3.4.91 \(\int \cot (x) \sqrt {a+b \tan ^4(x)} \, dx\) [391]

3.4.91.1 Optimal result

Integrand size = 15, antiderivative size = 102 \[ \int \cot (x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right ) \]

output

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

3.4.91.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96 \[ \int \cot (x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {1}{2} \left (\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )+\sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )\right ) \]

input

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

output

(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + Sqrt[a + b]*Ar 
cTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - Sqrt[a]*ArcTa 
nh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2
 

3.4.91.3 Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.96, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4153, 1579, 606, 243, 73, 221, 719, 224, 219, 488, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 1579

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

\(\Big \downarrow \) 606

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

input

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

output

(Sqrt[b]*ArcTanh[(Sqrt[b]*Tan[x]^2)/Sqrt[a + b*Tan[x]^4]] + Sqrt[a + b]*Ar 
cTanh[(a - b*Tan[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tan[x]^4])] - Sqrt[a]*ArcTa 
nh[Sqrt[a + b*Tan[x]^4]/Sqrt[a]])/2
 

3.4.91.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 488

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

rule 606

Int[(((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] : 
> Simp[a/c   Int[(c + d*x)^(n + 1)*((a + b*x^2)^(p - 1)/x), x], x] - Simp[1 
/c   Int[(c + d*x)^n*(a*d - b*c*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d}, x] && GtQ[p, 0] && ILtQ[n, 0]
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

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 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]))
 

3.4.91.4 Maple [F]

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

input

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

output

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

3.4.91.5 Fricas [A] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 1021, normalized size of antiderivative = 10.01 \[ \int \cot (x) \sqrt {a+b \tan ^4(x)} \, dx=\text {Too large to display} \]

input

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

output

[1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 + 
a) + 1/4*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)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + a)*sqr 
t(a) + 2*a)/tan(x)^4), -1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/ 
(b*tan(x)^2)) + 1/4*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)/(t 
an(x)^4 + 2*tan(x)^2 + 1)) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x) 
^4 + a)*sqrt(a) + 2*a)/tan(x)^4), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a) 
*sqrt(-a)/a) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)*sqrt( 
b)*tan(x)^2 + a) + 1/4*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)), 1/2*sqrt(-a)*arctan(sqrt(b*tan(x)^4 + a)*sq 
rt(-a)/a) - 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-b)/(b*tan(x)^2) 
) + 1/4*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)), 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*sqrt(-a - b)/ 
(b*tan(x)^2 - a)) + 1/4*sqrt(b)*log(2*b*tan(x)^4 + 2*sqrt(b*tan(x)^4 + a)* 
sqrt(b)*tan(x)^2 + a) + 1/4*sqrt(a)*log((b*tan(x)^4 - 2*sqrt(b*tan(x)^4 + 
a)*sqrt(a) + 2*a)/tan(x)^4), 1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + ...
 

3.4.91.6 Sympy [F]

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

input

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

output

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

3.4.91.7 Maxima [F]

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

input

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

output

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

3.4.91.8 Giac [F(-2)]

Exception generated. \[ \int \cot (x) \sqrt {a+b \tan ^4(x)} \, dx=\text {Exception raised: TypeError} \]

input

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

3.4.91.9 Mupad [F(-1)]

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

input

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

output

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