left (3 \, x\right )} - {\left (a + b\right )} e^{x}\right )}}{b^{2} e^{\left (4 \, x\right )} + b^{2} + 2 \, {\left (2 \, a b + b^{2}\right )} e^{\left (2 \, x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(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 + b)*e^(3*x) - (a + b)*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 = 1.37, size = 257, normalized size = 7.14 \[ \frac {{\mathrm {e}}^{-x}}{2\,b}+\frac {{\mathrm {e}}^x}{2\,b}+\frac {\left (2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{3\,x}\,\left (a^3\,\sqrt {a\,b^3}+b^3\,\sqrt {a\,b^3}+3\,a\,b^2\,\sqrt {a\,b^3}+3\,a^2\,b\,\sqrt {a\,b^3}\right )}{2\,a\,b\,{\left ({\left (a+b\right )}^2\right )}^{3/2}}+\frac {a\,b^4\,{\mathrm {e}}^x\,\sqrt {a\,b^3}\,\left (\frac {8\,{\left (a^2+2\,a\,b+b^2\right )}^{3/2}}{a\,b^6\,{\left (a+b\right )}^3}+\frac {2\,\left (a^3\,\sqrt {a\,b^3}+b^3\,\sqrt {a\,b^3}+3\,a\,b^2\,\sqrt {a\,b^3}+3\,a^2\,b\,\sqrt {a\,b^3}\right )}{a^2\,b^5\,\sqrt {a\,b^3}\,{\left ({\left (a+b\right )}^2\right )}^{3/2}}\right )}{4}\right )-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,{\left (a+b\right )}^3\,\sqrt {a\,b^3}}{2\,a\,b\,{\left ({\left (a+b\right )}^2\right )}^{3/2}}\right )\right )\,\sqrt {a^2+2\,a\,b+b^2}}{2\,\sqrt {a\,b^3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^3/(a + b*cosh(x)^2),x)

[Out]

exp(-x)/(2*b) + exp(x)/(2*b) + ((2*atan((exp(3*x)*(a^3*(a*b^3)^(1/2) + b^3*(a*b^3)^(1/2) + 3*a*b^2*(a*b^3)^(1/
2) + 3*a^2*b*(a*b^3)^(1/2)))/(2*a*b*((a + b)^2)^(3/2)) + (a*b^4*exp(x)*(a*b^3)^(1/2)*((8*(2*a*b + a^2 + b^2)^(
3/2))/(a*b^6*(a + b)^3) + (2*(a^3*(a*b^3)^(1/2) + b^3*(a*b^3)^(1/2) + 3*a*b^2*(a*b^3)^(1/2) + 3*a^2*b*(a*b^3)^
(1/2)))/(a^2*b^5*(a*b^3)^(1/2)*((a + b)^2)^(3/2))))/4) - 2*atan((exp(x)*(a + b)^3*(a*b^3)^(1/2))/(2*a*b*((a +
b)^2)^(3/2))))*(2*a*b + a^2 + b^2)^(1/2))/(2*(a*b^3)^(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(sinh(x)**3/(a+b*cosh(x)**2),x)

[Out]

Timed out

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