3.50 \(\int (-1-\cosh ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=101 \[ \frac{2 i \sqrt{\cosh ^2(x)+1} \text{EllipticF}\left (\frac{\pi }{2}+i x,-1\right )}{3 \sqrt{-\cosh ^2(x)-1}}-\frac{1}{3} \sinh (x) \cosh (x) \sqrt{-\cosh ^2(x)-1}+\frac{2 i \sqrt{-\cosh ^2(x)-1} E\left (\left .i x+\frac{\pi }{2}\right |-1\right )}{\sqrt{\cosh ^2(x)+1}} \]

[Out]

((2*I)*Sqrt[-1 - Cosh[x]^2]*EllipticE[Pi/2 + I*x, -1])/Sqrt[1 + Cosh[x]^2] + (((2*I)/3)*Sqrt[1 + Cosh[x]^2]*El
lipticF[Pi/2 + I*x, -1])/Sqrt[-1 - Cosh[x]^2] - (Cosh[x]*Sqrt[-1 - Cosh[x]^2]*Sinh[x])/3

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Rubi [A]  time = 0.0923304, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ -\frac{1}{3} \sinh (x) \cosh (x) \sqrt{-\cosh ^2(x)-1}+\frac{2 i \sqrt{\cosh ^2(x)+1} F\left (\left .i x+\frac{\pi }{2}\right |-1\right )}{3 \sqrt{-\cosh ^2(x)-1}}+\frac{2 i \sqrt{-\cosh ^2(x)-1} E\left (\left .i x+\frac{\pi }{2}\right |-1\right )}{\sqrt{\cosh ^2(x)+1}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 - Cosh[x]^2)^(3/2),x]

[Out]

((2*I)*Sqrt[-1 - Cosh[x]^2]*EllipticE[Pi/2 + I*x, -1])/Sqrt[1 + Cosh[x]^2] + (((2*I)/3)*Sqrt[1 + Cosh[x]^2]*El
lipticF[Pi/2 + I*x, -1])/Sqrt[-1 - Cosh[x]^2] - (Cosh[x]*Sqrt[-1 - Cosh[x]^2]*Sinh[x])/3

Rule 3180

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
e + f*x]^2)^(p - 1))/(2*f*p), x] + Dist[1/(2*p), Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*
a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]

Rule 3172

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[
B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /;
FreeQ[{a, b, e, f, A, B}, x]

Rule 3178

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + (b*Sin
[e + f*x]^2)/a], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rule 3177

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[e + f*x, -(b/a)])/f, x]
/; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3183

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[1 + (b*Sin[e + f*x]^2)/a]/Sqrt[a +
b*Sin[e + f*x]^2], Int[1/Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rule 3182

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1*EllipticF[e + f*x, -(b/a)])/(Sqrt[a]*
f), x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx &=-\frac{1}{3} \cosh (x) \sqrt{-1-\cosh ^2(x)} \sinh (x)+\frac{1}{3} \int \frac{4+6 \cosh ^2(x)}{\sqrt{-1-\cosh ^2(x)}} \, dx\\ &=-\frac{1}{3} \cosh (x) \sqrt{-1-\cosh ^2(x)} \sinh (x)-\frac{2}{3} \int \frac{1}{\sqrt{-1-\cosh ^2(x)}} \, dx-2 \int \sqrt{-1-\cosh ^2(x)} \, dx\\ &=-\frac{1}{3} \cosh (x) \sqrt{-1-\cosh ^2(x)} \sinh (x)-\frac{\left (2 \sqrt{-1-\cosh ^2(x)}\right ) \int \sqrt{1+\cosh ^2(x)} \, dx}{\sqrt{1+\cosh ^2(x)}}-\frac{\left (2 \sqrt{1+\cosh ^2(x)}\right ) \int \frac{1}{\sqrt{1+\cosh ^2(x)}} \, dx}{3 \sqrt{-1-\cosh ^2(x)}}\\ &=\frac{2 i \sqrt{-1-\cosh ^2(x)} E\left (\left .\frac{\pi }{2}+i x\right |-1\right )}{\sqrt{1+\cosh ^2(x)}}+\frac{2 i \sqrt{1+\cosh ^2(x)} F\left (\left .\frac{\pi }{2}+i x\right |-1\right )}{3 \sqrt{-1-\cosh ^2(x)}}-\frac{1}{3} \cosh (x) \sqrt{-1-\cosh ^2(x)} \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0700247, size = 78, normalized size = 0.77 \[ \frac{8 i \sqrt{\cosh (2 x)+3} \text{EllipticF}\left (i x,\frac{1}{2}\right )+6 \sinh (2 x)+\sinh (4 x)-48 i \sqrt{\cosh (2 x)+3} E\left (i x\left |\frac{1}{2}\right .\right )}{12 \sqrt{2} \sqrt{-\cosh (2 x)-3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Cosh[x]^2)^(3/2),x]

