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]

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

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Rubi [A]  time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.09, 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]

-1/3*(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])/d

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fricas [B]  time = 0.51, size = 139, normalized size = 2.28 \[ -\frac {\sqrt {\frac {1}{2}} {\left (a \cosh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{3} - 9 \, a \cosh \left (d x + c\right )^{2} + 3 \, {\left (a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right )^{2} - 9 \, a \cosh \left (d x + c\right ) + 3 \, {\left (a \cosh \left (d x + c\right )^{2} - 6 \, a \cosh \left (d x + c\right ) - 3 \, a\right )} \sinh \left (d x + c\right ) + a\right )} \sqrt {-\frac {a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{3 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \]

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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giac [B]  time = 0.16, size = 119, normalized size = 1.95 \[ \frac {\sqrt {2} {\left (\sqrt {-a} a e^{\left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 9 \, \sqrt {-a} a e^{\left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 9 \, \sqrt {-a} a e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + \sqrt {-a} a e^{\left (-\frac {3}{2} \, d x - \frac {3}{2} \, c\right )} \mathrm {sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )}}{6 \, d} \]

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

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maple [A]  time = 0.25, size = 56, normalized size = 0.92 \[ \frac {8 \sinh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} \cosh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\cosh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{3 \sqrt {-2 a \left (\sinh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]

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*a*sinh(1/2*d*x+1/2*c)^2)^(1/2)/d

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maxima [B]  time = 0.43, size = 124, normalized size = 2.03 \[ \frac {3 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-d x - c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {3}{2}}} + \frac {3 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, d x - 3 \, c\right )}}{6 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}}}{6 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac {3}{2}}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a-a\,\mathrm {cosh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*cosh(c + d*x))^(3/2),x)

[Out]

int((a - a*cosh(c + d*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- a \cosh {\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]

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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