3.1.49 \(\int (-1+\cosh ^2(x))^{3/2} \, dx\) [49]

Optimal. Leaf size=29 \[ -\frac {2}{3} \coth (x) \sqrt {\sinh ^2(x)}+\frac {1}{3} \coth (x) \sinh ^2(x)^{3/2} \]

[Out]

1/3*coth(x)*(sinh(x)^2)^(3/2)-2/3*coth(x)*(sinh(x)^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3255, 3282, 3286, 2718} \begin {gather*} \frac {1}{3} \sinh ^2(x)^{3/2} \coth (x)-\frac {2}{3} \sqrt {\sinh ^2(x)} \coth (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Cosh[x]^2)^(3/2),x]

[Out]

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

Rule 2718

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

Rule 3255

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3282

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 3286

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

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{12} (-9 \cosh (x)+\cosh (3 x)) \text {csch}(x) \sqrt {\sinh ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Cosh[x]^2)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.94, size = 21, normalized size = 0.72

method	result	size
default	\(\frac {\sqrt {-\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \cosh \left (x \right ) \left (\cosh ^{2}\left (x \right )-3\right )}{3 \sinh \left (x \right )}\)	\(21\)
risch	\(\frac {{\mathrm e}^{4 x} \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{24 \,{\mathrm e}^{2 x}-24}-\frac {3 \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}\, {\mathrm e}^{2 x}}{8 \left ({\mathrm e}^{2 x}-1\right )}-\frac {3 \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{8 \left ({\mathrm e}^{2 x}-1\right )}+\frac {{\mathrm e}^{-2 x} \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{24 \,{\mathrm e}^{2 x}-24}\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^2-1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 23, normalized size = 0.79 \begin {gather*} -\frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {3}{8} \, e^{\left (-x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {3}{8} \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cosh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 19, normalized size = 0.66 \begin {gather*} \frac {1}{12} \, \cosh \left (x\right )^{3} + \frac {1}{4} \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \frac {3}{4} \, \cosh \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cosh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/12*cosh(x)^3 + 1/4*cosh(x)*sinh(x)^2 - 3/4*cosh(x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\cosh ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cosh(x)**2)**(3/2),x)

[Out]

Integral((cosh(x)**2 - 1)**(3/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (21) = 42\).
time = 0.42, size = 66, normalized size = 2.28 \begin {gather*} -\frac {1}{24} \, {\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 )} + \frac {1}{24} \, e^{\left (3 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - \frac {3}{8} \, e^{x} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+cosh(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) + 1/24*e^(3*x)*sgn(e^(3*x) - e^x) - 3/8*e^x
*sgn(e^(3*x) - e^x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left ({\mathrm {cosh}\left (x\right )}^2-1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)^2 - 1)^(3/2),x)

[Out]

int((cosh(x)^2 - 1)^(3/2), x)

________________________________________________________________________________________