3.1.94 \(\int (-1-\sinh ^2(x))^{3/2} \, dx\) [94]

Optimal. Leaf size=33 \[ -\frac {2}{3} \sqrt {-\cosh ^2(x)} \tanh (x)+\frac {1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x) \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3255, 3282, 3286, 2717} \begin {gather*} \frac {1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac {2}{3} \sqrt {-\cosh ^2(x)} \tanh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - Sinh[x]^2)^(3/2),x]

[Out]

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

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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-\sinh ^2(x)\right )^{3/2} \, dx &=\int \left (-\cosh ^2(x)\right )^{3/2} \, dx\\ &=\frac {1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac {2}{3} \int \sqrt {-\cosh ^2(x)} \, dx\\ &=\frac {1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac {1}{3} \left (2 \sqrt {-\cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=-\frac {2}{3} \sqrt {-\cosh ^2(x)} \tanh (x)+\frac {1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.76 \begin {gather*} -\frac {1}{12} \sqrt {-\cosh ^2(x)} \text {sech}(x) (9 \sinh (x)+\sinh (3 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Sinh[x]^2)^(3/2),x]

[Out]

-1/12*(Sqrt[-Cosh[x]^2]*Sech[x]*(9*Sinh[x] + Sinh[3*x]))

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Maple [A]
time = 0.71, size = 21, normalized size = 0.64

method	result	size
default	\(\frac {\cosh \left (x \right ) \sinh \left (x \right ) \left (\cosh ^{2}\left (x \right )+2\right )}{3 \sqrt {-\left (\cosh ^{2}\left (x \right )\right )}}\)	\(21\)
risch	\(-\frac {{\mathrm e}^{4 x} \sqrt {-\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{24 \left (1+{\mathrm e}^{2 x}\right )}-\frac {3 \sqrt {-\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}\, {\mathrm e}^{2 x}}{8 \left (1+{\mathrm e}^{2 x}\right )}+\frac {3 \sqrt {-\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{8 \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{-2 x} \sqrt {-\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{24+24 \,{\mathrm e}^{2 x}}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (25) = 50\).
time = 0.50, size = 53, normalized size = 1.61 \begin {gather*} \frac {3 \, e^{\left (-2 \, x\right )}}{8 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac {3}{2}}} - \frac {3 \, e^{\left (-4 \, x\right )}}{8 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac {3}{2}}} - \frac {e^{\left (-6 \, x\right )}}{24 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac {3}{2}}} + \frac {1}{24 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*e^(-2*x)/(-e^(-2*x))^(3/2) - 3/8*e^(-4*x)/(-e^(-2*x))^(3/2) - 1/24*e^(-6*x)/(-e^(-2*x))^(3/2) + 1/24/(-e^(
-2*x))^(3/2)

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Fricas [C] Result contains complex when optimal does not.
time = 0.55, size = 26, normalized size = 0.79 \begin {gather*} \frac {1}{24} \, {\left (-i \, e^{\left (6 \, x\right )} - 9 i \, e^{\left (4 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} + i\right )} e^{\left (-3 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(-I*e^(6*x) - 9*I*e^(4*x) + 9*I*e^(2*x) + I)*e^(-3*x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \sinh ^{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-sinh(x)**2)**(3/2),x)

[Out]

Integral((-sinh(x)**2 - 1)**(3/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{24} i \, {\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - \frac {1}{24} i \, e^{\left (3 \, x\right )} - \frac {3}{8} i \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-sinh(x)^2)^(3/2),x, algorithm="giac")

[Out]

1/24*I*(9*e^(2*x) + 1)*e^(-3*x) - 1/24*I*e^(3*x) - 3/8*I*e^x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- sinh(x)^2 - 1)^(3/2),x)

[Out]

int((- sinh(x)^2 - 1)^(3/2), x)

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