\(\int \cot ^3(x) (a+b \tan ^4(x))^{3/2} \, dx\) [396]

Optimal result

Integrand size = 17, antiderivative size = 140 \[ \int \cot ^3(x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=-\frac {1}{2} b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )+\frac {1}{2} b \sqrt {a+b \tan ^4(x)}-\frac {1}{2} a \cot ^2(x) \sqrt {a+b \tan ^4(x)} \] Output:

-1/2*b^(3/2)*arctanh(b^(1/2)*tan(x)^2/(a+b*tan(x)^4)^(1/2))-1/2*(a+b)^(3/2 
)*arctanh((a-b*tan(x)^2)/(a+b)^(1/2)/(a+b*tan(x)^4)^(1/2))+1/2*a^(3/2)*arc 
tanh((a+b*tan(x)^4)^(1/2)/a^(1/2))+1/2*b*(a+b*tan(x)^4)^(1/2)-1/2*a*cot(x) 
^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 = 2.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.77 \[ \int \cot ^3(x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\frac {-\sqrt {a} \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a}}\right ) \sqrt {a+b \tan ^4(x)}-2 a \cot ^2(x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{2},\frac {1}{2},-\frac {b \tan ^4(x)}{a}\right ) \sqrt {a+b \tan ^4(x)}+\sqrt {1+\frac {b \tan ^4(x)}{a}} \left (-2 \sqrt {b} (a+b) \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-2 (a+b)^{3/2} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )+2 b \sqrt {a+b \tan ^4(x)}-b \tan ^2(x) \sqrt {a+b \tan ^4(x)}\right )}{4 \sqrt {1+\frac {b \tan ^4(x)}{a}}} \] Input:

Integrate[Cot[x]^3*(a + b*Tan[x]^4)^(3/2),x]
 

Output:

(-(Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Tan[x]^2)/Sqrt[a]]*Sqrt[a + b*Tan[x]^4 
]) - 2*a*Cot[x]^2*Hypergeometric2F1[-3/2, -1/2, 1/2, -((b*Tan[x]^4)/a)]*Sq 
rt[a + b*Tan[x]^4] + Sqrt[1 + (b*Tan[x]^4)/a]*(-2*Sqrt[b]*(a + b)*ArcTanh[ 
(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] - 2*(a + b)^(3/2)*ArcTanh[(a - b* 
Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] + 2*a^(3/2)*ArcTanh[Sqrt[a + 
 b*Tan[x]^4]/Sqrt[a]] + 2*b*Sqrt[a + b*Tan[x]^4] - b*Tan[x]^2*Sqrt[a + b*T 
an[x]^4]))/(4*Sqrt[1 + (b*Tan[x]^4)/a])
 

Rubi [A] (warning: unable to verify)

Time = 0.52 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.59, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, 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.

\(\displaystyle \int \cot ^3(x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \tan (x)^4\right )^{3/2}}{\tan (x)^3}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \int \frac {\cot ^3(x) \left (a+b \tan ^4(x)\right )^{3/2}}{\tan ^2(x)+1}d\tan (x)\)

\(\Big \downarrow \) 1579

\(\displaystyle \frac {1}{2} \int \frac {\cot ^2(x) \left (b \tan ^4(x)+a\right )^{3/2}}{\tan ^2(x)+1}d\tan ^2(x)\)

\(\Big \downarrow \) 617

\(\displaystyle \frac {1}{2} \int \left (\left (b \tan ^4(x)+a\right )^{3/2} \cot ^2(x)-\left (b \tan ^4(x)+a\right )^{3/2} \cot (x)+\frac {\left (b \tan ^4(x)+a\right )^{3/2}}{\tan ^2(x)+1}\right )d\tan ^2(x)\)

\(\Big \downarrow \) 2009

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

Input:

Int[Cot[x]^3*(a + b*Tan[x]^4)^(3/2),x]
 

Output:

