3.123 \(\int (a \cosh ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=34 \[ \frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {2}{3} a \tanh (x) \sqrt {a \cosh ^2(x)} \]

[Out]

1/3*(a*cosh(x)^2)^(3/2)*tanh(x)+2/3*a*(a*cosh(x)^2)^(1/2)*tanh(x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3203, 3207, 2637} \[ \frac {1}{3} \tanh (x) \left (a \cosh ^2(x)\right )^{3/2}+\frac {2}{3} a \tanh (x) \sqrt {a \cosh ^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cosh[x]^2)^(3/2),x]

[Out]

(2*a*Sqrt[a*Cosh[x]^2]*Tanh[x])/3 + ((a*Cosh[x]^2)^(3/2)*Tanh[x])/3

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3203

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x]
+ Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] &&  !IntegerQ[p] &&
GtQ[p, 1]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \left (a \cosh ^2(x)\right )^{3/2} \, dx &=\frac {1}{3} \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {1}{3} (2 a) \int \sqrt {a \cosh ^2(x)} \, dx\\ &=\frac {1}{3} \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)+\frac {1}{3} \left (2 a \sqrt {a \cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=\frac {2}{3} a \sqrt {a \cosh ^2(x)} \tanh (x)+\frac {1}{3} \left (a \cosh ^2(x)\right )^{3/2} \tanh (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 26, normalized size = 0.76 \[ \frac {1}{12} a (9 \sinh (x)+\sinh (3 x)) \text {sech}(x) \sqrt {a \cosh ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sqrt[a*Cosh[x]^2]*Sech[x]*(9*Sinh[x] + Sinh[3*x]))/12

________________________________________________________________________________________

fricas [B]  time = 0.81, size = 222, normalized size = 6.53 \[ \frac {{\left (6 \, a \cosh \relax (x) e^{x} \sinh \relax (x)^{5} + a e^{x} \sinh \relax (x)^{6} + 3 \, {\left (5 \, a \cosh \relax (x)^{2} + 3 \, a\right )} e^{x} \sinh \relax (x)^{4} + 4 \, {\left (5 \, a \cosh \relax (x)^{3} + 9 \, a \cosh \relax (x)\right )} e^{x} \sinh \relax (x)^{3} + 3 \, {\left (5 \, a \cosh \relax (x)^{4} + 18 \, a \cosh \relax (x)^{2} - 3 \, a\right )} e^{x} \sinh \relax (x)^{2} + 6 \, {\left (a \cosh \relax (x)^{5} + 6 \, a \cosh \relax (x)^{3} - 3 \, a \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (a \cosh \relax (x)^{6} + 9 \, a \cosh \relax (x)^{4} - 9 \, a \cosh \relax (x)^{2} - a\right )} e^{x}\right )} \sqrt {a e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} + a} e^{\left (-x\right )}}{24 \, {\left (\cosh \relax (x)^{3} e^{\left (2 \, x\right )} + {\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{3} + \cosh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 3 \, {\left (\cosh \relax (x)^{2} e^{\left (2 \, x\right )} + \cosh \relax (x)^{2}\right )} \sinh \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(6*a*cosh(x)*e^x*sinh(x)^5 + a*e^x*sinh(x)^6 + 3*(5*a*cosh(x)^2 + 3*a)*e^x*sinh(x)^4 + 4*(5*a*cosh(x)^3 +
9*a*cosh(x))*e^x*sinh(x)^3 + 3*(5*a*cosh(x)^4 + 18*a*cosh(x)^2 - 3*a)*e^x*sinh(x)^2 + 6*(a*cosh(x)^5 + 6*a*co
sh(x)^3 - 3*a*cosh(x))*e^x*sinh(x) + (a*cosh(x)^6 + 9*a*cosh(x)^4 - 9*a*cosh(x)^2 - a)*e^x)*sqrt(a*e^(4*x) + 2
*a*e^(2*x) + a)*e^(-x)/(cosh(x)^3*e^(2*x) + (e^(2*x) + 1)*sinh(x)^3 + cosh(x)^3 + 3*(cosh(x)*e^(2*x) + cosh(x)
)*sinh(x)^2 + 3*(cosh(x)^2*e^(2*x) + cosh(x)^2)*sinh(x))

________________________________________________________________________________________

giac [A]  time = 0.12, size = 29, normalized size = 0.85 \[ -\frac {1}{24} \, {\left ({\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}\right )} a^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((9*e^(2*x) + 1)*e^(-3*x) - e^(3*x) - 9*e^x)*a^(3/2)

________________________________________________________________________________________

maple [A]  time = 0.18, size = 24, normalized size = 0.71 \[ \frac {a^{2} \cosh \relax (x ) \sinh \relax (x ) \left (\cosh ^{2}\relax (x )+2\right )}{3 \sqrt {a \left (\cosh ^{2}\relax (x )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*cosh(x)*sinh(x)*(cosh(x)^2+2)/(a*cosh(x)^2)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 35, normalized size = 1.03 \[ \frac {1}{24} \, a^{\frac {3}{2}} e^{\left (3 \, x\right )} - \frac {3}{8} \, a^{\frac {3}{2}} e^{\left (-x\right )} - \frac {1}{24} \, a^{\frac {3}{2}} e^{\left (-3 \, x\right )} + \frac {3}{8} \, a^{\frac {3}{2}} e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*a^(3/2)*e^(3*x) - 3/8*a^(3/2)*e^(-x) - 1/24*a^(3/2)*e^(-3*x) + 3/8*a^(3/2)*e^x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a\,{\mathrm {cosh}\relax (x)}^2\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*cosh(x)^2)^(3/2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*cosh(x)**2)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________