\(\int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx\) [324]

Optimal result

Integrand size = 25, antiderivative size = 116 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{2 a^{3/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} f}-\frac {\cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 a f} \] Output:

1/2*(2*a+b)*arctanh((a+b*tan(f*x+e)^2)^(1/2)/a^(1/2))/a^(3/2)/f-arctanh((a 
+b*tan(f*x+e)^2)^(1/2)/(a-b)^(1/2))/(a-b)^(1/2)/f-1/2*cot(f*x+e)^2*(a+b*ta 
n(f*x+e)^2)^(1/2)/a/f
 

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.16 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\left (2 a^2-a b-b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a} \left (-2 a \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )+(-a+b) \cot ^2(e+f x) \sqrt {a+b \tan ^2(e+f x)}\right )}{2 a^{3/2} (a-b) f} \] Input:

Integrate[Cot[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]
 

Output:

((2*a^2 - a*b - b^2)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]] + Sqrt[a] 
*(-2*a*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]] + (-a + 
 b)*Cot[e + f*x]^2*Sqrt[a + b*Tan[e + f*x]^2]))/(2*a^(3/2)*(a - b)*f)
 

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4153, 354, 114, 27, 174, 73, 221}

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.

\(\displaystyle \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (e+f x)^3 \sqrt {a+b \tan (e+f x)^2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \frac {\int \frac {\cot ^3(e+f x)}{\left (\tan ^2(e+f x)+1\right ) \sqrt {b \tan ^2(e+f x)+a}}d\tan (e+f x)}{f}\)

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 114

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {\frac {2 (2 a+b) \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}}d\sqrt {b \tan ^2(e+f x)+a}}{b}-\frac {4 a \int \frac {1}{\frac {\tan ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \tan ^2(e+f x)+a}}{b}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {\frac {4 a \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {2 (2 a+b) \text {arctanh}\left (\frac {\sqrt {a+b \tan ^2(e+f x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{2 a}-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a}}{2 f}\)

Input:

Int[Cot[e + f*x]^3/Sqrt[a + b*Tan[e + f*x]^2],x]
 

Output:

(-1/2*((-2*(2*a + b)*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a]])/Sqrt[a] 
+ (4*a*ArcTanh[Sqrt[a + b*Tan[e + f*x]^2]/Sqrt[a - b]])/Sqrt[a - b])/a - ( 
Cot[e + f*x]*Sqrt[a + b*Tan[e + f*x]^2])/a)/(2*f)
 

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 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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

rule 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 3042

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

rule 4153

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], 
 x]}, Simp[c*(ff/f)   Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, 
n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio 
nalQ[n]))
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(874\) vs. \(2(98)=196\).

Time = 6.56 (sec) , antiderivative size = 875, normalized size of antiderivative = 7.54

method	result	size
default	\(\text {Expression too large to display}\)	\(875\)

Input:

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

Output:

1/2/f/a^(5/2)/(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^ 
2)^(1/2)/(a+b*tan(f*x+e)^2)^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/(1-cos 
(f*x+e))^2*(4*ln(4*(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e 
)+1)^2)^(1/2)*cos(f*x+e)+4*(a-b)^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(c 
os(f*x+e)+1)^2)^(1/2)+4*a*cos(f*x+e)-4*cos(f*x+e)*b)*a^(5/2)*(1-cos(f*x+e) 
)^2-((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(1-cos(f*x+e) 
)^2*a^(3/2)*(a-b)^(1/2)-2*ln(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+2*(1- 
cos(f*x+e))^2*b+2*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2) 
*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2))*a^2*(1-cos(f*x+e))^2*(a-b)^(1/2)-ln 
(2/(1-cos(f*x+e))^2*(-a*(1-cos(f*x+e))^2+2*(1-cos(f*x+e))^2*b+2*((a*cos(f* 
x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin( 
f*x+e)^2))*a*(1-cos(f*x+e))^2*(a-b)^(1/2)*b+2*ln(2/a^(1/2)*(a^(1/2)*((a*co 
s(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+((a*cos(f*x+ 
e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)-a*cos(f*x+e)+cos(f*x+ 
e)*b+b)/(cos(f*x+e)+1))*a^2*(1-cos(f*x+e))^2*(a-b)^(1/2)+ln(2/a^(1/2)*(a^( 
1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*cos(f*x+e)+( 
(a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/2)-a*cos(f*x+ 
e)+cos(f*x+e)*b+b)/(cos(f*x+e)+1))*a*(1-cos(f*x+e))^2*(a-b)^(1/2)*b+((a*co 
s(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(3/2)*(a-b)^(1/2)*sin 
(f*x+e)^2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 676, normalized size of antiderivative = 5.83 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(2*sqrt(a - b)*a^2*log((b*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + 
a)*sqrt(a - b) + 2*a - b)/(tan(f*x + e)^2 + 1))*tan(f*x + e)^2 + (2*a^2 - 
a*b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sq 
rt(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a) 
*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), 1/4*(4*a^2*sqrt(-a + b)*ar 
ctan(sqrt(-a + b)/sqrt(b*tan(f*x + e)^2 + a))*tan(f*x + e)^2 + (2*a^2 - a* 
b - b^2)*sqrt(a)*log((b*tan(f*x + e)^2 + 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt 
(a) + 2*a)/tan(f*x + e)^2)*tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*( 
a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), 1/2*(sqrt(a - b)*a^2*log((b* 
tan(f*x + e)^2 - 2*sqrt(b*tan(f*x + e)^2 + a)*sqrt(a - b) + 2*a - b)/(tan( 
f*x + e)^2 + 1))*tan(f*x + e)^2 - (2*a^2 - a*b - b^2)*sqrt(-a)*arctan(sqrt 
(-a)/sqrt(b*tan(f*x + e)^2 + a))*tan(f*x + e)^2 - sqrt(b*tan(f*x + e)^2 + 
a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + e)^2), 1/2*(2*a^2*sqrt(-a + b)* 
arctan(sqrt(-a + b)/sqrt(b*tan(f*x + e)^2 + a))*tan(f*x + e)^2 - (2*a^2 - 
a*b - b^2)*sqrt(-a)*arctan(sqrt(-a)/sqrt(b*tan(f*x + e)^2 + a))*tan(f*x + 
e)^2 - sqrt(b*tan(f*x + e)^2 + a)*(a^2 - a*b))/((a^3 - a^2*b)*f*tan(f*x + 
e)^2)]
 

