3.91 \(\int (a-a \cosh (x))^{3/2} (A+B \cosh (x)) \, dx\)

Optimal. Leaf size=71 \[ -\frac{8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt{a-a \cosh (x)}}-\frac{2}{15} a (5 A-3 B) \sinh (x) \sqrt{a-a \cosh (x)}+\frac{2}{5} B \sinh (x) (a-a \cosh (x))^{3/2} \]

[Out]

(-8*a^2*(5*A - 3*B)*Sinh[x])/(15*Sqrt[a - a*Cosh[x]]) - (2*a*(5*A - 3*B)*Sqrt[a - a*Cosh[x]]*Sinh[x])/15 + (2*
B*(a - a*Cosh[x])^(3/2)*Sinh[x])/5

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Rubi [A]  time = 0.0802921, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2751, 2647, 2646} \[ -\frac{8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt{a-a \cosh (x)}}-\frac{2}{15} a (5 A-3 B) \sinh (x) \sqrt{a-a \cosh (x)}+\frac{2}{5} B \sinh (x) (a-a \cosh (x))^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Cosh[x])^(3/2)*(A + B*Cosh[x]),x]

[Out]

(-8*a^2*(5*A - 3*B)*Sinh[x])/(15*Sqrt[a - a*Cosh[x]]) - (2*a*(5*A - 3*B)*Sqrt[a - a*Cosh[x]]*Sinh[x])/15 + (2*
B*(a - a*Cosh[x])^(3/2)*Sinh[x])/5

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

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 (x))^{3/2} (A+B \cosh (x)) \, dx &=\frac{2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)-\frac{1}{5} (-5 A+3 B) \int (a-a \cosh (x))^{3/2} \, dx\\ &=-\frac{2}{15} a (5 A-3 B) \sqrt{a-a \cosh (x)} \sinh (x)+\frac{2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)+\frac{1}{15} (4 a (5 A-3 B)) \int \sqrt{a-a \cosh (x)} \, dx\\ &=-\frac{8 a^2 (5 A-3 B) \sinh (x)}{15 \sqrt{a-a \cosh (x)}}-\frac{2}{15} a (5 A-3 B) \sqrt{a-a \cosh (x)} \sinh (x)+\frac{2}{5} B (a-a \cosh (x))^{3/2} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0971045, size = 47, normalized size = 0.66 \[ -\frac{1}{15} a \coth \left (\frac{x}{2}\right ) \sqrt{a-a \cosh (x)} (2 (5 A-9 B) \cosh (x)-50 A+3 B \cosh (2 x)+39 B) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Sqrt[a - a*Cosh[x]]*(-50*A + 39*B + 2*(5*A - 9*B)*Cosh[x] + 3*B*Cosh[2*x])*Coth[x/2])/15

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Maple [A]  time = 0.049, size = 55, normalized size = 0.8 \begin{align*}{\frac{8\,{a}^{2}}{15}\sinh \left ({\frac{x}{2}} \right ) \cosh \left ({\frac{x}{2}} \right ) \left ( 6\,B \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( 5\,A-3\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}-10\,A+6\,B \right ){\frac{1}{\sqrt{-2\, \left ( \sinh \left ( x/2 \right ) \right ) ^{2}a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/15*sinh(1/2*x)*a^2*cosh(1/2*x)*(6*B*sinh(1/2*x)^4+(5*A-3*B)*sinh(1/2*x)^2-10*A+6*B)/(-2*sinh(1/2*x)^2*a)^(1/
2)

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Maxima [B]  time = 1.6047, size = 269, normalized size = 3.79 \begin{align*} \frac{1}{6} \,{\left (\frac{9 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} + \frac{9 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} a^{\frac{3}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}}\right )} A + \frac{1}{20} \, B{\left (\frac{{\left (5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )} - 15 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )} - \sqrt{2} a^{\frac{3}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )} + 15 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )} + \sqrt{2} a^{\frac{3}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{3}{2}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(9*sqrt(2)*a^(3/2)*e^(-x)/(-e^(-x))^(3/2) + 9*sqrt(2)*a^(3/2)*e^(-2*x)/(-e^(-x))^(3/2) - sqrt(2)*a^(3/2)*e
^(-3*x)/(-e^(-x))^(3/2) - sqrt(2)*a^(3/2)/(-e^(-x))^(3/2))*A + 1/20*B*((5*sqrt(2)*a^(3/2)*e^(-x) - 15*sqrt(2)*
a^(3/2)*e^(-2*x) - 5*sqrt(2)*a^(3/2)*e^(-3*x) - sqrt(2)*a^(3/2))*e^x/(-e^(-x))^(3/2) - (5*sqrt(2)*a^(3/2)*e^(-
x) + 15*sqrt(2)*a^(3/2)*e^(-2*x) - 5*sqrt(2)*a^(3/2)*e^(-3*x) + sqrt(2)*a^(3/2)*e^(-4*x))/(-e^(-x))^(3/2))

