3.49 \(\int (a-a \cosh (c+d x))^{3/2} \, dx\)

Optimal. Leaf size=61 \[ -\frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a-a \cosh (c+d x)}}-\frac{2 a \sinh (c+d x) \sqrt{a-a \cosh (c+d x)}}{3 d} \]

[Out]

(-8*a^2*Sinh[c + d*x])/(3*d*Sqrt[a - a*Cosh[c + d*x]]) - (2*a*Sqrt[a - a*Cosh[c + d*x]]*Sinh[c + d*x])/(3*d)

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Rubi [A]  time = 0.031382, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2647, 2646} \[ -\frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a-a \cosh (c+d x)}}-\frac{2 a \sinh (c+d x) \sqrt{a-a \cosh (c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Cosh[c + d*x])^(3/2),x]

[Out]

(-8*a^2*Sinh[c + d*x])/(3*d*Sqrt[a - a*Cosh[c + d*x]]) - (2*a*Sqrt[a - a*Cosh[c + d*x]]*Sinh[c + d*x])/(3*d)

Rule 2647

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^(n -
1))/(d*n), x] + Dist[(a*(2*n - 1))/n, Int[(a + b*Sin[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && Eq
Q[a^2 - b^2, 0] && IGtQ[n - 1/2, 0]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int (a-a \cosh (c+d x))^{3/2} \, dx &=-\frac{2 a \sqrt{a-a \cosh (c+d x)} \sinh (c+d x)}{3 d}+\frac{1}{3} (4 a) \int \sqrt{a-a \cosh (c+d x)} \, dx\\ &=-\frac{8 a^2 \sinh (c+d x)}{3 d \sqrt{a-a \cosh (c+d x)}}-\frac{2 a \sqrt{a-a \cosh (c+d x)} \sinh (c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.09078, size = 56, normalized size = 0.92 \[ -\frac{a \left (\cosh \left (\frac{3}{2} (c+d x)\right )-9 \cosh \left (\frac{1}{2} (c+d x)\right )\right ) \text{csch}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cosh (c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[a - a*Cosh[c + d*x]]*(-9*Cosh[(c + d*x)/2] + Cosh[(3*(c + d*x))/2])*Csch[(c + d*x)/2])/(3*d)

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Maple [A]  time = 0.036, size = 56, normalized size = 0.9 \begin{align*}{\frac{8\,{a}^{2}}{3\,d}\sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}-3 \right ){\frac{1}{\sqrt{-2\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*sinh(1/2*d*x+1/2*c)*a^2*cosh(1/2*d*x+1/2*c)*(cosh(1/2*d*x+1/2*c)^2-3)/(-2*sinh(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [B]  time = 1.60667, size = 167, normalized size = 2.74 \begin{align*} \frac{3 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-d x - c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{3}{2}}} + \frac{3 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, d x - 3 \, c\right )}}{6 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} a^{\frac{3}{2}}}{6 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*sqrt(2)*a^(3/2)*e^(-d*x - c)/(d*(-e^(-d*x - c))^(3/2)) + 3/2*sqrt(2)*a^(3/2)*e^(-2*d*x - 2*c)/(d*(-e^(-d*x
- c))^(3/2)) - 1/6*sqrt(2)*a^(3/2)*e^(-3*d*x - 3*c)/(d*(-e^(-d*x - c))^(3/2)) - 1/6*sqrt(2)*a^(3/2)/(d*(-e^(-
d*x - c))^(3/2))

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Fricas [B]  time = 1.8818, size = 387, normalized size = 6.34 \begin{align*} -\frac{\sqrt{\frac{1}{2}}{\left (a \cosh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{3} - 9 \, a \cosh \left (d x + c\right )^{2} + 3 \,{\left (a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 3 \,{\left (a \cosh \left (d x + c\right )^{2} - 6 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt{-\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{3 \,{\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(1/2)*(a*cosh(d*x + c)^3 + a*sinh(d*x + c)^3 - 9*a*cosh(d*x + c)^2 + 3*(a*cosh(d*x + c) - 3*a)*sinh(d
*x + c)^2 - 9*a*cosh(d*x + c) + 3*(a*cosh(d*x + c)^2 - 6*a*cosh(d*x + c) - 3*a)*sinh(d*x + c) + a)*sqrt(-a/(co
sh(d*x + c) + sinh(d*x + c)))/(d*cosh(d*x + c) + d*sinh(d*x + c))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(d*x+c))**(3/2),x)

[Out]

Integral((-a*cosh(c + d*x) + a)**(3/2), x)

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Giac [B]  time = 1.18524, size = 171, normalized size = 2.8 \begin{align*} \frac{\sqrt{2}{\left (\sqrt{-a e^{\left (d x + c\right )}} a e^{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 9 \, \sqrt{-a e^{\left (d x + c\right )}} a \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + \frac{{\left (9 \, a^{3} e^{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - a^{3} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )} e^{\left (-d x - c\right )}}{\sqrt{-a e^{\left (d x + c\right )}} a}\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(2)*(sqrt(-a*e^(d*x + c))*a*e^(d*x + c)*sgn(-e^(d*x + c) + 1) - 9*sqrt(-a*e^(d*x + c))*a*sgn(-e^(d*x +
c) + 1) + (9*a^3*e^(d*x + c)*sgn(-e^(d*x + c) + 1) - a^3*sgn(-e^(d*x + c) + 1))*e^(-d*x - c)/(sqrt(-a*e^(d*x
+ c))*a))/d