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

Optimal. Leaf size=68 \[ \frac{8 a^2 (5 A+3 B) \sinh (x)}{15 \sqrt{a \cosh (x)+a}}+\frac{2}{15} a (5 A+3 B) \sinh (x) \sqrt{a \cosh (x)+a}+\frac{2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0738993, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2751, 2647, 2646} \[ \frac{8 a^2 (5 A+3 B) \sinh (x)}{15 \sqrt{a \cosh (x)+a}}+\frac{2}{15} a (5 A+3 B) \sinh (x) \sqrt{a \cosh (x)+a}+\frac{2}{5} B \sinh (x) (a \cosh (x)+a)^{3/2} \]

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 = 0.0872422, size = 46, normalized size = 0.68 \[ \frac{1}{15} a \tanh \left (\frac{x}{2}\right ) \sqrt{a (\cosh (x)+1)} (2 (5 A+9 B) \cosh (x)+50 A+3 B \cosh (2 x)+39 B) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a*(1 + Cosh[x])]*(50*A + 39*B + 2*(5*A + 9*B)*Cosh[x] + 3*B*Cosh[2*x])*Tanh[x/2])/15

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 57, normalized size = 0.8 \begin{align*}{\frac{4\,{a}^{2}\sqrt{2}}{15}\cosh \left ({\frac{x}{2}} \right ) \sinh \left ({\frac{x}{2}} \right ) \left ( 6\,B \left ( \sinh \left ( x/2 \right ) \right ) ^{4}+ \left ( 5\,A+15\,B \right ) \left ( \sinh \left ({\frac{x}{2}} \right ) \right ) ^{2}+15\,A+15\,B \right ){\frac{1}{\sqrt{a \left ( \cosh \left ({\frac{x}{2}} \right ) \right ) ^{2}}}}} \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]

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

________________________________________________________________________________________

Maxima [B]  time = 1.61723, size = 220, normalized size = 3.24 \begin{align*} \frac{1}{6} \,{\left (\sqrt{2} a^{\frac{3}{2}} e^{\left (\frac{3}{2} \, x\right )} + 9 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (\frac{1}{2} \, x\right )} - 9 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-\frac{1}{2} \, x\right )} - \sqrt{2} a^{\frac{3}{2}} e^{\left (-\frac{3}{2} \, x\right )}\right )} A + \frac{1}{20} \,{\left ({\left (\sqrt{2} a^{\frac{3}{2}} e^{\left (-x\right )} + 5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-2 \, x\right )} + 15 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-3 \, x\right )} - 5 \, \sqrt{2} a^{\frac{3}{2}} e^{\left (-4 \, x\right )}\right )} e^{\left (\frac{7}{2} \, x\right )} +{\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}} e^{\left (-4 \, x\right )}\right )} e^{\left (\frac{3}{2} \, x\right )}\right )} B \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*(sqrt(2)*a^(3/2)*e^(3/2*x) + 9*sqrt(2)*a^(3/2)*e^(1/2*x) - 9*sqrt(2)*a^(3/2)*e^(-1/2*x) - sqrt(2)*a^(3/2)*
e^(-3/2*x))*A + 1/20*((sqrt(2)*a^(3/2)*e^(-x) + 5*sqrt(2)*a^(3/2)*e^(-2*x) + 15*sqrt(2)*a^(3/2)*e^(-3*x) - 5*s
qrt(2)*a^(3/2)*e^(-4*x))*e^(7/2*x) + (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^(3/2*x))*B

________________________________________________________________________________________

Fricas [B]  time = 2.17396, size = 805, normalized size = 11.84 \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)

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [B]  time = 1.15364, size = 153, normalized size = 2.25 \begin{align*} -\frac{1}{60} \, \sqrt{2}{\left (\frac{{\left (90 \, A a^{4} e^{\left (2 \, x\right )} + 60 \, B a^{4} e^{\left (2 \, x\right )} + 10 \, A a^{4} e^{x} + 15 \, B a^{4} e^{x} + 3 \, B a^{4}\right )} e^{\left (-\frac{5}{2} \, x\right )}}{a^{\frac{5}{2}}} - \frac{3 \, B a^{\frac{13}{2}} e^{\left (\frac{5}{2} \, x\right )} + 10 \, A a^{\frac{13}{2}} e^{\left (\frac{3}{2} \, x\right )} + 15 \, B a^{\frac{13}{2}} e^{\left (\frac{3}{2} \, x\right )} + 90 \, A a^{\frac{13}{2}} e^{\left (\frac{1}{2} \, x\right )} + 60 \, B a^{\frac{13}{2}} e^{\left (\frac{1}{2} \, x\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) + 60*B*a^4*e^(2*x) + 10*A*a^4*e^x + 15*B*a^4*e^x + 3*B*a^4)*e^(-5/2*x)/a^(5/2
) - (3*B*a^(13/2)*e^(5/2*x) + 10*A*a^(13/2)*e^(3/2*x) + 15*B*a^(13/2)*e^(3/2*x) + 90*A*a^(13/2)*e^(1/2*x) + 60
*B*a^(13/2)*e^(1/2*x))/a^5)