3.1.50 \(\int (-1-\cosh ^2(x))^{3/2} \, dx\) [50]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3259, 3251, 3257, 3256, 3262, 3261} \begin {gather*} -\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}} \end {gather*}

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 3251

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 3256

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

Rule 3257

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 3259

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[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 3261

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

Rule 3262

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]

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*}

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Mathematica [A]
time = 0.05, size = 78, normalized size = 0.77 \begin {gather*} \frac {-48 i \sqrt {3+\cosh (2 x)} E\left (i x\left |\frac {1}{2}\right .\right )+8 i \sqrt {3+\cosh (2 x)} F\left (i x\left |\frac {1}{2}\right .\right )+6 \sinh (2 x)+\sinh (4 x)}{12 \sqrt {2} \sqrt {-3-\cosh (2 x)}} \end {gather*}

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 = 1.08, size = 96, normalized size = 0.95

method	result	size
default	\(-\frac {\sqrt {-\left (\cosh ^{2}\left (x \right )+1\right ) \left (\sinh ^{2}\left (x \right )\right )}\, \left (-\left (\cosh ^{5}\left (x \right )\right )+2 \sqrt {-\left (\sinh ^{2}\left (x \right )\right )}\, \sqrt {\cosh ^{2}\left (x \right )+1}\, \EllipticF \left (\cosh \left (x \right ), i\right )-6 \sqrt {-\left (\sinh ^{2}\left (x \right )\right )}\, \sqrt {\cosh ^{2}\left (x \right )+1}\, \EllipticE \left (\cosh \left (x \right ), i\right )+\cosh \left (x \right )\right )}{3 \sqrt {1-\left (\cosh ^{4}\left (x \right )\right )}\, \sinh \left (x \right ) \sqrt {-1-\left (\cosh ^{2}\left (x \right )\right )}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-(cosh(x)^2+1)*sinh(x)^2)^(1/2)*(-cosh(x)^5+2*(-sinh(x)^2)^(1/2)*(cosh(x)^2+1)^(1/2)*EllipticF(cosh(x),I
)-6*(-sinh(x)^2)^(1/2)*(cosh(x)^2+1)^(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.08, size = 143, normalized size = 1.42 \begin {gather*} \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 {gather*}

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]
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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \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)

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