\(\int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx\) [25]

Optimal result

Integrand size = 17, antiderivative size = 85 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=-\sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {(3 a-b) \sqrt {a+b \cot ^2(x)} \tan (x)}{3 a}+\frac {1}{3} \sqrt {a+b \cot ^2(x)} \tan ^3(x) \] Output:

-(a-b)^(1/2)*arctan((a-b)^(1/2)*cot(x)/(a+b*cot(x)^2)^(1/2))-1/3*(3*a-b)*( 
a+b*cot(x)^2)^(1/2)*tan(x)/a+1/3*(a+b*cot(x)^2)^(1/2)*tan(x)^3
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 3.41 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.05 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\frac {1}{3} \sqrt {a+b \cot ^2(x)} \left (1+\frac {b \cot ^2(x)}{a}\right ) \sin ^2(x) \left (-\frac {4 (a-b) \cos ^2(x) \left (a+b \cot ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {3}{2},\frac {(a-b) \cos ^2(x)}{a}\right )}{a^2}+\frac {\left (a-2 b \cot ^2(x)\right ) \csc ^2(x) \left (\arcsin \left (\sqrt {\frac {(a-b) \cos ^2(x)}{a}}\right ) \sqrt {\frac {(a-b) \cos ^2(x)}{a}}+\sqrt {\frac {b \cos ^2(x)}{a}+\sin ^2(x)}\right )}{\left (a+b \cot ^2(x)\right ) \sqrt {\frac {b \cos ^2(x)}{a}+\sin ^2(x)}}\right ) \tan ^3(x) \] Input:

Integrate[Sqrt[a + b*Cot[x]^2]*Tan[x]^4,x]
 

Output:

(Sqrt[a + b*Cot[x]^2]*(1 + (b*Cot[x]^2)/a)*Sin[x]^2*((-4*(a - b)*Cos[x]^2* 
(a + b*Cot[x]^2)*Hypergeometric2F1[2, 2, 3/2, ((a - b)*Cos[x]^2)/a])/a^2 + 
 ((a - 2*b*Cot[x]^2)*Csc[x]^2*(ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Sqrt[((a 
 - b)*Cos[x]^2)/a] + Sqrt[(b*Cos[x]^2)/a + Sin[x]^2]))/((a + b*Cot[x]^2)*S 
qrt[(b*Cos[x]^2)/a + Sin[x]^2]))*Tan[x]^3)/3
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3042, 4153, 377, 25, 445, 27, 291, 216}

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.

\(\displaystyle \int \tan ^4(x) \sqrt {a+b \cot ^2(x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \tan \left (x+\frac {\pi }{2}\right )^2}}{\tan \left (x+\frac {\pi }{2}\right )^4}dx\)

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{3} \left (-3 \sqrt {a-b} \arctan \left (\frac {\sqrt {a-b} \cot (x)}{\sqrt {a+b \cot ^2(x)}}\right )-\frac {(3 a-b) \tan (x) \sqrt {a+b \cot ^2(x)}}{a}\right )+\frac {1}{3} \tan ^3(x) \sqrt {a+b \cot ^2(x)}\)

Input:

Int[Sqrt[a + b*Cot[x]^2]*Tan[x]^4,x]
 

Output:

(Sqrt[a + b*Cot[x]^2]*Tan[x]^3)/3 + (-3*Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Co 
t[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a - b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/a)/3
 

Defintions of rubi rules used

rule 25

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

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 291

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

rule 377

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

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

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. \(313\) vs. \(2(71)=142\).

