3.141 \(\int (a \sinh ^2(x))^{3/2} \, dx\)

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

[Out]

(-2*a*Coth[x]*Sqrt[a*Sinh[x]^2])/3 + (Coth[x]*(a*Sinh[x]^2)^(3/2))/3

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Rubi [A]  time = 0.0239005, 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, 2638} \[ \frac{1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac{2}{3} a \coth (x) \sqrt{a \sinh ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*Coth[x]*Sqrt[a*Sinh[x]^2])/3 + (Coth[x]*(a*Sinh[x]^2)^(3/2))/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 2638

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

Rubi steps

\begin{align*} \int \left (a \sinh ^2(x)\right )^{3/2} \, dx &=\frac{1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac{1}{3} (2 a) \int \sqrt{a \sinh ^2(x)} \, dx\\ &=\frac{1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}-\frac{1}{3} \left (2 a \text{csch}(x) \sqrt{a \sinh ^2(x)}\right ) \int \sinh (x) \, dx\\ &=-\frac{2}{3} a \coth (x) \sqrt{a \sinh ^2(x)}+\frac{1}{3} \coth (x) \left (a \sinh ^2(x)\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0361036, size = 26, normalized size = 0.76 \[ \frac{1}{12} a (\cosh (3 x)-9 \cosh (x)) \text{csch}(x) \sqrt{a \sinh ^2(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 24, normalized size = 0.7 \begin{align*}{\frac{{a}^{2}\sinh \left ( x \right ) \cosh \left ( x \right ) \left ( \left ( \sinh \left ( x \right ) \right ) ^{2}-2 \right ) }{3}{\frac{1}{\sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.81804, 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*sinh(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 = 1.79921, 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*sinh(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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sinh ^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sinh(x)**2)**(3/2), x)

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Giac [B]  time = 1.18584, size = 95, normalized size = 2.79 \begin{align*} -\frac{1}{24} \,{\left ({\left (9 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) + 9 \, e^{x} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} a^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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