\(\int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx\) [392]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {3 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )}{2 f}-\frac {3 \sqrt {a \cos ^2(e+f x)}}{2 f}-\frac {\left (a \cos ^2(e+f x)\right )^{3/2} \csc ^2(e+f x)}{2 a f} \] Output:

3/2*a^(1/2)*arctanh((a*cos(f*x+e)^2)^(1/2)/a^(1/2))/f-3/2*(a*cos(f*x+e)^2) 
^(1/2)/f-1/2*(a*cos(f*x+e)^2)^(3/2)*csc(f*x+e)^2/a/f
 

Mathematica [A] (verified)

Time = 1.66 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.11 \[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a \cos ^2(e+f x)} \left (\frac {\text {arctanh}\left (\sqrt {\cos ^2(e+f x)}\right )}{\sqrt {\cos ^2(e+f x)}}+(-2+\cos (2 (e+f x))) \csc ^2(e+f x)\right )}{2 f} \] Input:

Integrate[Cot[e + f*x]^3*Sqrt[a - a*Sin[e + f*x]^2],x]
 

Output:

(2*Sqrt[a]*ArcTanh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]] + Sqrt[a*Cos[e + f*x]^2 
]*(ArcTanh[Sqrt[Cos[e + f*x]^2]]/Sqrt[Cos[e + f*x]^2] + (-2 + Cos[2*(e + f 
*x)])*Csc[e + f*x]^2))/(2*f)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3655, 3042, 25, 3684, 8, 51, 60, 73, 219}

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.

\(\displaystyle \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a-a \sin (e+f x)^2}}{\tan (e+f x)^3}dx\)

\(\Big \downarrow \) 3655

\(\displaystyle \int \cot ^3(e+f x) \sqrt {a \cos ^2(e+f x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \tan \left (e+f x+\frac {\pi }{2}\right )^3 \left (-\sqrt {a \sin \left (e+f x+\frac {\pi }{2}\right )^2}\right )dx\)

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3684

\(\displaystyle -\frac {\int \frac {\cos ^2(e+f x) \sqrt {a \cos ^2(e+f x)}}{\left (1-\cos ^2(e+f x)\right )^2}d\cos ^2(e+f x)}{2 f}\)

\(\Big \downarrow \) 8

\(\displaystyle -\frac {\int \frac {\left (a \cos ^2(e+f x)\right )^{3/2}}{\left (1-\cos ^2(e+f x)\right )^2}d\cos ^2(e+f x)}{2 a f}\)

\(\Big \downarrow \) 51

\(\displaystyle -\frac {\frac {\left (a \cos ^2(e+f x)\right )^{3/2}}{1-\cos ^2(e+f x)}-\frac {3}{2} a \int \frac {\sqrt {a \cos ^2(e+f x)}}{1-\cos ^2(e+f x)}d\cos ^2(e+f x)}{2 a f}\)

\(\Big \downarrow \) 60

\(\displaystyle -\frac {\frac {\left (a \cos ^2(e+f x)\right )^{3/2}}{1-\cos ^2(e+f x)}-\frac {3}{2} a \left (a \int \frac {1}{\sqrt {a \cos ^2(e+f x)} \left (1-\cos ^2(e+f x)\right )}d\cos ^2(e+f x)-2 \sqrt {a \cos ^2(e+f x)}\right )}{2 a f}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\frac {\left (a \cos ^2(e+f x)\right )^{3/2}}{1-\cos ^2(e+f x)}-\frac {3}{2} a \left (2 \int \frac {1}{1-\frac {\cos ^4(e+f x)}{a}}d\sqrt {a \cos ^2(e+f x)}-2 \sqrt {a \cos ^2(e+f x)}\right )}{2 a f}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\left (a \cos ^2(e+f x)\right )^{3/2}}{1-\cos ^2(e+f x)}-\frac {3}{2} a \left (2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cos ^2(e+f x)}}{\sqrt {a}}\right )-2 \sqrt {a \cos ^2(e+f x)}\right )}{2 a f}\)

Input:

Int[Cot[e + f*x]^3*Sqrt[a - a*Sin[e + f*x]^2],x]
 

Output:

-1/2*((a*Cos[e + f*x]^2)^(3/2)/(1 - Cos[e + f*x]^2) - (3*a*(2*Sqrt[a]*ArcT 
anh[Sqrt[a*Cos[e + f*x]^2]/Sqrt[a]] - 2*Sqrt[a*Cos[e + f*x]^2]))/2)/(a*f)
 

Defintions of rubi rules used

rule 8

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m   Int[u*(a* 
x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

