\(\int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx\) [97]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=-\frac {3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f}+\frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {5 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {\cot (e+f x) (a+a \sin (e+f x))^{3/2}}{f} \] Output:

-3*a^(3/2)*arctanh(a^(1/2)*cos(f*x+e)/(a+a*sin(f*x+e))^(1/2))/f+11/3*a^2*c 
os(f*x+e)/f/(a+a*sin(f*x+e))^(1/2)+5/3*a*cos(f*x+e)*(a+a*sin(f*x+e))^(1/2) 
/f-cot(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f
 

Mathematica [A] (verified)

Time = 6.37 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.93 \[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=-\frac {a \csc ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sin (e+f x))} \left (14 \cos \left (\frac {1}{2} (e+f x)\right )-9 \cos \left (\frac {3}{2} (e+f x)\right )+\cos \left (\frac {5}{2} (e+f x)\right )-14 \sin \left (\frac {1}{2} (e+f x)\right )+9 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-9 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-9 \sin \left (\frac {3}{2} (e+f x)\right )-\sin \left (\frac {5}{2} (e+f x)\right )\right )}{3 f \left (1+\cot \left (\frac {1}{2} (e+f x)\right )\right ) \left (\csc \left (\frac {1}{4} (e+f x)\right )-\sec \left (\frac {1}{4} (e+f x)\right )\right ) \left (\csc \left (\frac {1}{4} (e+f x)\right )+\sec \left (\frac {1}{4} (e+f x)\right )\right )} \] Input:

Integrate[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2),x]
 

Output:

-1/3*(a*Csc[(e + f*x)/2]^4*Sqrt[a*(1 + Sin[e + f*x])]*(14*Cos[(e + f*x)/2] 
 - 9*Cos[(3*(e + f*x))/2] + Cos[(5*(e + f*x))/2] - 14*Sin[(e + f*x)/2] + 9 
*Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] - 9*Log[1 - Cos 
[(e + f*x)/2] + Sin[(e + f*x)/2]]*Sin[e + f*x] - 9*Sin[(3*(e + f*x))/2] - 
Sin[(5*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4] - Sec[( 
e + f*x)/4])*(Csc[(e + f*x)/4] + Sec[(e + f*x)/4]))
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 3195, 27, 3042, 3455, 27, 3042, 3460, 3042, 3252, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3195

\(\displaystyle \frac {\int \frac {1}{2} \csc (e+f x) (3 a-5 a \sin (e+f x)) (\sin (e+f x) a+a)^{3/2}dx}{a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \csc (e+f x) (3 a-5 a \sin (e+f x)) (\sin (e+f x) a+a)^{3/2}dx}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {(3 a-5 a \sin (e+f x)) (\sin (e+f x) a+a)^{3/2}}{\sin (e+f x)}dx}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {2}{3} \int \frac {1}{2} \csc (e+f x) \sqrt {\sin (e+f x) a+a} \left (9 a^2-11 a^2 \sin (e+f x)\right )dx+\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \int \csc (e+f x) \sqrt {\sin (e+f x) a+a} \left (9 a^2-11 a^2 \sin (e+f x)\right )dx+\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int \frac {\sqrt {\sin (e+f x) a+a} \left (9 a^2-11 a^2 \sin (e+f x)\right )}{\sin (e+f x)}dx+\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 3460

\(\displaystyle \frac {\frac {1}{3} \left (9 a^2 \int \csc (e+f x) \sqrt {\sin (e+f x) a+a}dx+\frac {22 a^3 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )+\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (9 a^2 \int \frac {\sqrt {\sin (e+f x) a+a}}{\sin (e+f x)}dx+\frac {22 a^3 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}\right )+\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {1}{3} \left (\frac {22 a^3 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-\frac {18 a^3 \int \frac {1}{a-\frac {a^2 \cos ^2(e+f x)}{\sin (e+f x) a+a}}d\frac {a \cos (e+f x)}{\sqrt {\sin (e+f x) a+a}}}{f}\right )+\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {10 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}+\frac {1}{3} \left (\frac {22 a^3 \cos (e+f x)}{f \sqrt {a \sin (e+f x)+a}}-\frac {18 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}\right )}{2 a}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f}\)

Input:

Int[Cot[e + f*x]^2*(a + a*Sin[e + f*x])^(3/2),x]
 

Output:

-((Cot[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/f) + ((10*a^2*Cos[e + f*x]*Sqr 
t[a + a*Sin[e + f*x]])/(3*f) + ((-18*a^(5/2)*ArcTanh[(Sqrt[a]*Cos[e + f*x] 
)/Sqrt[a + a*Sin[e + f*x]]])/f + (22*a^3*Cos[e + f*x])/(f*Sqrt[a + a*Sin[e 
 + f*x]]))/3)/(2*a)
 

