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]

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

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Rubi [A]  time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, 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 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]

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)]

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*}

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Mathematica [A]  time = 0.11, 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]

-1/15*(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])

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fricas [B]  time = 0.52, size = 279, normalized size = 3.93 \[ -\frac {\sqrt {\frac {1}{2}} {\left (3 \, B a \cosh \relax (x)^{5} + 3 \, B a \sinh \relax (x)^{5} + 5 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x)^{4} - 30 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x)^{3} + 5 \, {\left (3 \, B a \cosh \relax (x) + {\left (2 \, A - 3 \, B\right )} a\right )} \sinh \relax (x)^{4} - 30 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x)^{2} + 10 \, {\left (3 \, B a \cosh \relax (x)^{2} + 2 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x) - 3 \, {\left (3 \, A - 2 \, B\right )} a\right )} \sinh \relax (x)^{3} + 5 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x) + 30 \, {\left (B a \cosh \relax (x)^{3} + {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x)^{2} - 3 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x) - {\left (3 \, A - 2 \, B\right )} a\right )} \sinh \relax (x)^{2} + 3 \, B a + 5 \, {\left (3 \, B a \cosh \relax (x)^{4} + 4 \, {\left (2 \, A - 3 \, B\right )} a \cosh \relax (x)^{3} - 18 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x)^{2} - 12 \, {\left (3 \, A - 2 \, B\right )} a \cosh \relax (x) + {\left (2 \, A - 3 \, B\right )} a\right )} \sinh \relax (x)\right )} \sqrt {-\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{30 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]

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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giac [B]  time = 0.15, size = 212, normalized size = 2.99 \[ \frac {1}{60} \, \sqrt {2} {\left (\frac {{\left (90 \, A a^{4} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 60 \, B a^{4} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) - 10 \, A a^{4} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{4} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 3 \, B a^{4} \mathrm {sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-2 \, x\right )}}{\sqrt {-a e^{x}} a^{2}} + \frac {3 \, \sqrt {-a e^{x}} B a^{6} e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{x} + 1\right ) + 10 \, \sqrt {-a e^{x}} A a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 15 \, \sqrt {-a e^{x}} B a^{6} e^{x} \mathrm {sgn}\left (-e^{x} + 1\right ) - 90 \, \sqrt {-a e^{x}} A a^{6} \mathrm {sgn}\left (-e^{x} + 1\right ) + 60 \, \sqrt {-a e^{x}} B a^{6} \mathrm {sgn}\left (-e^{x} + 1\right )}{a^{5}}\right )} \]

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)

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maple [A]  time = 0.32, size = 55, normalized size = 0.77 \[ \frac {8 \sinh \left (\frac {x}{2}\right ) a^{2} \cosh \left (\frac {x}{2}\right ) \left (6 B \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (5 A -3 B \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )-10 A +6 B \right )}{15 \sqrt {-2 a \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )}} \]

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*a*sinh(1/2*x)^2)^(1/
2)

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maxima [B]  time = 0.47, size = 199, normalized size = 2.80 \[ \frac {1}{6} \, {\left (\frac {9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} + \frac {9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {\sqrt {2} a^{\frac {3}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}}\right )} A + \frac {1}{20} \, B {\left (\frac {{\left (5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} - 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - \sqrt {2} a^{\frac {3}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} + 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} + \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac {3}{2}}}\right )} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (- a \left (\cosh {\relax (x )} - 1\right )\right )^{\frac {3}{2}} \left (A + B \cosh {\relax (x )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-a*(cosh(x) - 1))**(3/2)*(A + B*cosh(x)), x)

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