3.593 \(\int \frac{\sinh (x)}{(-9+4 \cosh ^2(x))^{5/2}} \, dx\)

Optimal. Leaf size=37 \[ \frac{2 \cosh (x)}{243 \sqrt{4 \cosh ^2(x)-9}}-\frac{\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]

[Out]

-Cosh[x]/(27*(-9 + 4*Cosh[x]^2)^(3/2)) + (2*Cosh[x])/(243*Sqrt[-9 + 4*Cosh[x]^2])

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Rubi [A]  time = 0.0452293, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3190, 192, 191} \[ \frac{2 \cosh (x)}{243 \sqrt{4 \cosh ^2(x)-9}}-\frac{\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(-9 + 4*Cosh[x]^2)^(5/2),x]

[Out]

-Cosh[x]/(27*(-9 + 4*Cosh[x]^2)^(3/2)) + (2*Cosh[x])/(243*Sqrt[-9 + 4*Cosh[x]^2])

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 192

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

Rule 191

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

Rubi steps

\begin{align*} \int \frac{\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (-9+4 x^2\right )^{5/2}} \, dx,x,\cosh (x)\right )\\ &=-\frac{\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}-\frac{2}{27} \operatorname{Subst}\left (\int \frac{1}{\left (-9+4 x^2\right )^{3/2}} \, dx,x,\cosh (x)\right )\\ &=-\frac{\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}+\frac{2 \cosh (x)}{243 \sqrt{-9+4 \cosh ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0709346, size = 26, normalized size = 0.7 \[ \frac{\cosh (x) (4 \cosh (2 x)-23)}{243 (2 \cosh (2 x)-7)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(-9 + 4*Cosh[x]^2)^(5/2),x]

[Out]

(Cosh[x]*(-23 + 4*Cosh[2*x]))/(243*(-7 + 2*Cosh[2*x])^(3/2))

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Maple [A]  time = 0.012, size = 30, normalized size = 0.8 \begin{align*} -{\frac{\cosh \left ( x \right ) }{27} \left ( -9+4\, \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\cosh \left ( x \right ) }{243}{\frac{1}{\sqrt{-9+4\, \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(-9+4*cosh(x)^2)^(5/2),x)

[Out]

-1/27*cosh(x)/(-9+4*cosh(x)^2)^(3/2)+2/243*cosh(x)/(-9+4*cosh(x)^2)^(1/2)

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Maxima [B]  time = 1.03513, size = 169, normalized size = 4.57 \begin{align*} -\frac{1855 \, e^{\left (-2 \, x\right )} - 8485 \, e^{\left (-4 \, x\right )} + 5285 \, e^{\left (-6 \, x\right )} - 980 \, e^{\left (-8 \, x\right )} + 56 \, e^{\left (-10 \, x\right )} - 106}{12150 \,{\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}{\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} + \frac{980 \, e^{\left (-2 \, x\right )} - 5285 \, e^{\left (-4 \, x\right )} + 8485 \, e^{\left (-6 \, x\right )} - 1855 \, e^{\left (-8 \, x\right )} + 106 \, e^{\left (-10 \, x\right )} - 56}{12150 \,{\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}{\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(-9+4*cosh(x)^2)^(5/2),x, algorithm="maxima")

[Out]

-1/12150*(1855*e^(-2*x) - 8485*e^(-4*x) + 5285*e^(-6*x) - 980*e^(-8*x) + 56*e^(-10*x) - 106)/((3*e^(-x) + e^(-
2*x) + 1)^(5/2)*(-3*e^(-x) + e^(-2*x) + 1)^(5/2)) + 1/12150*(980*e^(-2*x) - 5285*e^(-4*x) + 8485*e^(-6*x) - 18
55*e^(-8*x) + 106*e^(-10*x) - 56)/((3*e^(-x) + e^(-2*x) + 1)^(5/2)*(-3*e^(-x) + e^(-2*x) + 1)^(5/2))

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Fricas [B]  time = 2.05894, size = 1585, normalized size = 42.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(-9+4*cosh(x)^2)^(5/2),x, algorithm="fricas")

[Out]

1/486*(2*cosh(x)^8 + 16*cosh(x)*sinh(x)^7 + 2*sinh(x)^8 + 28*(2*cosh(x)^2 - 1)*sinh(x)^6 - 28*cosh(x)^6 + 56*(
2*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(70*cosh(x)^4 - 210*cosh(x)^2 + 51)*sinh(x)^4 + 102*cosh(x)^4 + 8*(14*c
osh(x)^5 - 70*cosh(x)^3 + 51*cosh(x))*sinh(x)^3 + 4*(14*cosh(x)^6 - 105*cosh(x)^4 + 153*cosh(x)^2 - 7)*sinh(x)
^2 - 28*cosh(x)^2 + 8*(2*cosh(x)^7 - 21*cosh(x)^5 + 51*cosh(x)^3 - 7*cosh(x))*sinh(x) + (2*cosh(x)^6 + 12*cosh
(x)*sinh(x)^5 + 2*sinh(x)^6 + 3*(10*cosh(x)^2 - 7)*sinh(x)^4 - 21*cosh(x)^4 + 4*(10*cosh(x)^3 - 21*cosh(x))*si
nh(x)^3 + 3*(10*cosh(x)^4 - 42*cosh(x)^2 - 7)*sinh(x)^2 - 21*cosh(x)^2 + 6*(2*cosh(x)^5 - 14*cosh(x)^3 - 7*cos
h(x))*sinh(x) + 2)*sqrt((2*cosh(x)^2 + 2*sinh(x)^2 - 7)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) + 2)/(cos
h(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 14*(2*cosh(x)^2 - 1)*sinh(x)^6 - 14*cosh(x)^6 + 28*(2*cosh(x)^3 - 3
*cosh(x))*sinh(x)^5 + (70*cosh(x)^4 - 210*cosh(x)^2 + 51)*sinh(x)^4 + 51*cosh(x)^4 + 4*(14*cosh(x)^5 - 70*cosh
(x)^3 + 51*cosh(x))*sinh(x)^3 + 2*(14*cosh(x)^6 - 105*cosh(x)^4 + 153*cosh(x)^2 - 7)*sinh(x)^2 - 14*cosh(x)^2
+ 4*(2*cosh(x)^7 - 21*cosh(x)^5 + 51*cosh(x)^3 - 7*cosh(x))*sinh(x) + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(-9+4*cosh(x)**2)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.14142, size = 54, normalized size = 1.46 \begin{align*} \frac{{\left ({\left (2 \, e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} + 2}{486 \,{\left (e^{\left (4 \, x\right )} - 7 \, e^{\left (2 \, x\right )} + 1\right )}^{\frac{3}{2}}} - \frac{1}{243} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(-9+4*cosh(x)^2)^(5/2),x, algorithm="giac")

[Out]

1/486*(((2*e^(2*x) - 21)*e^(2*x) - 21)*e^(2*x) + 2)/(e^(4*x) - 7*e^(2*x) + 1)^(3/2) - 1/243