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]

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

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Rubi [A]  time = 0.0225666, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 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
]])

Rule 2637

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

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 = 0.0087829, 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

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Maple [A]  time = 0.037, size = 24, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}\cosh \left ( x \right ) \sinh \left ( x \right ) \left ( \left ( \cosh \left ( x \right ) \right ) ^{2}+2 \right ) }{3}{\frac{1}{\sqrt{a \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

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)

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Maxima [A]  time = 1.59739, size = 47, normalized size = 1.38 \begin{align*} \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} \end{align*}

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

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Fricas [B]  time = 2.42448, size = 672, normalized size = 19.76 \begin{align*} \frac{{\left (6 \, a \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{5} + a e^{x} \sinh \left (x\right )^{6} + 3 \,{\left (5 \, a \cosh \left (x\right )^{2} + 3 \, a\right )} e^{x} \sinh \left (x\right )^{4} + 4 \,{\left (5 \, a \cosh \left (x\right )^{3} + 9 \, a \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, a \cosh \left (x\right )^{4} + 18 \, a \cosh \left (x\right )^{2} - 3 \, a\right )} e^{x} \sinh \left (x\right )^{2} + 6 \,{\left (a \cosh \left (x\right )^{5} + 6 \, a \cosh \left (x\right )^{3} - 3 \, a \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) +{\left (a \cosh \left (x\right )^{6} + 9 \, a \cosh \left (x\right )^{4} - 9 \, a \cosh \left (x\right )^{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 \left (x\right )^{3} e^{\left (2 \, x\right )} +{\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{3} + \cosh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 3 \,{\left (\cosh \left (x\right )^{2} e^{\left (2 \, x\right )} + \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )\right )}} \end{align*}

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))

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

[Out]

Timed out

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Giac [A]  time = 1.14336, size = 39, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

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)