3.135 \(\int x (a+a \cosh (x))^{3/2} \, dx\)

Optimal. Leaf size=89 \[ -\frac{8}{9} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{16}{3} a \sqrt{a \cosh (x)+a}+\frac{4}{3} a x \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{8}{3} a x \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]

[Out]

(-16*a*Sqrt[a + a*Cosh[x]])/3 - (8*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/9 + (4*a*x*Cosh[x/2]*Sqrt[a + a*Cosh[x]]
*Sinh[x/2])/3 + (8*a*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/3

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Rubi [A]  time = 0.0749648, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3319, 3310, 3296, 2638} \[ -\frac{8}{9} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{16}{3} a \sqrt{a \cosh (x)+a}+\frac{4}{3} a x \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{8}{3} a x \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + a*Cosh[x])^(3/2),x]

[Out]

(-16*a*Sqrt[a + a*Cosh[x]])/3 - (8*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/9 + (4*a*x*Cosh[x/2]*Sqrt[a + a*Cosh[x]]
*Sinh[x/2])/3 + (8*a*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/3

Rule 3319

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[((2*a)^IntPart[n
]*(a + b*Sin[e + f*x])^FracPart[n])/Sin[e/2 + (a*Pi)/(4*b) + (f*x)/2]^(2*FracPart[n]), Int[(c + d*x)^m*Sin[e/2
+ (a*Pi)/(4*b) + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[n
+ 1/2] && (GtQ[n, 0] || IGtQ[m, 0])

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x (a+a \cosh (x))^{3/2} \, dx &=\left (2 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x \cosh ^3\left (\frac{x}{2}\right ) \, dx\\ &=-\frac{8}{9} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{1}{3} \left (4 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x \cosh \left (\frac{x}{2}\right ) \, dx\\ &=-\frac{8}{9} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{8}{3} a x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )-\frac{1}{3} \left (8 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \sinh \left (\frac{x}{2}\right ) \, dx\\ &=-\frac{16}{3} a \sqrt{a+a \cosh (x)}-\frac{8}{9} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{8}{3} a x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0838374, size = 56, normalized size = 0.63 \[ \frac{1}{9} a \text{sech}\left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} \left (3 x \left (9 \sinh \left (\frac{x}{2}\right )+\sinh \left (\frac{3 x}{2}\right )\right )-54 \cosh \left (\frac{x}{2}\right )-2 \cosh \left (\frac{3 x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + a*Cosh[x])^(3/2),x]

[Out]

(a*Sqrt[a*(1 + Cosh[x])]*Sech[x/2]*(-54*Cosh[x/2] - 2*Cosh[(3*x)/2] + 3*x*(9*Sinh[x/2] + Sinh[(3*x)/2])))/9

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+a\cosh \left ( x \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.7426, size = 124, normalized size = 1.39 \begin{align*} -\frac{1}{18} \,{\left (3 \, \sqrt{2} a^{\frac{3}{2}} x + 2 \, \sqrt{2} a^{\frac{3}{2}} -{\left (3 \, \sqrt{2} a^{\frac{3}{2}} x - 2 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{\left (3 \, x\right )} - 27 \,{\left (\sqrt{2} a^{\frac{3}{2}} x - 2 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{\left (2 \, x\right )} + 27 \,{\left (\sqrt{2} a^{\frac{3}{2}} x + 2 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{x}\right )} e^{\left (-\frac{3}{2} \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(3*sqrt(2)*a^(3/2)*x + 2*sqrt(2)*a^(3/2) - (3*sqrt(2)*a^(3/2)*x - 2*sqrt(2)*a^(3/2))*e^(3*x) - 27*(sqrt(
2)*a^(3/2)*x - 2*sqrt(2)*a^(3/2))*e^(2*x) + 27*(sqrt(2)*a^(3/2)*x + 2*sqrt(2)*a^(3/2))*e^x)*e^(-3/2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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Giac [A]  time = 1.21066, size = 111, normalized size = 1.25 \begin{align*} \frac{1}{18} \, \sqrt{2}{\left (3 \, a^{\frac{3}{2}} x e^{\left (\frac{3}{2} \, x\right )} + 27 \, a^{\frac{3}{2}} x e^{\left (\frac{1}{2} \, x\right )} - 27 \, a^{\frac{3}{2}} x e^{\left (-\frac{1}{2} \, x\right )} - 3 \, a^{\frac{3}{2}} x e^{\left (-\frac{3}{2} \, x\right )} - 2 \, a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} - 54 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - 54 \, a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, x\right )} - 2 \, a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, x\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*sqrt(2)*(3*a^(3/2)*x*e^(3/2*x) + 27*a^(3/2)*x*e^(1/2*x) - 27*a^(3/2)*x*e^(-1/2*x) - 3*a^(3/2)*x*e^(-3/2*x
) - 2*a^(3/2)*e^(3/2*x) - 54*a^(3/2)*e^(1/2*x) - 54*a^(3/2)*e^(-1/2*x) - 2*a^(3/2)*e^(-3/2*x))