Integrand size = 13, antiderivative size = 56 \[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=-\frac {1}{2} \text {arcsinh}\left (\tan ^2(x)\right )-\frac {\text {arctanh}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {1+\tan ^4(x)}}\right )}{\sqrt {2}}+\frac {1}{2} \sqrt {1+\tan ^4(x)} \] Output:
-1/2*arcsinh(tan(x)^2)-1/2*arctanh(1/2*(1-tan(x)^2)*2^(1/2)/(1+tan(x)^4)^( 1/2))*2^(1/2)+1/2*(1+tan(x)^4)^(1/2)
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.32 \[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\frac {\left (-2 \sqrt {2} \text {arcsinh}(\cos (2 x)) \cos ^2(x)-2 \text {arctanh}\left (\frac {2 \sin ^2(x)}{\sqrt {3+\cos (4 x)}}\right ) \cos ^2(x)+\sqrt {3+\cos (4 x)}\right ) \sqrt {1+\tan ^4(x)}}{2 \sqrt {3+\cos (4 x)}} \] Input:
Integrate[Tan[x]*Sqrt[1 + Tan[x]^4],x]
Output:
((-2*Sqrt[2]*ArcSinh[Cos[2*x]]*Cos[x]^2 - 2*ArcTanh[(2*Sin[x]^2)/Sqrt[3 + Cos[4*x]]]*Cos[x]^2 + Sqrt[3 + Cos[4*x]])*Sqrt[1 + Tan[x]^4])/(2*Sqrt[3 + Cos[4*x]])
Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 4153, 1577, 493, 719, 222, 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.
\(\displaystyle \int \tan (x) \sqrt {\tan ^4(x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (x) \sqrt {\tan (x)^4+1}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int \frac {\tan (x) \sqrt {\tan ^4(x)+1}}{\tan ^2(x)+1}d\tan (x)\) |
\(\Big \downarrow \) 1577 |
\(\displaystyle \frac {1}{2} \int \frac {\sqrt {\tan ^4(x)+1}}{\tan ^2(x)+1}d\tan ^2(x)\) |
\(\Big \downarrow \) 493 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1-\tan ^2(x)}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^4(x)+1}}d\tan ^2(x)+\sqrt {\tan ^4(x)+1}\right )\) |
\(\Big \downarrow \) 719 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {1}{\sqrt {\tan ^4(x)+1}}d\tan ^2(x)+2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^4(x)+1}}d\tan ^2(x)+\sqrt {\tan ^4(x)+1}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^4(x)+1}}d\tan ^2(x)-\text {arcsinh}\left (\tan ^2(x)\right )+\sqrt {\tan ^4(x)+1}\right )\) |
\(\Big \downarrow \) 488 |
\(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{2-\tan ^4(x)}d\frac {1-\tan ^2(x)}{\sqrt {\tan ^4(x)+1}}-\text {arcsinh}\left (\tan ^2(x)\right )+\sqrt {\tan ^4(x)+1}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\text {arcsinh}\left (\tan ^2(x)\right )-\sqrt {2} \text {arctanh}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )+\sqrt {\tan ^4(x)+1}\right )\) |
Input:
Int[Tan[x]*Sqrt[1 + Tan[x]^4],x]
Output:
(-ArcSinh[Tan[x]^2] - Sqrt[2]*ArcTanh[(1 - Tan[x]^2)/(Sqrt[2]*Sqrt[1 + Tan [x]^4])] + Sqrt[1 + Tan[x]^4])/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/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
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[((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]
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]
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 = 0.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {\sqrt {\left (1+\tan \left (x \right )^{2}\right )^{2}-2 \tan \left (x \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (\tan \left (x \right )^{2}\right )}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-2 \tan \left (x \right )^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (1+\tan \left (x \right )^{2}\right )^{2}-2 \tan \left (x \right )^{2}}}\right )}{2}\) | \(64\) |
default | \(\frac {\sqrt {\left (1+\tan \left (x \right )^{2}\right )^{2}-2 \tan \left (x \right )^{2}}}{2}-\frac {\operatorname {arcsinh}\left (\tan \left (x \right )^{2}\right )}{2}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (-2 \tan \left (x \right )^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (1+\tan \left (x \right )^{2}\right )^{2}-2 \tan \left (x \right )^{2}}}\right )}{2}\) | \(64\) |
Input:
int((1+tan(x)^4)^(1/2)*tan(x),x,method=_RETURNVERBOSE)
Output:
1/2*((1+tan(x)^2)^2-2*tan(x)^2)^(1/2)-1/2*arcsinh(tan(x)^2)-1/2*2^(1/2)*ar ctanh(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. 88 vs. \(2 (43) = 86\).