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

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 20.40 (sec) , antiderivative size = 1404, normalized size of antiderivative = 10.03 \[ \int \cot ^3(x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate(cot(x)^3*(a+b*tan(x)^4)^(3/2),x, algorithm="fricas")
 

Output:

[1/4*(b^(3/2)*log(2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + 
 a)*tan(x)^2 + (a + b)^(3/2)*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))*tan(x)^2 + a^(3/2)*log((b*tan(x)^4 + 2*sqrt(b*tan( 
x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4)*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*ta 
n(x)^2 - a))/tan(x)^2, 1/4*(2*sqrt(-b)*b*arctan(sqrt(-b)*tan(x)^2/sqrt(b*t 
an(x)^4 + a))*tan(x)^2 + (a + b)^(3/2)*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))*tan(x)^2 + a^(3/2)*log((b*tan(x)^4 + 2*s 
qrt(b*tan(x)^4 + a)*sqrt(a) + 2*a)/tan(x)^4)*tan(x)^2 + 2*sqrt(b*tan(x)^4 
+ a)*(b*tan(x)^2 - a))/tan(x)^2, -1/4*(2*sqrt(-a)*a*arctan(sqrt(-a)/sqrt(b 
*tan(x)^4 + a))*tan(x)^2 - b^(3/2)*log(2*b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + 
a)*sqrt(b)*tan(x)^2 + a)*tan(x)^2 - (a + b)^(3/2)*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))*tan(x)^2 - 2*sqrt(b*tan(x)^4 
+ a)*(b*tan(x)^2 - a))/tan(x)^2, 1/4*(2*sqrt(-b)*b*arctan(sqrt(-b)*tan(x)^ 
2/sqrt(b*tan(x)^4 + a))*tan(x)^2 - 2*sqrt(-a)*a*arctan(sqrt(-a)/sqrt(b*tan 
(x)^4 + a))*tan(x)^2 + (a + b)^(3/2)*log(((a*b + 2*b^2)*tan(x)^4 - 2*a*b*t 
an(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))*tan(x)^2 + 2*sqrt(b*tan(x)^4 + a)*(b*ta...
 

Sympy [F]

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

integrate(cot(x)**3*(a+b*tan(x)**4)**(3/2),x)
 

Output:

Integral((a + b*tan(x)**4)**(3/2)*cot(x)**3, x)
 

Maxima [F]

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

integrate(cot(x)^3*(a+b*tan(x)^4)^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*tan(x)^4 + a)^(3/2)*cot(x)^3, x)
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.24 \[ \int \cot ^3(x) \left (a+b \tan ^4(x)\right )^{3/2} \, dx=-\frac {a^{2} \arctan \left (-\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {1}{2} \, b^{\frac {3}{2}} \log \left ({\left | -\sqrt {b} \tan \left (x\right )^{2} + \sqrt {b \tan \left (x\right )^{4} + a} \right |}\right ) + \frac {a^{2} \sqrt {b}}{{\left (\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a}\right )}^{2} - a} + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a} b + \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \arctan \left (-\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{\sqrt {-a - b}} \] Input:

integrate(cot(x)^3*(a+b*tan(x)^4)^(3/2),x, algorithm="giac")
 

Output:

-a^2*arctan(-(sqrt(b)*tan(x)^2 - sqrt(b*tan(x)^4 + a))/sqrt(-a))/sqrt(-a) 
+ 1/2*b^(3/2)*log(abs(-sqrt(b)*tan(x)^2 + sqrt(b*tan(x)^4 + a))) + a^2*sqr 
t(b)/((sqrt(b)*tan(x)^2 - sqrt(b*tan(x)^4 + a))^2 - a) + 1/2*sqrt(b*tan(x) 
^4 + a)*b + (a^2 + 2*a*b + b^2)*arctan(-(sqrt(b)*tan(x)^2 - sqrt(b*tan(x)^ 
4 + a) + sqrt(b))/sqrt(-a - b))/sqrt(-a - b)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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