Sympy [F]

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

integrate(cot(f*x+e)**3/(a+b*tan(f*x+e)**2)**(1/2),x)
 

Output:

Integral(cot(e + f*x)**3/sqrt(a + b*tan(e + f*x)**2), x)
 

Maxima [F]

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

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

Output:

integrate(cot(f*x + e)^3/sqrt(b*tan(f*x + e)^2 + a), x)
 

Giac [F(-2)]

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

integrate(cot(f*x+e)^3/(a+b*tan(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:Error: Bad Argument Type
 

Mupad [B] (verification not implemented)

Time = 8.10 (sec) , antiderivative size = 830, normalized size of antiderivative = 7.16 \[ \int \frac {\cot ^3(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

(atan((((((2*a*b^4*f^2 + 2*a^2*b^3*f^2)/(2*a^2*f^3) - ((a + b*tan(e + f*x) 
^2)^(1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2*f^2))/(8*a^2*f^3*(a - b)^(1/2)))/(2 
*f*(a - b)^(1/2)) - ((a + b*tan(e + f*x)^2)^(1/2)*(4*a*b^3 + b^4 + 8*a^2*b 
^2))/(4*a^2*f^2))*1i)/(f*(a - b)^(1/2)) - ((((2*a*b^4*f^2 + 2*a^2*b^3*f^2) 
/(2*a^2*f^3) + ((a + b*tan(e + f*x)^2)^(1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2* 
f^2))/(8*a^2*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) + ((a + b*tan(e + f*x 
)^2)^(1/2)*(4*a*b^3 + b^4 + 8*a^2*b^2))/(4*a^2*f^2))*1i)/(f*(a - b)^(1/2)) 
)/((((2*a*b^4*f^2 + 2*a^2*b^3*f^2)/(2*a^2*f^3) - ((a + b*tan(e + f*x)^2)^( 
1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2*f^2))/(8*a^2*f^3*(a - b)^(1/2)))/(2*f*(a 
 - b)^(1/2)) - ((a + b*tan(e + f*x)^2)^(1/2)*(4*a*b^3 + b^4 + 8*a^2*b^2))/ 
(4*a^2*f^2))/(f*(a - b)^(1/2)) + (((2*a*b^4*f^2 + 2*a^2*b^3*f^2)/(2*a^2*f^ 
3) + ((a + b*tan(e + f*x)^2)^(1/2)*(16*a^2*b^3*f^2 - 32*a^3*b^2*f^2))/(8*a 
^2*f^3*(a - b)^(1/2)))/(2*f*(a - b)^(1/2)) + ((a + b*tan(e + f*x)^2)^(1/2) 
*(4*a*b^3 + b^4 + 8*a^2*b^2))/(4*a^2*f^2))/(f*(a - b)^(1/2)) - (a*b^3 + b^ 
4/2)/(a^2*f^3)))*1i)/(f*(a - b)^(1/2)) - (b*(a + b*tan(e + f*x)^2)^(1/2))/ 
(2*a*(f*(a + b*tan(e + f*x)^2) - a*f)) + (atanh((b^6*(a + b*tan(e + f*x)^2 
)^(1/2))/(4*(a^3)^(1/2)*((3*a*b^4)/2 + (5*b^5)/4 + b^6/(4*a))) + (3*b^4*(a 
 + b*tan(e + f*x)^2)^(1/2))/(2*(a^3)^(1/2)*((3*b^4)/(2*a) + (5*b^5)/(4*a^2 
) + b^6/(4*a^3))) + (5*b^5*(a + b*tan(e + f*x)^2)^(1/2))/(4*(a^3)^(1/2)*(( 
3*b^4)/2 + (5*b^5)/(4*a) + b^6/(4*a^2))))*(2*a + b))/(2*f*(a^3)^(1/2))
 

Reduce [F]

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

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

Output:

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