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 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 3195

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)/tan[(e_.) + (f_.)*(x_)]^2, 
 x_Symbol] :> Simp[-(a + b*Sin[e + f*x])^m/(f*Tan[e + f*x]), x] + Simp[1/a 
  Int[(a + b*Sin[e + f*x])^m*((b*m - a*(m + 1)*Sin[e + f*x])/Sin[e + f*x]), 
 x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1 
/2] &&  !LtQ[m, -1]
 

rule 3252

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)]), x_Symbol] :> Simp[-2*(b/f)   Subst[Int[1/(b*c + a*d - d*x^2), 
x], x, b*(Cos[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3455

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n 
 + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 
) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + 
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1 
] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

rule 3460

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( 
f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp 
[-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt[a + 
b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b 
*d*(2*n + 3))   Int[Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - 
 b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (-1+\sin \left (f x +e \right )\right )}\, \left (12 \sqrt {a -\sin \left (f x +e \right ) a}\, \sin \left (f x +e \right ) a^{\frac {3}{2}}-2 \left (a -\sin \left (f x +e \right ) a \right )^{\frac {3}{2}} \sin \left (f x +e \right ) \sqrt {a}-3 \sqrt {a -\sin \left (f x +e \right ) a}\, a^{\frac {3}{2}}-9 \,\operatorname {arctanh}\left (\frac {\sqrt {a -\sin \left (f x +e \right ) a}}{\sqrt {a}}\right ) \sin \left (f x +e \right ) a^{2}\right )}{3 \sin \left (f x +e \right ) \sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +\sin \left (f x +e \right ) a}\, f}\)	\(154\)

Input:

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

Output:

1/3*(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(12*(a-sin(f*x+e)*a)^(1/2)*s 
in(f*x+e)*a^(3/2)-2*(a-sin(f*x+e)*a)^(3/2)*sin(f*x+e)*a^(1/2)-3*(a-sin(f*x 
+e)*a)^(1/2)*a^(3/2)-9*arctanh((a-sin(f*x+e)*a)^(1/2)/a^(1/2))*sin(f*x+e)* 
a^2)/sin(f*x+e)/a^(1/2)/cos(f*x+e)/(a+sin(f*x+e)*a)^(1/2)/f
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (105) = 210\).

Time = 0.10 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.60 \[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\frac {9 \, {\left (a \cos \left (f x + e\right )^{2} - {\left (a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, {\left (2 \, a \cos \left (f x + e\right )^{3} - 8 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - {\left (2 \, a \cos \left (f x + e\right )^{2} + 10 \, a \cos \left (f x + e\right ) + 11 \, a\right )} \sin \left (f x + e\right ) + 11 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - {\left (f \cos \left (f x + e\right ) + f\right )} \sin \left (f x + e\right ) - f\right )}} \] Input:

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

Output:

1/12*(9*(a*cos(f*x + e)^2 - (a*cos(f*x + e) + a)*sin(f*x + e) - a)*sqrt(a) 
*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^2 + (cos(f*x 
 + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqr 
t(a) - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f* 
x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f* 
x + e) - cos(f*x + e) - 1)) + 4*(2*a*cos(f*x + e)^3 - 8*a*cos(f*x + e)^2 + 
 a*cos(f*x + e) - (2*a*cos(f*x + e)^2 + 10*a*cos(f*x + e) + 11*a)*sin(f*x 
+ e) + 11*a)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^2 - (f*cos(f*x + e) 
 + f)*sin(f*x + e) - f)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.51 \[ \int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {2} {\left (16 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 48 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {12 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1}\right )} \sqrt {a}}{12 \, f} \] Input:

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

Output:

1/12*sqrt(2)*(16*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f 
*x + 1/2*e)^3 - 9*sqrt(2)*a*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 
 1/2*e))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*f*x + 1/2*e)))*sgn(cos(-1/4*p 
i + 1/2*f*x + 1/2*e)) - 48*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4* 
pi + 1/2*f*x + 1/2*e) - 12*a*sgn(cos(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4* 
pi + 1/2*f*x + 1/2*e)/(2*sin(-1/4*pi + 1/2*f*x + 1/2*e)^2 - 1))*sqrt(a)/f
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(a)*a*(int(sqrt(sin(e + f*x) + 1)*cot(e + f*x)**2*sin(e + f*x),x) + in 
t(sqrt(sin(e + f*x) + 1)*cot(e + f*x)**2,x))