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Fricas [B]  time = 2.15376, size = 807, normalized size = 11.37 \begin{align*} -\frac{\sqrt{\frac{1}{2}}{\left (3 \, B a \cosh \left (x\right )^{5} + 3 \, B a \sinh \left (x\right )^{5} + 5 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{4} - 30 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{3} + 5 \,{\left (3 \, B a \cosh \left (x\right ) +{\left (2 \, A - 3 \, B\right )} a\right )} \sinh \left (x\right )^{4} - 30 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{2} + 10 \,{\left (3 \, B a \cosh \left (x\right )^{2} + 2 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right ) - 3 \,{\left (3 \, A - 2 \, B\right )} a\right )} \sinh \left (x\right )^{3} + 5 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right ) + 30 \,{\left (B a \cosh \left (x\right )^{3} +{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{2} - 3 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right ) -{\left (3 \, A - 2 \, B\right )} a\right )} \sinh \left (x\right )^{2} + 3 \, B a + 5 \,{\left (3 \, B a \cosh \left (x\right )^{4} + 4 \,{\left (2 \, A - 3 \, B\right )} a \cosh \left (x\right )^{3} - 18 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right )^{2} - 12 \,{\left (3 \, A - 2 \, B\right )} a \cosh \left (x\right ) +{\left (2 \, A - 3 \, B\right )} a\right )} \sinh \left (x\right )\right )} \sqrt{-\frac{a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{30 \,{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*sqrt(1/2)*(3*B*a*cosh(x)^5 + 3*B*a*sinh(x)^5 + 5*(2*A - 3*B)*a*cosh(x)^4 - 30*(3*A - 2*B)*a*cosh(x)^3 +
5*(3*B*a*cosh(x) + (2*A - 3*B)*a)*sinh(x)^4 - 30*(3*A - 2*B)*a*cosh(x)^2 + 10*(3*B*a*cosh(x)^2 + 2*(2*A - 3*B)
*a*cosh(x) - 3*(3*A - 2*B)*a)*sinh(x)^3 + 5*(2*A - 3*B)*a*cosh(x) + 30*(B*a*cosh(x)^3 + (2*A - 3*B)*a*cosh(x)^
2 - 3*(3*A - 2*B)*a*cosh(x) - (3*A - 2*B)*a)*sinh(x)^2 + 3*B*a + 5*(3*B*a*cosh(x)^4 + 4*(2*A - 3*B)*a*cosh(x)^
3 - 18*(3*A - 2*B)*a*cosh(x)^2 - 12*(3*A - 2*B)*a*cosh(x) + (2*A - 3*B)*a)*sinh(x))*sqrt(-a/(cosh(x) + sinh(x)
))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^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(x))**(3/2)*(A+B*cosh(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.25391, size = 286, normalized size = 4.03 \begin{align*} \frac{1}{60} \, \sqrt{2}{\left (\frac{{\left (90 \, A a^{4} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 60 \, B a^{4} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 10 \, A a^{4} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{4} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{4} \mathrm{sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-2 \, x\right )}}{\sqrt{-a e^{x}} a^{2}} + \frac{3 \, \sqrt{-a e^{x}} B a^{6} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) + 10 \, \sqrt{-a e^{x}} A a^{6} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 15 \, \sqrt{-a e^{x}} B a^{6} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 90 \, \sqrt{-a e^{x}} A a^{6} \mathrm{sgn}\left (-e^{x} + 1\right ) + 60 \, \sqrt{-a e^{x}} B a^{6} \mathrm{sgn}\left (-e^{x} + 1\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*sqrt(2)*((90*A*a^4*e^(2*x)*sgn(-e^x + 1) - 60*B*a^4*e^(2*x)*sgn(-e^x + 1) - 10*A*a^4*e^x*sgn(-e^x + 1) +
15*B*a^4*e^x*sgn(-e^x + 1) - 3*B*a^4*sgn(-e^x + 1))*e^(-2*x)/(sqrt(-a*e^x)*a^2) + (3*sqrt(-a*e^x)*B*a^6*e^(2*x
)*sgn(-e^x + 1) + 10*sqrt(-a*e^x)*A*a^6*e^x*sgn(-e^x + 1) - 15*sqrt(-a*e^x)*B*a^6*e^x*sgn(-e^x + 1) - 90*sqrt(
-a*e^x)*A*a^6*sgn(-e^x + 1) + 60*sqrt(-a*e^x)*B*a^6*sgn(-e^x + 1))/a^5)