3.48 \(\int (a-a \cosh (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=92 \[ -\frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a-a \cosh (c+d x)}}-\frac{16 a^2 \sinh (c+d x) \sqrt{a-a \cosh (c+d x)}}{15 d}-\frac{2 a \sinh (c+d x) (a-a \cosh (c+d x))^{3/2}}{5 d} \]

[Out]

(-64*a^3*Sinh[c + d*x])/(15*d*Sqrt[a - a*Cosh[c + d*x]]) - (16*a^2*Sqrt[a - a*Cosh[c + d*x]]*Sinh[c + d*x])/(1
5*d) - (2*a*(a - a*Cosh[c + d*x])^(3/2)*Sinh[c + d*x])/(5*d)

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Rubi [A]  time = 0.0520059, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2647, 2646} \[ -\frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a-a \cosh (c+d x)}}-\frac{16 a^2 \sinh (c+d x) \sqrt{a-a \cosh (c+d x)}}{15 d}-\frac{2 a \sinh (c+d x) (a-a \cosh (c+d x))^{3/2}}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Cosh[c + d*x])^(5/2),x]

[Out]

(-64*a^3*Sinh[c + d*x])/(15*d*Sqrt[a - a*Cosh[c + d*x]]) - (16*a^2*Sqrt[a - a*Cosh[c + d*x]]*Sinh[c + d*x])/(1
5*d) - (2*a*(a - a*Cosh[c + d*x])^(3/2)*Sinh[c + d*x])/(5*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))^{5/2} \, dx &=-\frac{2 a (a-a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d}+\frac{1}{5} (8 a) \int (a-a \cosh (c+d x))^{3/2} \, dx\\ &=-\frac{16 a^2 \sqrt{a-a \cosh (c+d x)} \sinh (c+d x)}{15 d}-\frac{2 a (a-a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d}+\frac{1}{15} \left (32 a^2\right ) \int \sqrt{a-a \cosh (c+d x)} \, dx\\ &=-\frac{64 a^3 \sinh (c+d x)}{15 d \sqrt{a-a \cosh (c+d x)}}-\frac{16 a^2 \sqrt{a-a \cosh (c+d x)} \sinh (c+d x)}{15 d}-\frac{2 a (a-a \cosh (c+d x))^{3/2} \sinh (c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.132919, size = 72, normalized size = 0.78 \[ \frac{a^2 \left (150 \cosh \left (\frac{1}{2} (c+d x)\right )-25 \cosh \left (\frac{3}{2} (c+d x)\right )+3 \cosh \left (\frac{5}{2} (c+d x)\right )\right ) \text{csch}\left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \cosh (c+d x)}}{30 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - a*Cosh[c + d*x])^(5/2),x]

[Out]

(a^2*Sqrt[a - a*Cosh[c + d*x]]*(150*Cosh[(c + d*x)/2] - 25*Cosh[(3*(c + d*x))/2] + 3*Cosh[(5*(c + d*x))/2])*Cs
ch[(c + d*x)/2])/(30*d)

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Maple [A]  time = 0.042, size = 71, normalized size = 0.8 \begin{align*} -{\frac{16\,{a}^{3}}{15\,d}\sinh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \cosh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 3\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-4\, \left ( \sinh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+8 \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))^(5/2),x)

[Out]

-16/15*sinh(1/2*d*x+1/2*c)*a^3*cosh(1/2*d*x+1/2*c)*(3*sinh(1/2*d*x+1/2*c)^4-4*sinh(1/2*d*x+1/2*c)^2+8)/(-2*sin
h(1/2*d*x+1/2*c)^2*a)^(1/2)/d

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Maxima [B]  time = 1.61936, size = 257, normalized size = 2.79 \begin{align*} \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-d x - c\right )}}{12 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, d x - 2 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, d x - 3 \, c\right )}}{2 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} + \frac{5 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, d x - 4 \, c\right )}}{12 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{\sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, d x - 5 \, c\right )}}{20 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} - \frac{\sqrt{2} a^{\frac{5}{2}}}{20 \, d \left (-e^{\left (-d x - c\right )}\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

5/12*sqrt(2)*a^(5/2)*e^(-d*x - c)/(d*(-e^(-d*x - c))^(5/2)) - 5/2*sqrt(2)*a^(5/2)*e^(-2*d*x - 2*c)/(d*(-e^(-d*
x - c))^(5/2)) - 5/2*sqrt(2)*a^(5/2)*e^(-3*d*x - 3*c)/(d*(-e^(-d*x - c))^(5/2)) + 5/12*sqrt(2)*a^(5/2)*e^(-4*d
*x - 4*c)/(d*(-e^(-d*x - c))^(5/2)) - 1/20*sqrt(2)*a^(5/2)*e^(-5*d*x - 5*c)/(d*(-e^(-d*x - c))^(5/2)) - 1/20*s
qrt(2)*a^(5/2)/(d*(-e^(-d*x - c))^(5/2))