Time = 4.10 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.69

method	result	size
default	\(-\frac {\sqrt {4}\, \sqrt {a +b \cot \left (x \right )^{2}}\, \left (\sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, a \left (4 \sin \left (x \right )+4 \tan \left (x \right )-\tan \left (x \right ) \sec \left (x \right )-\tan \left (x \right ) \sec \left (x \right )^{2}\right )+b \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, \left (-\sin \left (x \right )-\tan \left (x \right )\right )+3 \ln \left (4 \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )\right ) a^{2} \sin \left (x \right )-3 \ln \left (4 \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {-a +b}\, \cos \left (x \right )+4 \sqrt {-a +b}\, \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}-4 \cos \left (x \right ) a +4 b \cos \left (x \right )\right ) a b \sin \left (x \right )\right )}{6 a \sqrt {-a +b}\, \left (\cos \left (x \right )+1\right ) \sqrt {\frac {\cos \left (x \right )^{2} b +a \sin \left (x \right )^{2}}{\left (\cos \left (x \right )+1\right )^{2}}}}\)	\(314\)

Input:

int((a+b*cot(x)^2)^(1/2)*tan(x)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/6*4^(1/2)/a/(-a+b)^(1/2)*(a+b*cot(x)^2)^(1/2)/(cos(x)+1)/((cos(x)^2*b+a 
*sin(x)^2)/(cos(x)+1)^2)^(1/2)*(((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/ 
2)*(-a+b)^(1/2)*a*(4*sin(x)+4*tan(x)-tan(x)*sec(x)-tan(x)*sec(x)^2)+b*((co 
s(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*(-sin(x)-tan(x))+3*l 
n(4*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*cos(x)+4*(-a 
+b)^(1/2)*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)-4*cos(x)*a+4*b*cos( 
x))*a^2*sin(x)-3*ln(4*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)*(-a+b)^ 
(1/2)*cos(x)+4*(-a+b)^(1/2)*((cos(x)^2*b+a*sin(x)^2)/(cos(x)+1)^2)^(1/2)-4 
*cos(x)*a+4*b*cos(x))*a*b*sin(x))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.08 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\left [\frac {3 \, a \sqrt {-a + b} \log \left (-\frac {a^{2} \tan \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} - 4 \, a b\right )} \tan \left (x\right )^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {-a + b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 4 \, {\left (a \tan \left (x\right )^{3} - {\left (3 \, a - b\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{12 \, a}, -\frac {3 \, \sqrt {a - b} a \arctan \left (-\frac {{\left (a \tan \left (x\right )^{3} - {\left (a - 2 \, b\right )} \tan \left (x\right )\right )} \sqrt {a - b} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{2 \, {\left ({\left (a^{2} - a b\right )} \tan \left (x\right )^{2} + a b - b^{2}\right )}}\right ) - 2 \, {\left (a \tan \left (x\right )^{3} - {\left (3 \, a - b\right )} \tan \left (x\right )\right )} \sqrt {\frac {a \tan \left (x\right )^{2} + b}{\tan \left (x\right )^{2}}}}{6 \, a}\right ] \] Input:

integrate((a+b*cot(x)^2)^(1/2)*tan(x)^4,x, algorithm="fricas")
 

Output:

[1/12*(3*a*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + 
a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt( 
(a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + 4*(a*tan(x)^3 - 
 (3*a - b)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/a, -1/6*(3*sqrt(a - b) 
*a*arctan(-1/2*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(a - b)*sqrt((a*tan(x)^ 
2 + b)/tan(x)^2)/((a^2 - a*b)*tan(x)^2 + a*b - b^2)) - 2*(a*tan(x)^3 - (3* 
a - b)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/a]
 

Sympy [F]

\[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int \sqrt {a + b \cot ^{2}{\left (x \right )}} \tan ^{4}{\left (x \right )}\, dx \] Input:

integrate((a+b*cot(x)**2)**(1/2)*tan(x)**4,x)
 

Output:

Integral(sqrt(a + b*cot(x)**2)*tan(x)**4, x)
 

Maxima [F]

\[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int { \sqrt {b \cot \left (x\right )^{2} + a} \tan \left (x\right )^{4} \,d x } \] Input:

integrate((a+b*cot(x)^2)^(1/2)*tan(x)^4,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*cot(x)^2 + a)*tan(x)^4, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (71) = 142\).

