\(\int \frac {\tan ^2(x)}{\sqrt {a+b \tan ^4(x)}} \, dx\) [401]

Optimal result

Integrand size = 17, antiderivative size = 291 \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}+\frac {\sqrt [4]{a} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right ) \sqrt {\frac {a+b \tan ^4(x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right )^2}}}{2 \left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \sqrt {a+b \tan ^4(x)}}-\frac {\left (\sqrt {a}+\sqrt {b}\right ) \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right ) \sqrt {\frac {a+b \tan ^4(x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right )^2}}}{4 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt [4]{b} \sqrt {a+b \tan ^4(x)}} \] Output:

-1/2*arctan((a+b)^(1/2)*tan(x)/(a+b*tan(x)^4)^(1/2))/(a+b)^(1/2)+1/2*a^(1/ 
4)*InverseJacobiAM(2*arctan(b^(1/4)*tan(x)/a^(1/4)),1/2*2^(1/2))*(a^(1/2)+ 
b^(1/2)*tan(x)^2)*((a+b*tan(x)^4)/(a^(1/2)+b^(1/2)*tan(x)^2)^2)^(1/2)/(a^( 
1/2)-b^(1/2))/b^(1/4)/(a+b*tan(x)^4)^(1/2)-1/4*(a^(1/2)+b^(1/2))*EllipticP 
i(sin(2*arctan(b^(1/4)*tan(x)/a^(1/4))),-1/4*(a^(1/2)-b^(1/2))^2/a^(1/2)/b 
^(1/2),1/2*2^(1/2))*(a^(1/2)+b^(1/2)*tan(x)^2)*((a+b*tan(x)^4)/(a^(1/2)+b^ 
(1/2)*tan(x)^2)^2)^(1/2)/a^(1/4)/(a^(1/2)-b^(1/2))/b^(1/4)/(a+b*tan(x)^4)^ 
(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.42 \[ \int \frac {\tan ^2(x)}{\sqrt {a+b \tan ^4(x)}} \, dx=-\frac {i \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (x)\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {i \sqrt {a}}{\sqrt {b}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \tan (x)\right ),-1\right )\right ) \sqrt {1+\frac {b \tan ^4(x)}{a}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \sqrt {a+b \tan ^4(x)}} \] Input:

Integrate[Tan[x]^2/Sqrt[a + b*Tan[x]^4],x]
 

Output:

((-I)*(EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], -1] - Ellipt 
icPi[((-I)*Sqrt[a])/Sqrt[b], I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*Tan[x]], 
-1])*Sqrt[1 + (b*Tan[x]^4)/a])/(Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[a + b*Tan[x 
]^4])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3042, 4153, 1657, 27, 761, 2221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 1657

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

\(\displaystyle \frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right ) \sqrt {\frac {a+b \tan ^4(x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {a+b \tan ^4(x)}}-\frac {\int \frac {\sqrt {b} \tan ^2(x)+\sqrt {a}}{\left (\tan ^2(x)+1\right ) \sqrt {b \tan ^4(x)+a}}d\tan (x)}{\sqrt {a}-\sqrt {b}}\)

\(\Big \downarrow \) 2221

\(\displaystyle \frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right ) \sqrt {\frac {a+b \tan ^4(x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \tan (x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{b} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {a+b \tan ^4(x)}}-\frac {\frac {\left (\sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {a+b} \tan (x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}+\frac {\left (\sqrt {a}+\sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right ) \sqrt {\frac {a+b \tan ^4(x)}{\left (\sqrt {a}+\sqrt {b} \tan ^2(x)\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {a}-\sqrt {b}\right )^2}{4 \sqrt {a} \sqrt {b}},2 \arctan \left (\frac {\sqrt [4]{b} \tan (x)}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \sqrt {a+b \tan ^4(x)}}}{\sqrt {a}-\sqrt {b}}\)

Input:

Int[Tan[x]^2/Sqrt[a + b*Tan[x]^4],x]
 

Output:

(a^(1/4)*EllipticF[2*ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqr 
t[b]*Tan[x]^2)*Sqrt[(a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(2*( 
Sqrt[a] - Sqrt[b])*b^(1/4)*Sqrt[a + b*Tan[x]^4]) - (((Sqrt[a] - Sqrt[b])*A 
rcTan[(Sqrt[a + b]*Tan[x])/Sqrt[a + b*Tan[x]^4]])/(2*Sqrt[a + b]) + ((Sqrt 
[a] + Sqrt[b])*EllipticPi[-1/4*(Sqrt[a] - Sqrt[b])^2/(Sqrt[a]*Sqrt[b]), 2* 
ArcTan[(b^(1/4)*Tan[x])/a^(1/4)], 1/2]*(Sqrt[a] + Sqrt[b]*Tan[x]^2)*Sqrt[( 
a + b*Tan[x]^4)/(Sqrt[a] + Sqrt[b]*Tan[x]^2)^2])/(4*a^(1/4)*b^(1/4)*Sqrt[a 
 + b*Tan[x]^4]))/(Sqrt[a] - Sqrt[b])
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1657

Int[(x_)^2/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> 
With[{q = Rt[c/a, 2]}, Simp[(-a)*((e + d*q)/(c*d^2 - a*e^2))   Int[1/Sqrt[a 
 + c*x^4], x], x] + Simp[a*d*((e + d*q)/(c*d^2 - a*e^2))   Int[(1 + q*x^2)/ 
((d + e*x^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/ 
a] && NeQ[c*d^2 - a*e^2, 0]
 

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.95 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.62

method	result	size
derivativedivides	\(\frac {\sqrt {1-\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (x \right )^{4}}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (x \right )^{4}}}\)	\(179\)
default	\(\frac {\sqrt {1-\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (\tan \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (x \right )^{4}}}-\frac {\sqrt {1-\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, \tan \left (x \right )^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (\tan \left (x \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}}{\sqrt {b}}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {a +b \tan \left (x \right )^{4}}}\)	\(179\)

Input:

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

Output:

1/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tan(x)^2)^(1/2)*(1+I/a^(1 
/2)*b^(1/2)*tan(x)^2)^(1/2)/(a+b*tan(x)^4)^(1/2)*EllipticF(tan(x)*(I/a^(1/ 
2)*b^(1/2))^(1/2),I)-1/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*tan( 
x)^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*tan(x)^2)^(1/2)/(a+b*tan(x)^4)^(1/2)*Elli 
pticPi(tan(x)*(I/a^(1/2)*b^(1/2))^(1/2),I*a^(1/2)/b^(1/2),(-I/a^(1/2)*b^(1 
/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))
 

Fricas [F]

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

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

Output:

integral(tan(x)^2/sqrt(b*tan(x)^4 + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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