3.1.43 \(\int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx\) [43]

Optimal. Leaf size=34 \[ -\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {1+\tan ^4(x)}}\right )}{2 \sqrt {2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3751, 1262, 739, 212} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {\tan ^4(x)+1}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[x]/Sqrt[1 + Tan[x]^4],x]

[Out]

-1/2*ArcTanh[(1 - Tan[x]^2)/(Sqrt[2]*Sqrt[1 + Tan[x]^4])]/Sqrt[2]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 739

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

Rule 1262

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

Rule 3751

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]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^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] && RationalQ[n]))

Rubi steps

\begin {align*} \int \frac {\tan (x)}{\sqrt {1+\tan ^4(x)}} \, dx &=\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {1+x^2}} \, dx,x,\tan ^2(x)\right )\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {1-\tan ^2(x)}{\sqrt {1+\tan ^4(x)}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {1-\tan ^2(x)}{\sqrt {2} \sqrt {1+\tan ^4(x)}}\right )}{2 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 55, normalized size = 1.62 \begin {gather*} -\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)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[x]/Sqrt[1 + Tan[x]^4],x]

[Out]

-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])

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Maple [A]
time = 0.07, size = 37, normalized size = 1.09

method	result	size
derivativedivides	\(-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-2 \left (\tan ^{2}\left (x \right )\right )+2\right ) \sqrt {2}}{4 \sqrt {\left (1+\tan ^{2}\left (x \right )\right )^{2}-2 \left (\tan ^{2}\left (x \right )\right )}}\right )}{4}\)	\(37\)
default	\(-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (-2 \left (\tan ^{2}\left (x \right )\right )+2\right ) \sqrt {2}}{4 \sqrt {\left (1+\tan ^{2}\left (x \right )\right )^{2}-2 \left (\tan ^{2}\left (x \right )\right )}}\right )}{4}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (25) = 50\).
time = 3.71, size = 565, normalized size = 16.62 \begin {gather*} -\frac {1}{16} \, \sqrt {2} {\left (\log \left (4 \, \sqrt {2 \, {\left (6 \, \cos \left (4 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 36 \, \cos \left (4 \, x\right )^{2} + \sin \left (8 \, x\right )^{2} + 12 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) + 36 \, \sin \left (4 \, x\right )^{2} + 12 \, \cos \left (4 \, x\right ) + 1} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right )^{2} + 4 \, \sqrt {2 \, {\left (6 \, \cos \left (4 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 36 \, \cos \left (4 \, x\right )^{2} + \sin \left (8 \, x\right )^{2} + 12 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) + 36 \, \sin \left (4 \, x\right )^{2} + 12 \, \cos \left (4 \, x\right ) + 1} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right )^{2} + 32 \, {\left (2 \, {\left (6 \, \cos \left (4 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 36 \, \cos \left (4 \, x\right )^{2} + \sin \left (8 \, x\right )^{2} + 12 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) + 36 \, \sin \left (4 \, x\right )^{2} + 12 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right ) + 64\right ) + \log \left (4 \, \cos \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right )^{2} + 4 \, \sqrt {2 \, {\left (6 \, \cos \left (4 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 36 \, \cos \left (4 \, x\right )^{2} + \sin \left (8 \, x\right )^{2} + 12 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) + 36 \, \sin \left (4 \, x\right )^{2} + 12 \, \cos \left (4 \, x\right ) + 1} {\left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right )^{2}\right )} + 8 \, {\left (2 \, {\left (6 \, \cos \left (4 \, x\right ) + 1\right )} \cos \left (8 \, x\right ) + \cos \left (8 \, x\right )^{2} + 36 \, \cos \left (4 \, x\right )^{2} + \sin \left (8 \, x\right )^{2} + 12 \, \sin \left (8 \, x\right ) \sin \left (4 \, x\right ) + 36 \, \sin \left (4 \, x\right )^{2} + 12 \, \cos \left (4 \, x\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left (\cos \left (4 \, x\right ) + 3\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right ) + \sin \left (4 \, x\right ) \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (8 \, x\right ) + 6 \, \sin \left (4 \, x\right ), \cos \left (8 \, x\right ) + 6 \, \cos \left (4 \, x\right ) + 1\right )\right )\right )} + 24 \, \cos \left (4 \, x\right ) + 36\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*sqrt(2)*(log(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 + 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*arctan2(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*c
os(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*sin(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)*s
in(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*sin(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))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (25) = 50\).
time = 0.43, size = 186, normalized size = 5.47 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*sqrt(2)*log((577*tan(x)^16 - 1912*tan(x)^14 + 4124*tan(x)^12 - 6216*tan(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))*sqr
t(tan(x)^4 + 1) + 577)/(tan(x)^16 + 8*tan(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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan {\left (x \right )}}{\sqrt {\tan ^{4}{\left (x \right )} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tan(x)/sqrt(tan(x)**4 + 1), x)

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Giac [A]
time = 0.51, size = 50, normalized size = 1.47 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {tan}\left (x\right )}{\sqrt {{\mathrm {tan}\left (x\right )}^4+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(x)/(tan(x)^4 + 1)^(1/2),x)

[Out]

int(tan(x)/(tan(x)^4 + 1)^(1/2), x)

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