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

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

[Out]

-1280/9*a*(a+a*cosh(x))^(1/2)-16*a*x^2*(a+a*cosh(x))^(1/2)-64/27*a*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)-8/3*a*x^2
*cosh(1/2*x)^2*(a+a*cosh(x))^(1/2)+32/9*a*x*cosh(1/2*x)*sinh(1/2*x)*(a+a*cosh(x))^(1/2)+4/3*a*x^3*cosh(1/2*x)*
sinh(1/2*x)*(a+a*cosh(x))^(1/2)+640/9*a*x*(a+a*cosh(x))^(1/2)*tanh(1/2*x)+8/3*a*x^3*(a+a*cosh(x))^(1/2)*tanh(1
/2*x)

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Rubi [A]  time = 0.19, antiderivative size = 185, normalized size of antiderivative = 1.00, 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 2638

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

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 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 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 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])

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*}

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Mathematica [A]  time = 0.29, 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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fricas [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(x^3*(a+a*cosh(x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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giac [A]  time = 0.14, size = 192, normalized size = 1.04 \[ -\frac {1}{54} \, \sqrt {2} {\left (54 \, a^{\frac {3}{2}} x^{3} e^{\left (-\frac {1}{2} \, x\right )} + 9 \, a^{\frac {3}{2}} x^{3} e^{\left (-\frac {3}{2} \, x\right )} + 324 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {1}{2} \, x\right )} + 18 \, a^{\frac {3}{2}} x^{2} e^{\left (-\frac {3}{2} \, x\right )} + 1296 \, a^{\frac {3}{2}} x e^{\left (-\frac {1}{2} \, x\right )} + 24 \, a^{\frac {3}{2}} x e^{\left (-\frac {3}{2} \, x\right )} + 2592 \, a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} + 16 \, a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )} - {\left (9 \, a^{\frac {3}{2}} x^{3} - 18 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x - 16 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {3}{2} \, x\right )} - 81 \, {\left (a^{\frac {3}{2}} x^{3} - 6 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x - 48 \, a^{\frac {3}{2}}\right )} e^{\left (\frac {1}{2} \, x\right )} + 27 \, {\left (a^{\frac {3}{2}} x^{3} + 6 \, a^{\frac {3}{2}} x^{2} + 24 \, a^{\frac {3}{2}} x + 48 \, a^{\frac {3}{2}}\right )} e^{\left (-\frac {1}{2} \, x\right )}\right )} \]

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)*(54*a^(3/2)*x^3*e^(-1/2*x) + 9*a^(3/2)*x^3*e^(-3/2*x) + 324*a^(3/2)*x^2*e^(-1/2*x) + 18*a^(3/2)*
x^2*e^(-3/2*x) + 1296*a^(3/2)*x*e^(-1/2*x) + 24*a^(3/2)*x*e^(-3/2*x) + 2592*a^(3/2)*e^(-1/2*x) + 16*a^(3/2)*e^
(-3/2*x) - (9*a^(3/2)*x^3 - 18*a^(3/2)*x^2 + 24*a^(3/2)*x - 16*a^(3/2))*e^(3/2*x) - 81*(a^(3/2)*x^3 - 6*a^(3/2
)*x^2 + 24*a^(3/2)*x - 48*a^(3/2))*e^(1/2*x) + 27*(a^(3/2)*x^3 + 6*a^(3/2)*x^2 + 24*a^(3/2)*x + 48*a^(3/2))*e^
(-1/2*x))

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +a \cosh \relax (x )\right )^{\frac {3}{2}}\, dx \]

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 = 0.45, size = 180, normalized size = 0.97 \[ -\frac {1}{54} \, {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{3} + 18 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x + 16 \, \sqrt {2} a^{\frac {3}{2}} - {\left (9 \, \sqrt {2} a^{\frac {3}{2}} x^{3} - 18 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x - 16 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (3 \, x\right )} - 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} - 6 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x - 48 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{\left (2 \, x\right )} + 81 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + 6 \, \sqrt {2} a^{\frac {3}{2}} x^{2} + 24 \, \sqrt {2} a^{\frac {3}{2}} x + 48 \, \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}\right )} e^{\left (-\frac {3}{2} \, x\right )} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^3\,{\left (a+a\,\mathrm {cosh}\relax (x)\right )}^{3/2} \,d x \]

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

[Out]

Timed out

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