\(\int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx\) [25]

Optimal result

Integrand size = 35, antiderivative size = 435 \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {b \text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {a} e}+\frac {\sqrt {a-b+c} \text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {b \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {(b-2 c) \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{4 \sqrt {c} e}-\frac {\sqrt {c} \text {arctanh}\left (\frac {b+2 c \cot ^2(d+e x)}{2 \sqrt {c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 e}+\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x)}{2 e} \] Output:

-1/2*a^(1/2)*arctanh(1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b*cot(e*x+d)^2+c* 
cot(e*x+d)^4)^(1/2))/e+1/4*b*arctanh(1/2*(2*a+b*cot(e*x+d)^2)/a^(1/2)/(a+b 
*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/a^(1/2)/e+1/2*(a-b+c)^(1/2)*arctanh(1 
/2*(2*a-b+(b-2*c)*cot(e*x+d)^2)/(a-b+c)^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+ 
d)^4)^(1/2))/e+1/4*b*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x 
+d)^2+c*cot(e*x+d)^4)^(1/2))/c^(1/2)/e-1/4*(b-2*c)*arctanh(1/2*(b+2*c*cot( 
e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2))/c^(1/2)/e-1/2*c 
^(1/2)*arctanh(1/2*(b+2*c*cot(e*x+d)^2)/c^(1/2)/(a+b*cot(e*x+d)^2+c*cot(e* 
x+d)^4)^(1/2))/e+1/2*(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^2/ 
e
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.54 \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\frac {\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^2(d+e x) \left ((-2 a+b) \text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+2 \sqrt {a} \left (\sqrt {a-b+c} \text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )+\sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}\right )\right )}{4 \sqrt {a} e \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}} \] Input:

Integrate[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^3,x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.66 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4184, 1578, 1289, 2009}

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.

\(\displaystyle \int \tan ^3(d+e x) \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}}{\cot (d+e x)^3}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a} \tan ^3(d+e x)}{\cot ^2(d+e x)+1}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

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

Input:

Int[Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x]^3,x]
 

Output:

-1/2*(Sqrt[a]*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*Sqrt[a]*Sqrt[a + b*Cot[d 
 + e*x]^2 + c*Cot[d + e*x]^4])] - (b*ArcTanh[(2*a + b*Cot[d + e*x]^2)/(2*S 
qrt[a]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*Sqrt[a]) - Sqrt 
[a - b + c]*ArcTanh[(2*a - b + (b - 2*c)*Cot[d + e*x]^2)/(2*Sqrt[a - b + c 
]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])] - (b*ArcTanh[(b + 2*c*Co 
t[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/ 
(2*Sqrt[c]) + ((b - 2*c)*ArcTanh[(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[ 
a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4])])/(2*Sqrt[c]) + Sqrt[c]*ArcTanh[ 
(b + 2*c*Cot[d + e*x]^2)/(2*Sqrt[c]*Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + 
e*x]^4])] - Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4]*Tan[d + e*x])/e
 

Defintions of rubi rules used

rule 1289

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

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

Maple [F]

\[\int \sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}\, \tan \left (e x +d \right )^{3}d x\]

Input:

int((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x)
 

Output:

int((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x)
 

Fricas [A] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 1282, normalized size of antiderivative = 2.95 \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\text {Too large to display} \] Input:

integrate((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorith 
m="fricas")
 

Output:

[1/8*(4*a*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*t 
an(e*x + d)^2 - (2*a - b)*sqrt(a)*log(8*a^2*tan(e*x + d)^4 + 8*a*b*tan(e*x 
 + d)^2 + b^2 + 4*a*c + 4*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(a)* 
sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)) + 2*sqrt(a 
 - b + c)*a*log(((8*a^2 - 8*a*b + b^2 + 4*a*c)*tan(e*x + d)^4 + 2*(4*a*b - 
 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 + 4*((2 
*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(a - b + c)*sqrt((a 
*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4))/(tan(e*x + d)^4 + 
 2*tan(e*x + d)^2 + 1)))/(a*e), 1/8*(4*a*sqrt((a*tan(e*x + d)^4 + b*tan(e* 
x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d)^2 + 4*a*sqrt(-a + b - c)*arctan 
(-1/2*((2*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(-a + b - 
c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/((a^2 - 
a*b + a*c)*tan(e*x + d)^4 + (a*b - b^2 + b*c)*tan(e*x + d)^2 + (a - b)*c + 
 c^2)) - (2*a - b)*sqrt(a)*log(8*a^2*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 
 + b^2 + 4*a*c + 4*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(a)*sqrt((a 
*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)))/(a*e), 1/4*(2*a* 
sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)*tan(e*x + d 
)^2 + sqrt(-a)*(2*a - b)*arctan(1/2*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2 
)*sqrt(-a)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/ 
(a^2*tan(e*x + d)^4 + a*b*tan(e*x + d)^2 + a*c)) + sqrt(a - b + c)*a*lo...
 

Sympy [F]

\[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\int \sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}} \tan ^{3}{\left (d + e x \right )}\, dx \] Input:

integrate((a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2)*tan(e*x+d)**3,x)
 

Output:

Integral(sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4)*tan(d + e*x)**3, 
x)
 

Maxima [F]

\[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a} \tan \left (e x + d\right )^{3} \,d x } \] Input:

integrate((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorith 
m="maxima")
 

Output:

integrate(sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a)*tan(e*x + d)^3, x)
 

Giac [F(-2)]

Exception generated. \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\text {Exception raised: AttributeError} \] Input:

integrate((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x, algorith 
m="giac")
 

Output:

Exception raised: AttributeError >> type
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\int {\mathrm {tan}\left (d+e\,x\right )}^3\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a} \,d x \] Input:

int(tan(d + e*x)^3*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)
 

Output:

int(tan(d + e*x)^3*(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)
 

Reduce [F]

\[ \int \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)} \tan ^3(d+e x) \, dx=\int \sqrt {\cot \left (e x +d \right )^{4} c +\cot \left (e x +d \right )^{2} b +a}\, \tan \left (e x +d \right )^{3}d x \] Input:

int((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x)
 

Output:

int((a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2)*tan(e*x+d)^3,x)