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

Optimal. Leaf size=185 \[ -\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-16 a x^2 \sqrt{a \cosh (x)+a}+\frac{4}{3} a x^3 \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{8}{3} a x^3 \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{64}{27} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{1280}{9} a \sqrt{a \cosh (x)+a}+\frac{32}{9} a x \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{640}{9} a x \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]

[Out]

(-1280*a*Sqrt[a + a*Cosh[x]])/9 - 16*a*x^2*Sqrt[a + a*Cosh[x]] - (64*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/27 - (
8*a*x^2*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/3 + (32*a*x*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/9 + (4*a*x^3*Cos
h[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/3 + (640*a*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/9 + (8*a*x^3*Sqrt[a + a*Cosh
[x]]*Tanh[x/2])/3

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Rubi [A]  time = 0.19477, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3319, 3311, 3296, 2638, 3310} \[ -\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-16 a x^2 \sqrt{a \cosh (x)+a}+\frac{4}{3} a x^3 \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{8}{3} a x^3 \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{64}{27} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}-\frac{1280}{9} a \sqrt{a \cosh (x)+a}+\frac{32}{9} a x \sinh \left (\frac{x}{2}\right ) \cosh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a}+\frac{640}{9} a x \tanh \left (\frac{x}{2}\right ) \sqrt{a \cosh (x)+a} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + a*Cosh[x])^(3/2),x]

[Out]

(-1280*a*Sqrt[a + a*Cosh[x]])/9 - 16*a*x^2*Sqrt[a + a*Cosh[x]] - (64*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/27 - (
8*a*x^2*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]])/3 + (32*a*x*Cosh[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/9 + (4*a*x^3*Cos
h[x/2]*Sqrt[a + a*Cosh[x]]*Sinh[x/2])/3 + (640*a*x*Sqrt[a + a*Cosh[x]]*Tanh[x/2])/9 + (8*a*x^3*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 3311

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

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]

Rubi steps

\begin{align*} \int x^3 (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^3 \cosh ^3\left (\frac{x}{2}\right ) \, dx\\ &=-\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{4}{3} a x^3 \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^3 \cosh \left (\frac{x}{2}\right ) \, dx+\frac{1}{3} \left (16 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{64}{27} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}-\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{32}{9} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{4}{3} a x^3 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{8}{3} a x^3 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{1}{9} \left (32 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x \cosh \left (\frac{x}{2}\right ) \, dx-\left (8 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x^2 \sinh \left (\frac{x}{2}\right ) \, dx\\ &=-16 a x^2 \sqrt{a+a \cosh (x)}-\frac{64}{27} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}-\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{32}{9} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{4}{3} a x^3 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{64}{9} a x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{8}{3} a x^3 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )-\frac{1}{9} \left (64 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \sinh \left (\frac{x}{2}\right ) \, dx+\left (32 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int x \cosh \left (\frac{x}{2}\right ) \, dx\\ &=-\frac{128}{9} a \sqrt{a+a \cosh (x)}-16 a x^2 \sqrt{a+a \cosh (x)}-\frac{64}{27} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}-\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{32}{9} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{4}{3} a x^3 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{640}{9} a x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{8}{3} a x^3 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )-\left (64 a \sqrt{a+a \cosh (x)} \text{sech}\left (\frac{x}{2}\right )\right ) \int \sinh \left (\frac{x}{2}\right ) \, dx\\ &=-\frac{1280}{9} a \sqrt{a+a \cosh (x)}-16 a x^2 \sqrt{a+a \cosh (x)}-\frac{64}{27} a \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}-\frac{8}{3} a x^2 \cosh ^2\left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)}+\frac{32}{9} a x \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{4}{3} a x^3 \cosh \left (\frac{x}{2}\right ) \sqrt{a+a \cosh (x)} \sinh \left (\frac{x}{2}\right )+\frac{640}{9} a x \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )+\frac{8}{3} a x^3 \sqrt{a+a \cosh (x)} \tanh \left (\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.278957, size = 70, normalized size = 0.38 \[ \frac{2}{27} a \sqrt{a (\cosh (x)+1)} \left (-2 \left (117 x^2+968\right )+3 x \left (15 x^2+328\right ) \tanh \left (\frac{x}{2}\right )+\cosh (x) \left (3 x \left (3 x^2+8\right ) \tanh \left (\frac{x}{2}\right )-2 \left (9 x^2+8\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*Sqrt[a*(1 + Cosh[x])]*(-2*(968 + 117*x^2) + 3*x*(328 + 15*x^2)*Tanh[x/2] + Cosh[x]*(-2*(8 + 9*x^2) + 3*x*
(8 + 3*x^2)*Tanh[x/2])))/27

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

[Out]