Time = 0.19 (sec) , antiderivative size = 476, normalized size of antiderivative = 5.60 \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=-\frac {1}{6} \, {\left (3 \, \sqrt {-a + b} \log \left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2}\right ) - \frac {4 \, {\left (3 \, {\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{4} {\left (2 \, a - b\right )} \sqrt {-a + b} - 6 \, {\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} a^{2} \sqrt {-a + b} + {\left (4 \, a^{3} - a^{2} b\right )} \sqrt {-a + b}\right )}}{{\left ({\left (\sqrt {-a + b} \cos \left (x\right ) - \sqrt {-a \cos \left (x\right )^{2} + b \cos \left (x\right )^{2} + a}\right )}^{2} - a\right )}^{3}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) + \frac {{\left (3 \, a^{2} \sqrt {-a + b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 9 \, a^{2} \sqrt {b} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 15 \, a \sqrt {-a + b} b \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 21 \, a b^{\frac {3}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 12 \, \sqrt {-a + b} b^{2} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) - 12 \, b^{\frac {5}{2}} \log \left (-a - 2 \, \sqrt {-a + b} \sqrt {b} + 2 \, b\right ) + 8 \, a^{2} \sqrt {-a + b} - 18 \, a^{2} \sqrt {b} - 24 \, a \sqrt {-a + b} b + 30 \, a b^{\frac {3}{2}} + 12 \, \sqrt {-a + b} b^{2} - 12 \, b^{\frac {5}{2}}\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{6 \, {\left (a^{2} + 3 \, a \sqrt {-a + b} \sqrt {b} - 5 \, a b - 4 \, \sqrt {-a + b} b^{\frac {3}{2}} + 4 \, b^{2}\right )}} \] Input:

integrate((a+b*cot(x)^2)^(1/2)*tan(x)^4,x, algorithm="giac")
 

Output:

-1/6*(3*sqrt(-a + b)*log((sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x 
)^2 + a))^2) - 4*(3*(sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + 
 a))^4*(2*a - b)*sqrt(-a + b) - 6*(sqrt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 
+ b*cos(x)^2 + a))^2*a^2*sqrt(-a + b) + (4*a^3 - a^2*b)*sqrt(-a + b))/((sq 
rt(-a + b)*cos(x) - sqrt(-a*cos(x)^2 + b*cos(x)^2 + a))^2 - a)^3)*sgn(sin( 
x)) + 1/6*(3*a^2*sqrt(-a + b)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) - 9*a 
^2*sqrt(b)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) - 15*a*sqrt(-a + b)*b*lo 
g(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + 21*a*b^(3/2)*log(-a - 2*sqrt(-a + b 
)*sqrt(b) + 2*b) + 12*sqrt(-a + b)*b^2*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2 
*b) - 12*b^(5/2)*log(-a - 2*sqrt(-a + b)*sqrt(b) + 2*b) + 8*a^2*sqrt(-a + 
b) - 18*a^2*sqrt(b) - 24*a*sqrt(-a + b)*b + 30*a*b^(3/2) + 12*sqrt(-a + b) 
*b^2 - 12*b^(5/2))*sgn(sin(x))/(a^2 + 3*a*sqrt(-a + b)*sqrt(b) - 5*a*b - 4 
*sqrt(-a + b)*b^(3/2) + 4*b^2)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int {\mathrm {tan}\left (x\right )}^4\,\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a} \,d x \] Input:

int(tan(x)^4*(a + b*cot(x)^2)^(1/2),x)
 

Output:

int(tan(x)^4*(a + b*cot(x)^2)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a+b \cot ^2(x)} \tan ^4(x) \, dx=\int \sqrt {\cot \left (x \right )^{2} b +a}\, \tan \left (x \right )^{4}d x \] Input:

int((a+b*cot(x)^2)^(1/2)*tan(x)^4,x)
 

Output:

int(sqrt(cot(x)**2*b + a)*tan(x)**4,x)