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.57 \[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {3 \, \tan \left (x\right )^{4} - 2 \, \tan \left (x\right )^{2} + 2 \, \sqrt {\tan \left (x\right )^{4} + 1} {\left (\sqrt {2} \tan \left (x\right )^{2} - \sqrt {2}\right )} + 3}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {\tan \left (x\right )^{4} + 1} + \frac {1}{2} \, \log \left (-\tan \left (x\right )^{2} + \sqrt {\tan \left (x\right )^{4} + 1}\right ) \] Input:
integrate((1+tan(x)^4)^(1/2)*tan(x),x, algorithm="fricas")
Output:
1/4*sqrt(2)*log((3*tan(x)^4 - 2*tan(x)^2 + 2*sqrt(tan(x)^4 + 1)*(sqrt(2)*t an(x)^2 - sqrt(2)) + 3)/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/2*sqrt(tan(x)^4 + 1) + 1/2*log(-tan(x)^2 + sqrt(tan(x)^4 + 1))
\[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\int \sqrt {\tan ^{4}{\left (x \right )} + 1} \tan {\left (x \right )}\, dx \] Input:
integrate((1+tan(x)**4)**(1/2)*tan(x),x)
Output:
Integral(sqrt(tan(x)**4 + 1)*tan(x), x)
\[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\int { \sqrt {\tan \left (x\right )^{4} + 1} \tan \left (x\right ) \,d x } \] Input:
integrate((1+tan(x)^4)^(1/2)*tan(x),x, algorithm="maxima")
Output:
integrate(sqrt(tan(x)^4 + 1)*tan(x), x)
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.41 \[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (-\frac {\tan \left (x\right )^{2} + \sqrt {2} - \sqrt {\tan \left (x\right )^{4} + 1} + 1}{\tan \left (x\right )^{2} - \sqrt {2} - \sqrt {\tan \left (x\right )^{4} + 1} + 1}\right ) + \frac {1}{2} \, \sqrt {\tan \left (x\right )^{4} + 1} + \frac {1}{2} \, \log \left (-\tan \left (x\right )^{2} + \sqrt {\tan \left (x\right )^{4} + 1}\right ) \] Input:
integrate((1+tan(x)^4)^(1/2)*tan(x),x, algorithm="giac")
Output:
1/2*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)) + 1/2*sqrt(tan(x)^4 + 1) + 1/2*log(-ta n(x)^2 + sqrt(tan(x)^4 + 1))
Timed out. \[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\int \mathrm {tan}\left (x\right )\,\sqrt {{\mathrm {tan}\left (x\right )}^4+1} \,d x \] Input:
int(tan(x)*(tan(x)^4 + 1)^(1/2),x)
Output:
int(tan(x)*(tan(x)^4 + 1)^(1/2), x)
\[ \int \tan (x) \sqrt {1+\tan ^4(x)} \, dx=\int \sqrt {\tan \left (x \right )^{4}+1}\, \tan \left (x \right )d x \] Input:
int((1+tan(x)^4)^(1/2)*tan(x),x)
Output:
int(sqrt(tan(x)**4 + 1)*tan(x),x)