Integrand size = 13, antiderivative size = 34 \[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {1+\tan ^4(x)}}\right )}{2 \sqrt {2}} \] Output:
-1/4*arctanh(1/2*(1-tan(x)^2)*2^(1/2)/(1+tan(x)^4)^(1/2))*2^(1/2)
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.62 \[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=-\frac {\sqrt {3+\cos (4 x)} \log \left (\sqrt {2} \cos (2 x)+\sqrt {3+\cos (4 x)}\right ) \sec ^2(x)}{4 \sqrt {2} \sqrt {1+\tan ^4(x)}} \] Input:
Integrate[Tan[x]/Sqrt[1 + Tan[x]^4],x]
Output:
-1/4*(Sqrt[3 + Cos[4*x]]*Log[Sqrt[2]*Cos[2*x] + Sqrt[3 + Cos[4*x]]]*Sec[x] ^2)/(Sqrt[2]*Sqrt[1 + Tan[x]^4])
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 4153, 1577, 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.
\(\displaystyle \int \frac {\tan (x)}{\sqrt {\tan ^4(x)+1}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan (x)}{\sqrt {\tan (x)^4+1}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int \frac {\tan (x)}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^4(x)+1}}d\tan (x)\) |
\(\Big \downarrow \) 1577 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^4(x)+1}}d\tan ^2(x)\) |
\(\Big \downarrow \) 488 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{2-\tan ^4(x)}d\frac {1-\tan ^2(x)}{\sqrt {\tan ^4(x)+1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{2 \sqrt {2}}\) |
Input:
Int[Tan[x]/Sqrt[1 + Tan[x]^4],x]
Output:
-1/2*ArcTanh[(1 - Tan[x]^2)/(Sqrt[2]*Sqrt[1 + Tan[x]^4])]/Sqrt[2]
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])
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]
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]
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]))
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-2 \tan \left (x \right )^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (1+\tan \left (x \right )^{2}\right )^{2}-2 \tan \left (x \right )^{2}}}\right )}{4}\) | \(37\) |
default | \(-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-2 \tan \left (x \right )^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (1+\tan \left (x \right )^{2}\right )^{2}-2 \tan \left (x \right )^{2}}}\right )}{4}\) | \(37\) |
Input:
int(tan(x)/(1+tan(x)^4)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*2^(1/2)*arctanh(1/4*(-2*tan(x)^2+2)*2^(1/2)/((1+tan(x)^2)^2-2*tan(x)^ 2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.47 \[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=\frac {1}{32} \, \sqrt {2} \log \left (\frac {577 \, \tan \left (x\right )^{16} - 1912 \, \tan \left (x\right )^{14} + 4124 \, \tan \left (x\right )^{12} - 6216 \, \tan \left (x\right )^{10} + 7110 \, \tan \left (x\right )^{8} - 6216 \, \tan \left (x\right )^{6} + 4124 \, \tan \left (x\right )^{4} - 1912 \, \tan \left (x\right )^{2} + 8 \, {\left (51 \, \sqrt {2} \tan \left (x\right )^{14} - 169 \, \sqrt {2} \tan \left (x\right )^{12} + 339 \, \sqrt {2} \tan \left (x\right )^{10} - 465 \, \sqrt {2} \tan \left (x\right )^{8} + 465 \, \sqrt {2} \tan \left (x\right )^{6} - 339 \, \sqrt {2} \tan \left (x\right )^{4} + 169 \, \sqrt {2} \tan \left (x\right )^{2} - 51 \, \sqrt {2}\right )} \sqrt {\tan \left (x\right )^{4} + 1} + 577}{\tan \left (x\right )^{16} + 8 \, \tan \left (x\right )^{14} + 28 \, \tan \left (x\right )^{12} + 56 \, \tan \left (x\right )^{10} + 70 \, \tan \left (x\right )^{8} + 56 \, \tan \left (x\right )^{6} + 28 \, \tan \left (x\right )^{4} + 8 \, \tan \left (x\right )^{2} + 1}\right ) \] Input:
