3.90 \(\int (a-a \cosh (x))^{5/2} (A+B \cosh (x)) \, dx\)

Optimal. Leaf size=98 \[ -\frac{64 a^3 (7 A-5 B) \sinh (x)}{105 \sqrt{a-a \cosh (x)}}-\frac{16}{105} a^2 (7 A-5 B) \sinh (x) \sqrt{a-a \cosh (x)}-\frac{2}{35} a (7 A-5 B) \sinh (x) (a-a \cosh (x))^{3/2}+\frac{2}{7} B \sinh (x) (a-a \cosh (x))^{5/2} \]

[Out]

(-64*a^3*(7*A - 5*B)*Sinh[x])/(105*Sqrt[a - a*Cosh[x]]) - (16*a^2*(7*A - 5*B)*Sqrt[a - a*Cosh[x]]*Sinh[x])/105
- (2*a*(7*A - 5*B)*(a - a*Cosh[x])^(3/2)*Sinh[x])/35 + (2*B*(a - a*Cosh[x])^(5/2)*Sinh[x])/7

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Rubi [A]  time = 0.103554, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2751, 2647, 2646} \[ -\frac{64 a^3 (7 A-5 B) \sinh (x)}{105 \sqrt{a-a \cosh (x)}}-\frac{16}{105} a^2 (7 A-5 B) \sinh (x) \sqrt{a-a \cosh (x)}-\frac{2}{35} a (7 A-5 B) \sinh (x) (a-a \cosh (x))^{3/2}+\frac{2}{7} B \sinh (x) (a-a \cosh (x))^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a - a*Cosh[x])^(5/2)*(A + B*Cosh[x]),x]

[Out]

(-64*a^3*(7*A - 5*B)*Sinh[x])/(105*Sqrt[a - a*Cosh[x]]) - (16*a^2*(7*A - 5*B)*Sqrt[a - a*Cosh[x]]*Sinh[x])/105
- (2*a*(7*A - 5*B)*(a - a*Cosh[x])^(3/2)*Sinh[x])/35 + (2*B*(a - a*Cosh[x])^(5/2)*Sinh[x])/7

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))^{5/2} (A+B \cosh (x)) \, dx &=\frac{2}{7} B (a-a \cosh (x))^{5/2} \sinh (x)-\frac{1}{7} (-7 A+5 B) \int (a-a \cosh (x))^{5/2} \, dx\\ &=-\frac{2}{35} a (7 A-5 B) (a-a \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a-a \cosh (x))^{5/2} \sinh (x)+\frac{1}{35} (8 a (7 A-5 B)) \int (a-a \cosh (x))^{3/2} \, dx\\ &=-\frac{16}{105} a^2 (7 A-5 B) \sqrt{a-a \cosh (x)} \sinh (x)-\frac{2}{35} a (7 A-5 B) (a-a \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a-a \cosh (x))^{5/2} \sinh (x)+\frac{1}{105} \left (32 a^2 (7 A-5 B)\right ) \int \sqrt{a-a \cosh (x)} \, dx\\ &=-\frac{64 a^3 (7 A-5 B) \sinh (x)}{105 \sqrt{a-a \cosh (x)}}-\frac{16}{105} a^2 (7 A-5 B) \sqrt{a-a \cosh (x)} \sinh (x)-\frac{2}{35} a (7 A-5 B) (a-a \cosh (x))^{3/2} \sinh (x)+\frac{2}{7} B (a-a \cosh (x))^{5/2} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.134729, size = 61, normalized size = 0.62 \[ \frac{1}{210} a^2 \coth \left (\frac{x}{2}\right ) \sqrt{a-a \cosh (x)} ((505 B-392 A) \cosh (x)+6 (7 A-20 B) \cosh (2 x)+1246 A+15 B \cosh (3 x)-1040 B) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*Sqrt[a - a*Cosh[x]]*(1246*A - 1040*B + (-392*A + 505*B)*Cosh[x] + 6*(7*A - 20*B)*Cosh[2*x] + 15*B*Cosh[3*
x])*Coth[x/2])/210

