3.1.59 \(\int (-1-\cos ^2(x))^{3/2} \, dx\) [59]

3.1.59.1 Optimal result

Integrand size = 12, antiderivative size = 89 \[ \int \left (-1-\cos ^2(x)\right )^{3/2} \, dx=-\frac {2 \sqrt {-1-\cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{\sqrt {1+\cos ^2(x)}}-\frac {2 \sqrt {1+\cos ^2(x)} \operatorname {EllipticF}\left (\frac {\pi }{2}+x,-1\right )}{3 \sqrt {-1-\cos ^2(x)}}-\frac {1}{3} \cos (x) \sqrt {-1-\cos ^2(x)} \sin (x) \]

output

-1/3*cos(x)*sin(x)*(-1-cos(x)^2)^(1/2)+2*(sin(x)^2)^(1/2)/sin(x)*EllipticE 
(cos(x),I)*(-1-cos(x)^2)^(1/2)/(1+cos(x)^2)^(1/2)+2/3*(sin(x)^2)^(1/2)/sin 
(x)*EllipticF(cos(x),I)*(1+cos(x)^2)^(1/2)/(-1-cos(x)^2)^(1/2)
 

3.1.59.2 Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \left (-1-\cos ^2(x)\right )^{3/2} \, dx=\frac {48 \sqrt {3+\cos (2 x)} E\left (x\left |\frac {1}{2}\right .\right )-8 \sqrt {3+\cos (2 x)} \operatorname {EllipticF}\left (x,\frac {1}{2}\right )+6 \sin (2 x)+\sin (4 x)}{12 \sqrt {2} \sqrt {-3-\cos (2 x)}} \]

input

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

output

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

3.1.59.3 Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 92, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3659

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3651

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3657

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3656

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

\(\Big \downarrow \) 3662

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3661

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

input

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

output

(2*((-3*Sqrt[-1 - Cos[x]^2]*EllipticE[Pi/2 + x, -1])/Sqrt[1 + Cos[x]^2] - 
(Sqrt[1 + Cos[x]^2]*EllipticF[Pi/2 + x, -1])/Sqrt[-1 - Cos[x]^2]))/3 - (Co 
s[x]*Sqrt[-1 - Cos[x]^2]*Sin[x])/3
 

3.1.59.3.1 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]
 

3.1.59.4 Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {\sqrt {-\left (1+\cos ^{2}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}\, \left (-\left (\sin ^{4}\left (x \right )\right ) \cos \left (x \right )+10 i \sqrt {2-\left (\sin ^{2}\left (x \right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, F\left (i \cos \left (x \right ), i\right )-6 i \sqrt {2-\left (\sin ^{2}\left (x \right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, E\left (i \cos \left (x \right ), i\right )+2 \left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )\right )}{3 \sqrt {-1+\cos ^{4}\left (x \right )}\, \sin \left (x \right ) \sqrt {-1-\left (\cos ^{2}\left (x \right )\right )}}\)	\(110\)

input

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

output

1/3*(-(1+cos(x)^2)*sin(x)^2)^(1/2)*(-sin(x)^4*cos(x)+10*I*(2-sin(x)^2)^(1/ 
2)*(sin(x)^2)^(1/2)*EllipticF(I*cos(x),I)-6*I*(2-sin(x)^2)^(1/2)*(sin(x)^2 
)^(1/2)*EllipticE(I*cos(x),I)+2*sin(x)^2*cos(x))/(-1+cos(x)^4)^(1/2)/sin(x 
)/(-1-cos(x)^2)^(1/2)
 

3.1.59.5 Fricas [F]

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

input

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

output

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

3.1.59.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.1.59.7 Maxima [F]

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

input

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

output

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

3.1.59.8 Giac [F]

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

input

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

output

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

3.1.59.9 Mupad [F(-1)]

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

input

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

output

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