\(\int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx\) [390]

Optimal result

Integrand size = 15, antiderivative size = 90 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=-\frac {1}{2} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\frac {1}{2} \sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\frac {1}{2} \sqrt {a+b \tan ^4(x)} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {1}{2} \left (-\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )-\sqrt {a+b} \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )+\sqrt {a+b \tan ^4(x)}\right ) \] Input:

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4153, 1577, 493, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 1577

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

\(\Big \downarrow \) 493

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Defintions of rubi rules used

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 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 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 493

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

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 1577

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free 
Q[{a, c, d, e, p, q}, x]
 

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 [A] (verified)

Time = 0.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(\frac {\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tan \left (x \right )^{2}\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2}\)	\(139\)
default	\(\frac {\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tan \left (x \right )^{2}\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2}\)	\(139\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 475, normalized size of antiderivative = 5.28 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\left [\frac {1}{4} \, \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + \frac {1}{4} \, \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}, \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \frac {1}{4} \, \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}, -\frac {1}{2} \, \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + \frac {1}{4} \, \sqrt {b} \log \left (-2 \, b \tan \left (x\right )^{4} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {b} \tan \left (x\right )^{2} - a\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}, -\frac {1}{2} \, \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + \frac {1}{2} \, \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-b}}{b \tan \left (x\right )^{2}}\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a}\right ] \] Input:

integrate(tan(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*sqr 
t(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(b*tan(x)^4 + a), 1/2*sqrt(-b)*arctan(sqrt(b*ta 
n(x)^4 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/4*sqrt(a + b)*log(((a*b + 2*b^2)*ta 
n(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(b*tan(x)^4 + a) 
, -1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - 
 b)/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/4*sqrt(b)*log(-2*b*tan(x)^4 + 
2*sqrt(b*tan(x)^4 + a)*sqrt(b)*tan(x)^2 - a) + 1/2*sqrt(b*tan(x)^4 + a), - 
1/2*sqrt(-a - b)*arctan(sqrt(b*tan(x)^4 + a)*(b*tan(x)^2 - a)*sqrt(-a - b) 
/((a*b + b^2)*tan(x)^4 + a^2 + a*b)) + 1/2*sqrt(-b)*arctan(sqrt(b*tan(x)^4 
 + a)*sqrt(-b)/(b*tan(x)^2)) + 1/2*sqrt(b*tan(x)^4 + a)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99 \[ \int \tan (x) \sqrt {a+b \tan ^4(x)} \, dx=\frac {{\left (a + b\right )} \arctan \left (-\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{\sqrt {-a - b}} + \frac {1}{2} \, \sqrt {b} \log \left ({\left | -\sqrt {b} \tan \left (x\right )^{2} + \sqrt {b \tan \left (x\right )^{4} + a} \right |}\right ) + \frac {1}{2} \, \sqrt {b \tan \left (x\right )^{4} + a} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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