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

3.1.50.1 Optimal result

Integrand size = 12, antiderivative size = 101 \[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\frac {2 i \sqrt {-1-\cosh ^2(x)} E\left (\left .\frac {\pi }{2}+i x\right |-1\right )}{\sqrt {1+\cosh ^2(x)}}+\frac {2 i \sqrt {1+\cosh ^2(x)} \operatorname {EllipticF}\left (\frac {\pi }{2}+i x,-1\right )}{3 \sqrt {-1-\cosh ^2(x)}}-\frac {1}{3} \cosh (x) \sqrt {-1-\cosh ^2(x)} \sinh (x) \]

output

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

3.1.50.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77 \[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\frac {-48 i \sqrt {3+\cosh (2 x)} E\left (i x\left |\frac {1}{2}\right .\right )+8 i \sqrt {3+\cosh (2 x)} \operatorname {EllipticF}\left (i x,\frac {1}{2}\right )+6 \sinh (2 x)+\sinh (4 x)}{12 \sqrt {2} \sqrt {-3-\cosh (2 x)}} \]

input

Integrate[(-1 - Cosh[x]^2)^(3/2),x]
 

output

((-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 - Co 
sh[2*x]])
 

3.1.50.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3659, 27, 3042, 3651, 3042, 3657, 3042, 3656, 3662, 3042, 3661}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (-1-\sin \left (\frac {\pi }{2}+i x\right )^2\right )^{3/2}dx\)

\(\Big \downarrow \) 3659

\(\displaystyle \frac {1}{3} \int \frac {2 \left (3 \cosh ^2(x)+2\right )}{\sqrt {-\cosh ^2(x)-1}}dx-\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \int \frac {3 \cosh ^2(x)+2}{\sqrt {-\cosh ^2(x)-1}}dx-\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \int \frac {3 \sin \left (i x+\frac {\pi }{2}\right )^2+2}{\sqrt {-\sin \left (i x+\frac {\pi }{2}\right )^2-1}}dx\)

\(\Big \downarrow \) 3651

\(\displaystyle \frac {2}{3} \left (-\int \frac {1}{\sqrt {-\cosh ^2(x)-1}}dx-3 \int \sqrt {-\cosh ^2(x)-1}dx\right )-\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (-\int \frac {1}{\sqrt {-\sin \left (i x+\frac {\pi }{2}\right )^2-1}}dx-3 \int \sqrt {-\sin \left (i x+\frac {\pi }{2}\right )^2-1}dx\right )\)

\(\Big \downarrow \) 3657

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (-\frac {3 \sqrt {-\cosh ^2(x)-1} \int \sqrt {\cosh ^2(x)+1}dx}{\sqrt {\cosh ^2(x)+1}}-\int \frac {1}{\sqrt {-\sin \left (i x+\frac {\pi }{2}\right )^2-1}}dx\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (-\int \frac {1}{\sqrt {-\sin \left (i x+\frac {\pi }{2}\right )^2-1}}dx-\frac {3 \sqrt {-\cosh ^2(x)-1} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )^2+1}dx}{\sqrt {\cosh ^2(x)+1}}\right )\)

\(\Big \downarrow \) 3656

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (\frac {3 i \sqrt {-\cosh ^2(x)-1} E\left (\left .i x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cosh ^2(x)+1}}-\int \frac {1}{\sqrt {-\sin \left (i x+\frac {\pi }{2}\right )^2-1}}dx\right )\)

\(\Big \downarrow \) 3662

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (-\frac {\sqrt {\cosh ^2(x)+1} \int \frac {1}{\sqrt {\cosh ^2(x)+1}}dx}{\sqrt {-\cosh ^2(x)-1}}+\frac {3 i \sqrt {-\cosh ^2(x)-1} E\left (\left .i x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cosh ^2(x)+1}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (\frac {3 i \sqrt {-\cosh ^2(x)-1} E\left (\left .i x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cosh ^2(x)+1}}-\frac {\sqrt {\cosh ^2(x)+1} \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )^2+1}}dx}{\sqrt {-\cosh ^2(x)-1}}\right )\)

\(\Big \downarrow \) 3661

\(\displaystyle -\frac {1}{3} \sinh (x) \cosh (x) \sqrt {-\cosh ^2(x)-1}+\frac {2}{3} \left (\frac {i \sqrt {\cosh ^2(x)+1} \operatorname {EllipticF}\left (i x+\frac {\pi }{2},-1\right )}{\sqrt {-\cosh ^2(x)-1}}+\frac {3 i \sqrt {-\cosh ^2(x)-1} E\left (\left .i x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cosh ^2(x)+1}}\right )\)

input

Int[(-1 - Cosh[x]^2)^(3/2),x]
 

output

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

3.1.50.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3651

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

rule 3656

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 3657

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[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 3659

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C 
os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim 
p[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 3661

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

rule 3662

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

3.1.50.4 Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95

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

input

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

output

-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)
 

3.1.50.5 Fricas [F]

\[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\cosh \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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))
 

3.1.50.6 Sympy [F(-1)]

Timed out. \[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.1.50.7 Maxima [F]

\[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\cosh \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.1.50.8 Giac [F]

\[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\int { {\left (-\cosh \left (x\right )^{2} - 1\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.1.50.9 Mupad [F(-1)]

Timed out. \[ \int \left (-1-\cosh ^2(x)\right )^{3/2} \, dx=\int {\left (-{\mathrm {cosh}\left (x\right )}^2-1\right )}^{3/2} \,d x \]

input

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

output

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