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Maple [A]  time = 0.05, size = 69, normalized size = 0.7 \begin{align*} -{\frac{16\,{a}^{3}}{105}\sinh \left ({\frac{x}{2}} \right ) \cosh \left ({\frac{x}{2}} \right ) \left ( 30\,B \left ( \sinh \left ( x/2 \right ) \right ) ^{6}+ \left ( 21\,A-15\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{4}+ \left ( -28\,A+20\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}+56\,A-40\,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))^(5/2)*(A+B*cosh(x)),x)

[Out]

-16/105*sinh(1/2*x)*a^3*cosh(1/2*x)*(30*B*sinh(1/2*x)^6+(21*A-15*B)*sinh(1/2*x)^4+(-28*A+20*B)*sinh(1/2*x)^2+5
6*A-40*B)/(-2*sinh(1/2*x)^2*a)^(1/2)

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Maxima [B]  time = 1.67311, size = 389, normalized size = 3.97 \begin{align*} \frac{1}{60} \,{\left (\frac{25 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}} - \frac{150 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}} - \frac{150 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}} + \frac{25 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}} - \frac{3 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}} - \frac{3 \, \sqrt{2} a^{\frac{5}{2}}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}}\right )} A + \frac{1}{168} \, B{\left (\frac{{\left (21 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-x\right )} - 70 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, x\right )} + 210 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, x\right )} + 105 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, x\right )} - 7 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, x\right )} - 3 \, \sqrt{2} a^{\frac{5}{2}}\right )} e^{x}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}} - \frac{7 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-x\right )} - 105 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-2 \, x\right )} - 210 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-3 \, x\right )} + 70 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-4 \, x\right )} - 21 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-5 \, x\right )} + 3 \, \sqrt{2} a^{\frac{5}{2}} e^{\left (-6 \, x\right )}}{\left (-e^{\left (-x\right )}\right )^{\frac{5}{2}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(25*sqrt(2)*a^(5/2)*e^(-x)/(-e^(-x))^(5/2) - 150*sqrt(2)*a^(5/2)*e^(-2*x)/(-e^(-x))^(5/2) - 150*sqrt(2)*a
^(5/2)*e^(-3*x)/(-e^(-x))^(5/2) + 25*sqrt(2)*a^(5/2)*e^(-4*x)/(-e^(-x))^(5/2) - 3*sqrt(2)*a^(5/2)*e^(-5*x)/(-e
^(-x))^(5/2) - 3*sqrt(2)*a^(5/2)/(-e^(-x))^(5/2))*A + 1/168*B*((21*sqrt(2)*a^(5/2)*e^(-x) - 70*sqrt(2)*a^(5/2)
*e^(-2*x) + 210*sqrt(2)*a^(5/2)*e^(-3*x) + 105*sqrt(2)*a^(5/2)*e^(-4*x) - 7*sqrt(2)*a^(5/2)*e^(-5*x) - 3*sqrt(
2)*a^(5/2))*e^x/(-e^(-x))^(5/2) - (7*sqrt(2)*a^(5/2)*e^(-x) - 105*sqrt(2)*a^(5/2)*e^(-2*x) - 210*sqrt(2)*a^(5/
2)*e^(-3*x) + 70*sqrt(2)*a^(5/2)*e^(-4*x) - 21*sqrt(2)*a^(5/2)*e^(-5*x) + 3*sqrt(2)*a^(5/2)*e^(-6*x))/(-e^(-x)
)^(5/2))

