3.29 \(\int \frac {\cos ^5(x)}{a+b \cos ^2(x)} \, dx\)

Optimal. Leaf size=56 \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b} \]

[Out]

-(a-b)*sin(x)/b^2-1/3*sin(x)^3/b+a^2*arctanh(sin(x)*b^(1/2)/(a+b)^(1/2))/b^(5/2)/(a+b)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3186, 390, 208} \[ \frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^5/(a + b*Cos[x]^2),x]

[Out]

(a^2*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]) - ((a - b)*Sin[x])/b^2 - Sin[x]^3/(3*b)

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]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3186

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

Rubi steps

\begin {align*} \int \frac {\cos ^5(x)}{a+b \cos ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a+b-b x^2} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-\frac {a-b}{b^2}-\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b-b x^2\right )}\right ) \, dx,x,\sin (x)\right )\\ &=-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\sin (x)\right )}{b^2}\\ &=\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sin (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sin (x)}{b^2}-\frac {\sin ^3(x)}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 86, normalized size = 1.54 \[ \frac {\frac {6 a^2 \left (\log \left (\sqrt {a+b}+\sqrt {b} \sin (x)\right )-\log \left (\sqrt {a+b}-\sqrt {b} \sin (x)\right )\right )}{\sqrt {a+b}}+3 \sqrt {b} (3 b-4 a) \sin (x)+b^{3/2} \sin (3 x)}{12 b^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^5/(a + b*Cos[x]^2),x]

[Out]

((6*a^2*(-Log[Sqrt[a + b] - Sqrt[b]*Sin[x]] + Log[Sqrt[a + b] + Sqrt[b]*Sin[x]]))/Sqrt[a + b] + 3*Sqrt[b]*(-4*
a + 3*b)*Sin[x] + b^(3/2)*Sin[3*x])/(12*b^(5/2))

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fricas [A]  time = 0.65, size = 191, normalized size = 3.41 \[ \left [\frac {3 \, \sqrt {a b + b^{2}} a^{2} \log \left (-\frac {b \cos \relax (x)^{2} - 2 \, \sqrt {a b + b^{2}} \sin \relax (x) - a - 2 \, b}{b \cos \relax (x)^{2} + a}\right ) - 2 \, {\left (3 \, a^{2} b + a b^{2} - 2 \, b^{3} - {\left (a b^{2} + b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{6 \, {\left (a b^{3} + b^{4}\right )}}, -\frac {3 \, \sqrt {-a b - b^{2}} a^{2} \arctan \left (\frac {\sqrt {-a b - b^{2}} \sin \relax (x)}{a + b}\right ) + {\left (3 \, a^{2} b + a b^{2} - 2 \, b^{3} - {\left (a b^{2} + b^{3}\right )} \cos \relax (x)^{2}\right )} \sin \relax (x)}{3 \, {\left (a b^{3} + b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5/(a+b*cos(x)^2),x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(a*b + b^2)*a^2*log(-(b*cos(x)^2 - 2*sqrt(a*b + b^2)*sin(x) - a - 2*b)/(b*cos(x)^2 + a)) - 2*(3*a^
2*b + a*b^2 - 2*b^3 - (a*b^2 + b^3)*cos(x)^2)*sin(x))/(a*b^3 + b^4), -1/3*(3*sqrt(-a*b - b^2)*a^2*arctan(sqrt(
-a*b - b^2)*sin(x)/(a + b)) + (3*a^2*b + a*b^2 - 2*b^3 - (a*b^2 + b^3)*cos(x)^2)*sin(x))/(a*b^3 + b^4)]

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giac [A]  time = 0.16, size = 65, normalized size = 1.16 \[ -\frac {a^{2} \arctan \left (\frac {b \sin \relax (x)}{\sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} b^{2}} - \frac {b^{2} \sin \relax (x)^{3} + 3 \, a b \sin \relax (x) - 3 \, b^{2} \sin \relax (x)}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5/(a+b*cos(x)^2),x, algorithm="giac")

[Out]

-a^2*arctan(b*sin(x)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*b^2) - 1/3*(b^2*sin(x)^3 + 3*a*b*sin(x) - 3*b^2*sin(x
))/b^3

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maple [A]  time = 0.06, size = 50, normalized size = 0.89 \[ -\frac {\frac {\left (\sin ^{3}\relax (x )\right ) b}{3}+\sin \relax (x ) a -\sin \relax (x ) b}{b^{2}}+\frac {a^{2} \arctanh \left (\frac {\sin \relax (x ) b}{\sqrt {\left (a +b \right ) b}}\right )}{b^{2} \sqrt {\left (a +b \right ) b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5/(a+b*cos(x)^2),x)

[Out]

-1/b^2*(1/3*sin(x)^3*b+sin(x)*a-sin(x)*b)+1/b^2*a^2/((a+b)*b)^(1/2)*arctanh(sin(x)*b/((a+b)*b)^(1/2))

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maxima [A]  time = 0.69, size = 67, normalized size = 1.20 \[ -\frac {a^{2} \log \left (\frac {b \sin \relax (x) - \sqrt {{\left (a + b\right )} b}}{b \sin \relax (x) + \sqrt {{\left (a + b\right )} b}}\right )}{2 \, \sqrt {{\left (a + b\right )} b} b^{2}} - \frac {b \sin \relax (x)^{3} + 3 \, {\left (a - b\right )} \sin \relax (x)}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^5/(a+b*cos(x)^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.31, size = 51, normalized size = 0.91 \[ \frac {a^2\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sin \relax (x)}{\sqrt {a+b}}\right )}{b^{5/2}\,\sqrt {a+b}}-\frac {{\sin \relax (x)}^3}{3\,b}-\sin \relax (x)\,\left (\frac {a+b}{b^2}-\frac {2}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^5/(a + b*cos(x)^2),x)

[Out]

(a^2*atanh((b^(1/2)*sin(x))/(a + b)^(1/2)))/(b^(5/2)*(a + b)^(1/2)) - sin(x)^3/(3*b) - sin(x)*((a + b)/b^2 - 2
/b)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**5/(a+b*cos(x)**2),x)

[Out]

Timed out

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