3.71 \(\int \frac{1}{a-b \cos ^4(x)} \, dx\)

Optimal. Leaf size=101 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \cot (x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \cot (x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}} \]

[Out]

-ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Cot[x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) - ArcTan[(Sqrt[Sqrt[a] +
Sqrt[b]]*Cot[x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]])

________________________________________________________________________________________

Rubi [A]  time = 0.115908, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3209, 1166, 205} \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \cot (x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \cot (x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Cos[x]^4)^(-1),x]

[Out]

-ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Cot[x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]) - ArcTan[(Sqrt[Sqrt[a] +
Sqrt[b]]*Cot[x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]])

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rule 1166

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{a-b \cos ^4(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\cot (x)\right )\\ &=-\left (\frac{1}{2} \left (1-\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\cot (x)\right )\right )-\frac{1}{2} \left (1+\frac{\sqrt{b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \cot (x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \cot (x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}}\\ \end{align*}

Mathematica [A]  time = 0.188843, size = 109, normalized size = 1.08 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*Cos[x]^4)^(-1),x]

[Out]

ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - ArcTanh[(Sqrt[a]*Ta
n[x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]])

________________________________________________________________________________________

Maple [A]  time = 0.035, size = 64, normalized size = 0.6 \begin{align*} -{\frac{1}{2}{\it Artanh} \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) a}}}}+{\frac{1}{2}\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*cos(x)^4),x)

[Out]

-1/2/(((a*b)^(1/2)-a)*a)^(1/2)*arctanh(a*tan(x)/(((a*b)^(1/2)-a)*a)^(1/2))+1/2/(((a*b)^(1/2)+a)*a)^(1/2)*arcta
n(a*tan(x)/(((a*b)^(1/2)+a)*a)^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{b \cos \left (x\right )^{4} - a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate(1/(b*cos(x)^4 - a), x)

________________________________________________________________________________________

Fricas [B]  time = 2.52538, size = 1774, normalized size = 17.56 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b))*log(b*cos(x)^2 + 2*(a*b*cos(x)*sin
(x) - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b
+ a^3*b^2)) + 1)/(a^2 - a*b)) + (a^3 - a^2*b - 2*(a^3 - a^2*b)*cos(x)^2)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))) +
1/8*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b))*log(b*cos(x)^2 - 2*(a*b*cos(x)*sin(
x) - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b +
a^3*b^2)) + 1)/(a^2 - a*b)) + (a^3 - a^2*b - 2*(a^3 - a^2*b)*cos(x)^2)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))) + 1
/8*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(-b*cos(x)^2 + 2*(a*b*cos(x)*sin(x
) + (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a
^3*b^2)) - 1)/(a^2 - a*b)) + (a^3 - a^2*b - 2*(a^3 - a^2*b)*cos(x)^2)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))) - 1/8
*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(-b*cos(x)^2 - 2*(a*b*cos(x)*sin(x)
+ (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3
*b^2)) - 1)/(a^2 - a*b)) + (a^3 - a^2*b - 2*(a^3 - a^2*b)*cos(x)^2)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*cos(x)**4),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError