\(\int \frac {\cot (x)}{(a+b \tan ^4(x))^{5/2}} \, dx\) [407]

Optimal result

Integrand size = 15, antiderivative size = 149 \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {b \left (1-\tan ^2(x)\right )}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {b \left (3 (2 a+b)-(5 a+2 b) \tan ^2(x)\right )}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}} \] Output:

1/2*arctanh((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))/(a+b)^(5/2)-1 
/2*arctanh((a+b*tan(x)^4)^(1/2)/a^(1/2))/a^(5/2)+1/6*b*(1-tan(x)^2)/a/(a+b 
)/(a+b*tan(x)^4)^(3/2)+1/6*b*(6*a+3*b-(5*a+2*b)*tan(x)^2)/a^2/(a+b)^2/(a+b 
*tan(x)^4)^(1/2)
 

Mathematica [C] (verified)

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

Time = 1.10 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {1}{6} \left (\frac {3 \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{5/2}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \tan ^4(x)}{a}\right )}{a \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {a^2 (4 a+b)+3 a b (2 a+b) \tan ^2(x)+3 a^2 b \tan ^4(x)+b^2 (5 a+2 b) \tan ^6(x)}{a^2 (a+b)^2 \left (a+b \tan ^4(x)\right )^{3/2}}\right ) \] Input:

Integrate[Cot[x]/(a + b*Tan[x]^4)^(5/2),x]
 

Output:

((3*ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])])/(a + b)^ 
(5/2) + Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tan[x]^4)/a]/(a*(a + b*Tan 
[x]^4)^(3/2)) - (a^2*(4*a + b) + 3*a*b*(2*a + b)*Tan[x]^2 + 3*a^2*b*Tan[x] 
^4 + b^2*(5*a + 2*b)*Tan[x]^6)/(a^2*(a + b)^2*(a + b*Tan[x]^4)^(3/2)))/6
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4153, 1579, 617, 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.

\(\displaystyle \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (x) \left (a+b \tan (x)^4\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \int \frac {\cot (x)}{\left (\tan ^2(x)+1\right ) \left (a+b \tan ^4(x)\right )^{5/2}}d\tan (x)\)

\(\Big \downarrow \) 1579

\(\displaystyle \frac {1}{2} \int \frac {\cot (x)}{\left (\tan ^2(x)+1\right ) \left (b \tan ^4(x)+a\right )^{5/2}}d\tan ^2(x)\)

\(\Big \downarrow \) 617

\(\displaystyle \frac {1}{2} \int \left (\frac {\cot (x)}{\left (b \tan ^4(x)+a\right )^{5/2}}+\frac {1}{\left (-\tan ^2(x)-1\right ) \left (b \tan ^4(x)+a\right )^{5/2}}\right )d\tan ^2(x)\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {1}{a^2 \sqrt {a+b \tan ^4(x)}}-\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{3 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}}+\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{5/2}}+\frac {1}{3 a \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {a+b \tan ^2(x)}{3 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}\right )\)

Input:

Int[Cot[x]/(a + b*Tan[x]^4)^(5/2),x]
 

Output:

(ArcTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])]/(a + b)^(5/2 
) - ArcTanh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]]/a^(5/2) + 1/(3*a*(a + b*Tan[x]^4 
)^(3/2)) - (a + b*Tan[x]^2)/(3*a*(a + b)*(a + b*Tan[x]^4)^(3/2)) + 1/(a^2* 
Sqrt[a + b*Tan[x]^4]) - (3*a^2 + b*(5*a + 2*b)*Tan[x]^2)/(3*a^2*(a + b)^2* 
Sqrt[a + b*Tan[x]^4]))/2
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 1579

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

\[\int \frac {\cot \left (x \right )}{\left (a +b \tan \left (x \right )^{4}\right )^{\frac {5}{2}}}d x\]

Input:

int(cot(x)/(a+b*tan(x)^4)^(5/2),x)
 

Output:

int(cot(x)/(a+b*tan(x)^4)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 423 vs. \(2 (123) = 246\).

Time = 0.49 (sec) , antiderivative size = 1749, normalized size of antiderivative = 11.74 \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cot(x)/(a+b*tan(x)^4)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(3*(a^3*b^2*tan(x)^8 + 2*a^4*b*tan(x)^4 + a^5)*sqrt(a + b)*log(((a*b 
 + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - 
 a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 3*((a^3*b^2 
+ 3*a^2*b^3 + 3*a*b^4 + b^5)*tan(x)^8 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^ 
3 + 2*(a^4*b + 3*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4)*sqrt(a)*log(-(b*ta 
n(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4) - 2*((5*a^3*b^2 + 
 7*a^2*b^3 + 2*a*b^4)*tan(x)^6 - 7*a^4*b - 11*a^3*b^2 - 4*a^2*b^3 - 3*(2*a 
^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4 + 3*(2*a^4*b + 3*a^3*b^2 + a^2*b^3)*t 
an(x)^2)*sqrt(b*tan(x)^4 + a))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5 
)*tan(x)^8 + a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^7*b + 3*a^6*b^2 + 
3*a^5*b^3 + a^4*b^4)*tan(x)^4), 1/12*(6*((a^3*b^2 + 3*a^2*b^3 + 3*a*b^4 + 
b^5)*tan(x)^8 + a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3 + 2*(a^4*b + 3*a^3*b^2 
 + 3*a^2*b^3 + a*b^4)*tan(x)^4)*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sqrt( 
-a)/a) + 3*(a^3*b^2*tan(x)^8 + 2*a^4*b*tan(x)^4 + a^5)*sqrt(a + b)*log(((a 
*b + 2*b^2)*tan(x)^4 - 2*a*b*tan(x)^2 - 2*sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 
 - a)*sqrt(a + b) + 2*a^2 + a*b)/(tan(x)^4 + 2*tan(x)^2 + 1)) - 2*((5*a^3* 
b^2 + 7*a^2*b^3 + 2*a*b^4)*tan(x)^6 - 7*a^4*b - 11*a^3*b^2 - 4*a^2*b^3 - 3 
*(2*a^3*b^2 + 3*a^2*b^3 + a*b^4)*tan(x)^4 + 3*(2*a^4*b + 3*a^3*b^2 + a^2*b 
^3)*tan(x)^2)*sqrt(b*tan(x)^4 + a))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^ 
3*b^5)*tan(x)^8 + a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3 + 2*(a^7*b + 3*a^...
 

Sympy [F]

\[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int \frac {\cot {\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(cot(x)/(a+b*tan(x)**4)**(5/2),x)
 

Output:

Integral(cot(x)/(a + b*tan(x)**4)**(5/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(cot(x)/(a+b*tan(x)^4)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(cot(x)/(a+b*tan(x)^4)^(5/2),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 [F(-1)]

Timed out. \[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\text {Hanged} \] Input:

int(cot(x)/(a + b*tan(x)^4)^(5/2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {\cot (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\tan \left (x \right )^{4} b +a}\, \cot \left (x \right )}{\tan \left (x \right )^{12} b^{3}+3 \tan \left (x \right )^{8} a \,b^{2}+3 \tan \left (x \right )^{4} a^{2} b +a^{3}}d x \] Input:

int(cot(x)/(a+b*tan(x)^4)^(5/2),x)
 

Output:

int((sqrt(tan(x)**4*b + a)*cot(x))/(tan(x)**12*b**3 + 3*tan(x)**8*a*b**2 + 
 3*tan(x)**4*a**2*b + a**3),x)