3.94 \(\int (-1-\sinh ^2(x))^{3/2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0293963, antiderivative size = 33, normalized size of antiderivative = 1., 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 = {3176, 3203, 3207, 2637} \[ \frac{1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac{2}{3} \sqrt{-\cosh ^2(x)} \tanh (x) \]

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 3176

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 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 (-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*}

Mathematica [A]  time = 0.0063351, size = 25, normalized size = 0.76 \[ -\frac{1}{12} (9 \sinh (x)+\sinh (3 x)) \sqrt{-\cosh ^2(x)} \text{sech}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Sinh[x]^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.043, size = 21, normalized size = 0.6 \begin{align*}{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) \left ( \left ( \cosh \left ( x \right ) \right ) ^{2}+2 \right ) }{3}{\frac{1}{\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.54951, size = 72, normalized size = 2.18 \begin{align*} \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{align*}

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]  time = 1.80619, size = 81, normalized size = 2.45 \begin{align*} \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{align*}

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(-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((-1-sinh(x)**2)**(3/2),x)

[Out]

Timed out

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Giac [C]  time = 1.19878, size = 34, normalized size = 1.03 \begin{align*} \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{align*}

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