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

Optimal. Leaf size=28 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]]/(2*Sqrt[a])

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Rubi [A]  time = 0.0709482, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3229, 266, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[x]/Sqrt[a + b*Cos[x]^4],x]

[Out]

ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]]/(2*Sqrt[a])

Rule 3229

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(a + b*ff^(n/2)*x^(n/2))^p
)/(1 - ff*x)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] &
& IntegerQ[n/2]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0162997, size = 28, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]]/(2*Sqrt[a])

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Maple [A]  time = 0.033, size = 31, normalized size = 1.1 \begin{align*}{\frac{1}{2}\ln \left ({\frac{1}{ \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{a+b \left ( \cos \left ( x \right ) \right ) ^{4}} \right ) } \right ){\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/a^(1/2)*ln((2*a+2*a^(1/2)*(a+b*cos(x)^4)^(1/2))/cos(x)^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (x\right )}{\sqrt{b \cos \left (x\right )^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.88549, size = 189, normalized size = 6.75 \begin{align*} \left [\frac{\log \left (\frac{b \cos \left (x\right )^{4} + 2 \, \sqrt{b \cos \left (x\right )^{4} + a} \sqrt{a} + 2 \, a}{\cos \left (x\right )^{4}}\right )}{4 \, \sqrt{a}}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{b \cos \left (x\right )^{4} + a} \sqrt{-a}}{a}\right )}{2 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*log((b*cos(x)^4 + 2*sqrt(b*cos(x)^4 + a)*sqrt(a) + 2*a)/cos(x)^4)/sqrt(a), -1/2*sqrt(-a)*arctan(sqrt(b*co
s(x)^4 + a)*sqrt(-a)/a)/a]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan{\left (x \right )}}{\sqrt{a + b \cos ^{4}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15093, size = 32, normalized size = 1.14 \begin{align*} -\frac{\arctan \left (\frac{\sqrt{b \cos \left (x\right )^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(sqrt(b*cos(x)^4 + a)/sqrt(-a))/sqrt(-a)