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

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Maxima [A]  time = 1.67038, size = 243, normalized size = 1.31 \begin{align*} -\frac{1}{54} \,{\left (9 \, \sqrt{2} a^{\frac{3}{2}} x^{3} + 18 \, \sqrt{2} a^{\frac{3}{2}} x^{2} + 24 \, \sqrt{2} a^{\frac{3}{2}} x + 16 \, \sqrt{2} a^{\frac{3}{2}} -{\left (9 \, \sqrt{2} a^{\frac{3}{2}} x^{3} - 18 \, \sqrt{2} a^{\frac{3}{2}} x^{2} + 24 \, \sqrt{2} a^{\frac{3}{2}} x - 16 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \,{\left (\sqrt{2} a^{\frac{3}{2}} x^{3} - 6 \, \sqrt{2} a^{\frac{3}{2}} x^{2} + 24 \, \sqrt{2} a^{\frac{3}{2}} x - 48 \, \sqrt{2} a^{\frac{3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \,{\left (\sqrt{2} a^{\frac{3}{2}} x^{3} + 6 \, \sqrt{2} a^{\frac{3}{2}} x^{2} + 24 \, \sqrt{2} a^{\frac{3}{2}} x + 48 \, \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^3*(a+a*cosh(x))^(3/2),x, algorithm="maxima")

[Out]

-1/54*(9*sqrt(2)*a^(3/2)*x^3 + 18*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^(3/2)*x + 16*sqrt(2)*a^(3/2) - (9*sqrt(2)
*a^(3/2)*x^3 - 18*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^(3/2)*x - 16*sqrt(2)*a^(3/2))*e^(3*x) - 81*(sqrt(2)*a^(3/
2)*x^3 - 6*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^(3/2)*x - 48*sqrt(2)*a^(3/2))*e^(2*x) + 81*(sqrt(2)*a^(3/2)*x^3
+ 6*sqrt(2)*a^(3/2)*x^2 + 24*sqrt(2)*a^(3/2)*x + 48*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^3*(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**3*(a+a*cosh(x))**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.18834, size = 240, normalized size = 1.3 \begin{align*} \frac{1}{54} \, \sqrt{2}{\left (9 \, a^{\frac{3}{2}} x^{3} e^{\left (\frac{3}{2} \, x\right )} + 81 \, a^{\frac{3}{2}} x^{3} e^{\left (\frac{1}{2} \, x\right )} - 81 \, a^{\frac{3}{2}} x^{3} e^{\left (-\frac{1}{2} \, x\right )} - 9 \, a^{\frac{3}{2}} x^{3} e^{\left (-\frac{3}{2} \, x\right )} - 18 \, a^{\frac{3}{2}} x^{2} e^{\left (\frac{3}{2} \, x\right )} - 486 \, a^{\frac{3}{2}} x^{2} e^{\left (\frac{1}{2} \, x\right )} - 486 \, a^{\frac{3}{2}} x^{2} e^{\left (-\frac{1}{2} \, x\right )} - 18 \, a^{\frac{3}{2}} x^{2} e^{\left (-\frac{3}{2} \, x\right )} + 24 \, a^{\frac{3}{2}} x e^{\left (\frac{3}{2} \, x\right )} + 1944 \, a^{\frac{3}{2}} x e^{\left (\frac{1}{2} \, x\right )} - 1944 \, a^{\frac{3}{2}} x e^{\left (-\frac{1}{2} \, x\right )} - 24 \, a^{\frac{3}{2}} x e^{\left (-\frac{3}{2} \, x\right )} - 16 \, a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} - 3888 \, a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - 3888 \, a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, x\right )} - 16 \, 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^3*(a+a*cosh(x))^(3/2),x, algorithm="giac")

[Out]

1/54*sqrt(2)*(9*a^(3/2)*x^3*e^(3/2*x) + 81*a^(3/2)*x^3*e^(1/2*x) - 81*a^(3/2)*x^3*e^(-1/2*x) - 9*a^(3/2)*x^3*e
^(-3/2*x) - 18*a^(3/2)*x^2*e^(3/2*x) - 486*a^(3/2)*x^2*e^(1/2*x) - 486*a^(3/2)*x^2*e^(-1/2*x) - 18*a^(3/2)*x^2
*e^(-3/2*x) + 24*a^(3/2)*x*e^(3/2*x) + 1944*a^(3/2)*x*e^(1/2*x) - 1944*a^(3/2)*x*e^(-1/2*x) - 24*a^(3/2)*x*e^(
-3/2*x) - 16*a^(3/2)*e^(3/2*x) - 3888*a^(3/2)*e^(1/2*x) - 3888*a^(3/2)*e^(-1/2*x) - 16*a^(3/2)*e^(-3/2*x))