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Fricas [B]  time = 1.8034, size = 857, normalized size = 9.32 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (3 \, a^{2} \cosh \left (d x + c\right )^{5} + 3 \, a^{2} \sinh \left (d x + c\right )^{5} - 25 \, a^{2} \cosh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{3} + 5 \,{\left (3 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{4} + 150 \, a^{2} \cosh \left (d x + c\right )^{2} + 10 \,{\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} - 10 \, a^{2} \cosh \left (d x + c\right ) + 15 \, a^{2}\right )} \sinh \left (d x + c\right )^{3} - 25 \, a^{2} \cosh \left (d x + c\right ) + 30 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} - 5 \, a^{2} \cosh \left (d x + c\right )^{2} + 15 \, a^{2} \cosh \left (d x + c\right ) + 5 \, a^{2}\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} + 5 \,{\left (3 \, a^{2} \cosh \left (d x + c\right )^{4} - 20 \, a^{2} \cosh \left (d x + c\right )^{3} + 90 \, a^{2} \cosh \left (d x + c\right )^{2} + 60 \, a^{2} \cosh \left (d x + c\right ) - 5 \, a^{2}\right )} \sinh \left (d x + c\right )\right )} \sqrt{-\frac{a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}}}{30 \,{\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

1/30*sqrt(1/2)*(3*a^2*cosh(d*x + c)^5 + 3*a^2*sinh(d*x + c)^5 - 25*a^2*cosh(d*x + c)^4 + 150*a^2*cosh(d*x + c)
^3 + 5*(3*a^2*cosh(d*x + c) - 5*a^2)*sinh(d*x + c)^4 + 150*a^2*cosh(d*x + c)^2 + 10*(3*a^2*cosh(d*x + c)^2 - 1
0*a^2*cosh(d*x + c) + 15*a^2)*sinh(d*x + c)^3 - 25*a^2*cosh(d*x + c) + 30*(a^2*cosh(d*x + c)^3 - 5*a^2*cosh(d*
x + c)^2 + 15*a^2*cosh(d*x + c) + 5*a^2)*sinh(d*x + c)^2 + 3*a^2 + 5*(3*a^2*cosh(d*x + c)^4 - 20*a^2*cosh(d*x
+ c)^3 + 90*a^2*cosh(d*x + c)^2 + 60*a^2*cosh(d*x + c) - 5*a^2)*sinh(d*x + c))*sqrt(-a/(cosh(d*x + c) + sinh(d
*x + c)))/(d*cosh(d*x + c)^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^2)

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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((a-a*cosh(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.18001, size = 262, normalized size = 2.85 \begin{align*} -\frac{\sqrt{2}{\left (3 \, \sqrt{-a e^{\left (d x + c\right )}} a^{2} e^{\left (2 \, d x + 2 \, c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 25 \, \sqrt{-a e^{\left (d x + c\right )}} a^{2} e^{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 150 \, \sqrt{-a e^{\left (d x + c\right )}} a^{2} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - \frac{{\left (150 \, a^{5} e^{\left (2 \, d x + 2 \, c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) - 25 \, a^{5} e^{\left (d x + c\right )} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right ) + 3 \, a^{5} \mathrm{sgn}\left (-e^{\left (d x + c\right )} + 1\right )\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{\sqrt{-a e^{\left (d x + c\right )}} a^{2}}\right )}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*cosh(d*x+c))^(5/2),x, algorithm="giac")

[Out]

-1/60*sqrt(2)*(3*sqrt(-a*e^(d*x + c))*a^2*e^(2*d*x + 2*c)*sgn(-e^(d*x + c) + 1) - 25*sqrt(-a*e^(d*x + c))*a^2*
e^(d*x + c)*sgn(-e^(d*x + c) + 1) + 150*sqrt(-a*e^(d*x + c))*a^2*sgn(-e^(d*x + c) + 1) - (150*a^5*e^(2*d*x + 2
*c)*sgn(-e^(d*x + c) + 1) - 25*a^5*e^(d*x + c)*sgn(-e^(d*x + c) + 1) + 3*a^5*sgn(-e^(d*x + c) + 1))*e^(-2*d*x
- 2*c)/(sqrt(-a*e^(d*x + c))*a^2))/d