[Out]

((-48*I)*Sqrt[3 + Cosh[2*x]]*EllipticE[I*x, 1/2] + (8*I)*Sqrt[3 + Cosh[2*x]]*EllipticF[I*x, 1/2] + 6*Sinh[2*x]
+ Sinh[4*x])/(12*Sqrt[2]*Sqrt[-3 - Cosh[2*x]])

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Maple [A]  time = 0.27, size = 96, normalized size = 1. \begin{align*} -{\frac{1}{3\,\sinh \left ( x \right ) }\sqrt{- \left ( 1+ \left ( \cosh \left ( x \right ) \right ) ^{2} \right ) \left ( \sinh \left ( x \right ) \right ) ^{2}} \left ( - \left ( \cosh \left ( x \right ) \right ) ^{5}+2\,\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( \cosh \left ( x \right ) ,i \right ) -6\,\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}\sqrt{1+ \left ( \cosh \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( x \right ) ,i \right ) +\cosh \left ( x \right ) \right ){\frac{1}{\sqrt{1- \left ( \cosh \left ( x \right ) \right ) ^{4}}}}{\frac{1}{\sqrt{-1- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-(1+cosh(x)^2)*sinh(x)^2)^(1/2)*(-cosh(x)^5+2*(-sinh(x)^2)^(1/2)*(1+cosh(x)^2)^(1/2)*EllipticF(cosh(x),I
)-6*(-sinh(x)^2)^(1/2)*(1+cosh(x)^2)^(1/2)*EllipticE(cosh(x),I)+cosh(x))/(1-cosh(x)^4)^(1/2)/sinh(x)/(-1-cosh(
x)^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\cosh \left (x\right )^{2} - 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-cosh(x)^2 - 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{24 \,{\left (e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )}\right )}{\rm integral}\left (-\frac{4 \, \sqrt{-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1}{\left (5 \, e^{\left (2 \, x\right )} + 2 \, e^{x} + 5\right )}}{3 \,{\left (e^{\left (6 \, x\right )} - 2 \, e^{\left (5 \, x\right )} + 7 \, e^{\left (4 \, x\right )} - 12 \, e^{\left (3 \, x\right )} + 7 \, e^{\left (2 \, x\right )} - 2 \, e^{x} + 1\right )}}, x\right ) -{\left (e^{\left (5 \, x\right )} - e^{\left (4 \, x\right )} + 24 \, e^{\left (3 \, x\right )} + 24 \, e^{\left (2 \, x\right )} - e^{x} + 1\right )} \sqrt{-e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 1}}{24 \,{\left (e^{\left (4 \, x\right )} - e^{\left (3 \, x\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(24*(e^(4*x) - e^(3*x))*integral(-4/3*sqrt(-e^(4*x) - 6*e^(2*x) - 1)*(5*e^(2*x) + 2*e^x + 5)/(e^(6*x) - 2
*e^(5*x) + 7*e^(4*x) - 12*e^(3*x) + 7*e^(2*x) - 2*e^x + 1), x) - (e^(5*x) - e^(4*x) + 24*e^(3*x) + 24*e^(2*x)
- e^x + 1)*sqrt(-e^(4*x) - 6*e^(2*x) - 1))/(e^(4*x) - 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-cosh(x)**2)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\cosh \left (x\right )^{2} - 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-cosh(x)^2 - 1)^(3/2), x)