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

rule 3655

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A 
ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ 
[a + b, 0]
 

rule 3684

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_. 
), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1 
)/2)/(2*f)   Subst[Int[x^((m - 1)/2)*((b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m 
 + 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && Inte 
gerQ[(m - 1)/2] && IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17

method	result	size
default	\(\frac {a \cos \left (f x +e \right ) \left (4 \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}+2 \cos \left (f x +e \right )+\left (-3 \ln \left (\cos \left (f x +e \right )+1\right )+3 \ln \left (\cos \left (f x +e \right )-1\right )\right ) \sin \left (f x +e \right )^{2}\right )}{4 \sqrt {a \cos \left (f x +e \right )^{2}}\, \left (\cos \left (f x +e \right )-1\right ) \left (\cos \left (f x +e \right )+1\right ) f}\)	\(102\)
risch	\(-\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, \left (3 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) {\mathrm e}^{5 i \left (f x +e \right )}-3 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) {\mathrm e}^{5 i \left (f x +e \right )}+{\mathrm e}^{6 i \left (f x +e \right )}-6 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) {\mathrm e}^{3 i \left (f x +e \right )}+6 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) {\mathrm e}^{3 i \left (f x +e \right )}-3 \,{\mathrm e}^{4 i \left (f x +e \right )}+3 \ln \left ({\mathrm e}^{i f x}-{\mathrm e}^{-i e}\right ) {\mathrm e}^{i \left (f x +e \right )}-3 \ln \left ({\mathrm e}^{i f x}+{\mathrm e}^{-i e}\right ) {\mathrm e}^{i \left (f x +e \right )}-3 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{2 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{2}}\)	\(241\)

Input:

int(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4*a*cos(f*x+e)*(4*cos(f*x+e)*sin(f*x+e)^2+2*cos(f*x+e)+(-3*ln(cos(f*x+e) 
+1)+3*ln(cos(f*x+e)-1))*sin(f*x+e)^2)/(a*cos(f*x+e)^2)^(1/2)/(cos(f*x+e)-1 
)/(cos(f*x+e)+1)/f
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.01 \[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=-\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (4 \, \cos \left (f x + e\right )^{3} + 3 \, {\left (\cos \left (f x + e\right )^{2} - 1\right )} \log \left (-\frac {\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right ) - 6 \, \cos \left (f x + e\right )\right )}}{4 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )}} \] Input:

integrate(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="fricas")
 

Output:

-1/4*sqrt(a*cos(f*x + e)^2)*(4*cos(f*x + e)^3 + 3*(cos(f*x + e)^2 - 1)*log 
(-(cos(f*x + e) - 1)/(cos(f*x + e) + 1)) - 6*cos(f*x + e))/(f*cos(f*x + e) 
^3 - f*cos(f*x + e))
 

Sympy [F]

\[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot ^{3}{\left (e + f x \right )}\, dx \] Input:

integrate(cot(f*x+e)**3*(a-a*sin(f*x+e)**2)**(1/2),x)
 

Output:

Integral(sqrt(-a*(sin(e + f*x) - 1)*(sin(e + f*x) + 1))*cot(e + f*x)**3, x 
)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.14 \[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\frac {3 \, \sqrt {a} \log \left (\frac {2 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (f x + e\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (f x + e\right ) \right |}}\right ) - 3 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} - \frac {{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{a \sin \left (f x + e\right )^{2}}}{2 \, f} \] Input:

integrate(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="maxima")
 

Output:

1/2*(3*sqrt(a)*log(2*sqrt(-a*sin(f*x + e)^2 + a)*sqrt(a)/abs(sin(f*x + e)) 
 + 2*a/abs(sin(f*x + e))) - 3*sqrt(-a*sin(f*x + e)^2 + a) - (-a*sin(f*x + 
e)^2 + a)^(3/2)/(a*sin(f*x + e)^2))/f
 

Giac [F(-2)]

Exception generated. \[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^3\,\sqrt {a-a\,{\sin \left (e+f\,x\right )}^2} \,d x \] Input:

int(cot(e + f*x)^3*(a - a*sin(e + f*x)^2)^(1/2),x)
 

Output:

int(cot(e + f*x)^3*(a - a*sin(e + f*x)^2)^(1/2), x)
 

Reduce [F]

\[ \int \cot ^3(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx=\sqrt {a}\, \left (\int \sqrt {-\sin \left (f x +e \right )^{2}+1}\, \cot \left (f x +e \right )^{3}d x \right ) \] Input:

int(cot(f*x+e)^3*(a-a*sin(f*x+e)^2)^(1/2),x)
 

Output:

sqrt(a)*int(sqrt( - sin(e + f*x)**2 + 1)*cot(e + f*x)**3,x)