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Fricas [B]  time = 2.25447, size = 1554, normalized size = 15.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/420*sqrt(1/2)*(15*B*a^2*cosh(x)^7 + 15*B*a^2*sinh(x)^7 + 21*(2*A - 5*B)*a^2*cosh(x)^6 - 35*(10*A - 11*B)*a^2
*cosh(x)^5 + 525*(4*A - 3*B)*a^2*cosh(x)^4 + 21*(5*B*a^2*cosh(x) + (2*A - 5*B)*a^2)*sinh(x)^6 + 525*(4*A - 3*B
)*a^2*cosh(x)^3 + 7*(45*B*a^2*cosh(x)^2 + 18*(2*A - 5*B)*a^2*cosh(x) - 5*(10*A - 11*B)*a^2)*sinh(x)^5 - 35*(10
*A - 11*B)*a^2*cosh(x)^2 + 35*(15*B*a^2*cosh(x)^3 + 9*(2*A - 5*B)*a^2*cosh(x)^2 - 5*(10*A - 11*B)*a^2*cosh(x)
+ 15*(4*A - 3*B)*a^2)*sinh(x)^4 + 21*(2*A - 5*B)*a^2*cosh(x) + 35*(15*B*a^2*cosh(x)^4 + 12*(2*A - 5*B)*a^2*cos
h(x)^3 - 10*(10*A - 11*B)*a^2*cosh(x)^2 + 60*(4*A - 3*B)*a^2*cosh(x) + 15*(4*A - 3*B)*a^2)*sinh(x)^3 + 15*B*a^
2 + 35*(9*B*a^2*cosh(x)^5 + 9*(2*A - 5*B)*a^2*cosh(x)^4 - 10*(10*A - 11*B)*a^2*cosh(x)^3 + 90*(4*A - 3*B)*a^2*
cosh(x)^2 + 45*(4*A - 3*B)*a^2*cosh(x) - (10*A - 11*B)*a^2)*sinh(x)^2 + 7*(15*B*a^2*cosh(x)^6 + 18*(2*A - 5*B)
*a^2*cosh(x)^5 - 25*(10*A - 11*B)*a^2*cosh(x)^4 + 300*(4*A - 3*B)*a^2*cosh(x)^3 + 225*(4*A - 3*B)*a^2*cosh(x)^
2 - 10*(10*A - 11*B)*a^2*cosh(x) + 3*(2*A - 5*B)*a^2)*sinh(x))*sqrt(-a/(cosh(x) + sinh(x)))/(cosh(x)^3 + 3*cos
h(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)

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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))**(5/2)*(A+B*cosh(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.33842, size = 398, normalized size = 4.06 \begin{align*} \frac{1}{840} \, \sqrt{2}{\left (\frac{{\left (2100 \, A a^{6} e^{\left (3 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 1575 \, B a^{6} e^{\left (3 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 350 \, A a^{6} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) + 385 \, B a^{6} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) + 42 \, A a^{6} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) - 105 \, B a^{6} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 15 \, B a^{6} \mathrm{sgn}\left (-e^{x} + 1\right )\right )} e^{\left (-3 \, x\right )}}{\sqrt{-a e^{x}} a^{3}} - \frac{15 \, \sqrt{-a e^{x}} B a^{9} e^{\left (3 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) + 42 \, \sqrt{-a e^{x}} A a^{9} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 105 \, \sqrt{-a e^{x}} B a^{9} e^{\left (2 \, x\right )} \mathrm{sgn}\left (-e^{x} + 1\right ) - 350 \, \sqrt{-a e^{x}} A a^{9} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 385 \, \sqrt{-a e^{x}} B a^{9} e^{x} \mathrm{sgn}\left (-e^{x} + 1\right ) + 2100 \, \sqrt{-a e^{x}} A a^{9} \mathrm{sgn}\left (-e^{x} + 1\right ) - 1575 \, \sqrt{-a e^{x}} B a^{9} \mathrm{sgn}\left (-e^{x} + 1\right )}{a^{7}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/840*sqrt(2)*((2100*A*a^6*e^(3*x)*sgn(-e^x + 1) - 1575*B*a^6*e^(3*x)*sgn(-e^x + 1) - 350*A*a^6*e^(2*x)*sgn(-e
^x + 1) + 385*B*a^6*e^(2*x)*sgn(-e^x + 1) + 42*A*a^6*e^x*sgn(-e^x + 1) - 105*B*a^6*e^x*sgn(-e^x + 1) + 15*B*a^
6*sgn(-e^x + 1))*e^(-3*x)/(sqrt(-a*e^x)*a^3) - (15*sqrt(-a*e^x)*B*a^9*e^(3*x)*sgn(-e^x + 1) + 42*sqrt(-a*e^x)*
A*a^9*e^(2*x)*sgn(-e^x + 1) - 105*sqrt(-a*e^x)*B*a^9*e^(2*x)*sgn(-e^x + 1) - 350*sqrt(-a*e^x)*A*a^9*e^x*sgn(-e
^x + 1) + 385*sqrt(-a*e^x)*B*a^9*e^x*sgn(-e^x + 1) + 2100*sqrt(-a*e^x)*A*a^9*sgn(-e^x + 1) - 1575*sqrt(-a*e^x)
*B*a^9*sgn(-e^x + 1))/a^7)