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]

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 = 0.07, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 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 = 0.09, 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

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fricas [B]  time = 0.51, size = 279, normalized size = 4.10 \[ \frac {\sqrt {\frac {1}{2}} {\left (3 \, B a \cosh \relax (x)^{5} + 3 \, B a \sinh \relax (x)^{5} + 5 \, {\left (2 \, A + 3 \, B\right )} a \cosh \relax (x)^{4} + 30 \, {\left (3 \, A + 2 \, B\right )} a \cosh \relax (x)^{3} + 5 \, {\left (3 \, B a \cosh \relax (x) + {\left (2 \, A + 3 \, B\right )} a\right )} \sinh \relax (x)^{4} - 30 \, {\left (3 \, A + 2 \, B\right )} a \cosh \relax (x)^{2} + 10 \, {\left (3 \, B a \cosh \relax (x)^{2} + 2 \, {\left (2 \, A + 3 \, B\right )} a \cosh \relax (x) + 3 \, {\left (3 \, A + 2 \, B\right )} a\right )} \sinh \relax (x)^{3} - 5 \, {\left (2 \, A + 3 \, B\right )} a \cosh \relax (x) + 30 \, {\left (B a \cosh \relax (x)^{3} + {\left (2 \, A + 3 \, B\right )} a \cosh \relax (x)^{2} + 3 \, {\left (3 \, A + 2 \, B\right )} a \cosh \relax (x) - {\left (3 \, A + 2 \, B\right )} a\right )} \sinh \relax (x)^{2} - 3 \, B a + 5 \, {\left (3 \, B a \cosh \relax (x)^{4} + 4 \, {\left (2 \, A + 3 \, B\right )} a \cosh \relax (x)^{3} + 18 \, {\left (3 \, A + 2 \, B\right )} a \cosh \relax (x)^{2} - 12 \, {\left (3 \, A + 2 \, B\right )} a \cosh \relax (x) - {\left (2 \, A + 3 \, B\right )} a\right )} \sinh \relax (x)\right )} \sqrt {\frac {a}{\cosh \relax (x) + \sinh \relax (x)}}}{30 \, {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}\right )}} \]

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 = 0.13, size = 113, normalized size = 1.66 \[ -\frac {1}{60} \, \sqrt {2} {\left (\frac {{\left (90 \, A a^{4} e^{\left (2 \, x\right )} + 60 \, B a^{4} e^{\left (2 \, x\right )} + 10 \, A a^{4} e^{x} + 15 \, B a^{4} e^{x} + 3 \, B a^{4}\right )} e^{\left (-\frac {5}{2} \, x\right )}}{a^{\frac {5}{2}}} - \frac {3 \, B a^{\frac {13}{2}} e^{\left (\frac {5}{2} \, x\right )} + 10 \, A a^{\frac {13}{2}} e^{\left (\frac {3}{2} \, x\right )} + 15 \, B a^{\frac {13}{2}} e^{\left (\frac {3}{2} \, x\right )} + 90 \, A a^{\frac {13}{2}} e^{\left (\frac {1}{2} \, x\right )} + 60 \, B a^{\frac {13}{2}} e^{\left (\frac {1}{2} \, x\right )}}{a^{5}}\right )} \]

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)

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maple [A]  time = 0.20, size = 57, normalized size = 0.84 \[ \frac {4 \cosh \left (\frac {x}{2}\right ) a^{2} \sinh \left (\frac {x}{2}\right ) \left (6 B \left (\sinh ^{4}\left (\frac {x}{2}\right )\right )+\left (5 A +15 B \right ) \left (\sinh ^{2}\left (\frac {x}{2}\right )\right )+15 A +15 B \right ) \sqrt {2}}{15 \sqrt {a \left (\cosh ^{2}\left (\frac {x}{2}\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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maxima [B]  time = 0.46, size = 163, normalized size = 2.40 \[ \frac {1}{6} \, {\left (\sqrt {2} a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} + 9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} - 9 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-\frac {1}{2} \, x\right )} - \sqrt {2} a^{\frac {3}{2}} e^{\left (-\frac {3}{2} \, x\right )}\right )} A + \frac {1}{20} \, {\left ({\left (\sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} + 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} + 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}\right )} e^{\left (\frac {7}{2} \, x\right )} + {\left (5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-x\right )} - 15 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 5 \, \sqrt {2} a^{\frac {3}{2}} e^{\left (-3 \, x\right )} - \sqrt {2} a^{\frac {3}{2}} e^{\left (-4 \, x\right )}\right )} e^{\left (\frac {3}{2} \, x\right )}\right )} B \]

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

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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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