integrate(tan(x)/(1+tan(x)^4)^(1/2),x, algorithm="fricas")
Output:
1/32*sqrt(2)*log((577*tan(x)^16 - 1912*tan(x)^14 + 4124*tan(x)^12 - 6216*t an(x)^10 + 7110*tan(x)^8 - 6216*tan(x)^6 + 4124*tan(x)^4 - 1912*tan(x)^2 + 8*(51*sqrt(2)*tan(x)^14 - 169*sqrt(2)*tan(x)^12 + 339*sqrt(2)*tan(x)^10 - 465*sqrt(2)*tan(x)^8 + 465*sqrt(2)*tan(x)^6 - 339*sqrt(2)*tan(x)^4 + 169* sqrt(2)*tan(x)^2 - 51*sqrt(2))*sqrt(tan(x)^4 + 1) + 577)/(tan(x)^16 + 8*ta n(x)^14 + 28*tan(x)^12 + 56*tan(x)^10 + 70*tan(x)^8 + 56*tan(x)^6 + 28*tan (x)^4 + 8*tan(x)^2 + 1))
\[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=\int \frac {\tan {\left (x \right )}}{\sqrt {\tan ^{4}{\left (x \right )} + 1}}\, dx \] Input:
integrate(tan(x)/(1+tan(x)**4)**(1/2),x)
Output:
Integral(tan(x)/sqrt(tan(x)**4 + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 565, normalized size of antiderivative = 16.62 \[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=\text {Too large to display} \] Input:
integrate(tan(x)/(1+tan(x)^4)^(1/2),x, algorithm="maxima")
Output:
-1/16*sqrt(2)*(log(4*sqrt(2*(6*cos(4*x) + 1)*cos(8*x) + cos(8*x)^2 + 36*co s(4*x)^2 + sin(8*x)^2 + 12*sin(8*x)*sin(4*x) + 36*sin(4*x)^2 + 12*cos(4*x) + 1)*cos(1/2*arctan2(sin(8*x) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1))^2 + 4*sqrt(2*(6*cos(4*x) + 1)*cos(8*x) + cos(8*x)^2 + 36*cos(4*x)^2 + sin(8 *x)^2 + 12*sin(8*x)*sin(4*x) + 36*sin(4*x)^2 + 12*cos(4*x) + 1)*sin(1/2*ar ctan2(sin(8*x) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1))^2 + 32*(2*(6*cos( 4*x) + 1)*cos(8*x) + cos(8*x)^2 + 36*cos(4*x)^2 + sin(8*x)^2 + 12*sin(8*x) *sin(4*x) + 36*sin(4*x)^2 + 12*cos(4*x) + 1)^(1/4)*cos(1/2*arctan2(sin(8*x ) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1)) + 64) + log(4*cos(4*x)^2 + 4*s in(4*x)^2 + 4*sqrt(2*(6*cos(4*x) + 1)*cos(8*x) + cos(8*x)^2 + 36*cos(4*x)^ 2 + sin(8*x)^2 + 12*sin(8*x)*sin(4*x) + 36*sin(4*x)^2 + 12*cos(4*x) + 1)*( cos(1/2*arctan2(sin(8*x) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1))^2 + sin (1/2*arctan2(sin(8*x) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1))^2) + 8*(2* (6*cos(4*x) + 1)*cos(8*x) + cos(8*x)^2 + 36*cos(4*x)^2 + sin(8*x)^2 + 12*s in(8*x)*sin(4*x) + 36*sin(4*x)^2 + 12*cos(4*x) + 1)^(1/4)*((cos(4*x) + 3)* cos(1/2*arctan2(sin(8*x) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1)) + sin(4 *x)*sin(1/2*arctan2(sin(8*x) + 6*sin(4*x), cos(8*x) + 6*cos(4*x) + 1))) + 24*cos(4*x) + 36))
Time = 0.13 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.47 \[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (-\frac {\tan \left (x\right )^{2} + \sqrt {2} - \sqrt {\tan \left (x\right )^{4} + 1} + 1}{\tan \left (x\right )^{2} - \sqrt {2} - \sqrt {\tan \left (x\right )^{4} + 1} + 1}\right ) \] Input:
integrate(tan(x)/(1+tan(x)^4)^(1/2),x, algorithm="giac")
Output:
1/4*sqrt(2)*log(-(tan(x)^2 + sqrt(2) - sqrt(tan(x)^4 + 1) + 1)/(tan(x)^2 - sqrt(2) - sqrt(tan(x)^4 + 1) + 1))
Timed out. \[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=\int \frac {\mathrm {tan}\left (x\right )}{\sqrt {{\mathrm {tan}\left (x\right )}^4+1}} \,d x \] Input:
int(tan(x)/(tan(x)^4 + 1)^(1/2),x)
Output:
int(tan(x)/(tan(x)^4 + 1)^(1/2), x)
\[ \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx=\int \frac {\sqrt {\tan \left (x \right )^{4}+1}\, \tan \left (x \right )}{\tan \left (x \right )^{4}+1}d x \] Input:
int(tan(x)/(1+tan(x)^4)^(1/2),x)
Output:
int((sqrt(tan(x)**4 + 1)*tan(x))/(tan(x)**4 + 1),x)