3.22 \(\int \frac{\csc ^4(x)}{a+b \cos ^2(x)} \, dx\)

Optimal. Leaf size=61 \[ -\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{5/2}}-\frac{\cot ^3(x)}{3 (a+b)}-\frac{(a+2 b) \cot (x)}{(a+b)^2} \]

[Out]

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

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Rubi [A]  time = 0.0802187, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3191, 390, 205} \[ -\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{5/2}}-\frac{\cot ^3(x)}{3 (a+b)}-\frac{(a+2 b) \cot (x)}{(a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4/(a + b*Cos[x]^2),x]

[Out]

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

Rule 3191

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

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 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{\csc ^4(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a+2 b}{(a+b)^2}+\frac{x^2}{a+b}+\frac{b^2}{(a+b)^2 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=-\frac{(a+2 b) \cot (x)}{(a+b)^2}-\frac{\cot ^3(x)}{3 (a+b)}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{(a+b)^2}\\ &=-\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{5/2}}-\frac{(a+2 b) \cot (x)}{(a+b)^2}-\frac{\cot ^3(x)}{3 (a+b)}\\ \end{align*}

Mathematica [A]  time = 0.200132, size = 59, normalized size = 0.97 \[ \frac{b^2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{5/2}}-\frac{\cot (x) \left ((a+b) \csc ^2(x)+2 a+5 b\right )}{3 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4/(a + b*Cos[x]^2),x]

[Out]

(b^2*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) - (Cot[x]*(2*a + 5*b + (a + b)*Csc[x]^2))/(
3*(a + b)^2)

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Maple [A]  time = 0.038, size = 65, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}}{ \left ( a+b \right ) ^{2}}\arctan \left ({a\tan \left ( x \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( a+b \right ) a}}}}-{\frac{1}{ \left ( 3\,a+3\,b \right ) \left ( \tan \left ( x \right ) \right ) ^{3}}}-{\frac{a}{ \left ( a+b \right ) ^{2}\tan \left ( x \right ) }}-2\,{\frac{b}{ \left ( a+b \right ) ^{2}\tan \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2/(a+b)^2/((a+b)*a)^(1/2)*arctan(a*tan(x)/((a+b)*a)^(1/2))-1/3/(a+b)/tan(x)^3-1/(a+b)^2/tan(x)*a-2/(a+b)^2/t
an(x)*b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.97951, size = 936, normalized size = 15.34 \begin{align*} \left [\frac{4 \,{\left (2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{3} + 3 \,{\left (b^{2} \cos \left (x\right )^{2} - b^{2}\right )} \sqrt{-a^{2} - a b} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \,{\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (x\right )^{2} + 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (x\right )^{3} - a \cos \left (x\right )\right )} \sqrt{-a^{2} - a b} \sin \left (x\right ) + a^{2}}{b^{2} \cos \left (x\right )^{4} + 2 \, a b \cos \left (x\right )^{2} + a^{2}}\right ) \sin \left (x\right ) - 12 \,{\left (a^{3} + 3 \, a^{2} b + 2 \, a b^{2}\right )} \cos \left (x\right )}{12 \,{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} -{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}, \frac{2 \,{\left (2 \, a^{3} + 7 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (x\right )^{3} + 3 \,{\left (b^{2} \cos \left (x\right )^{2} - b^{2}\right )} \sqrt{a^{2} + a b} \arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (x\right )^{2} - a}{2 \, \sqrt{a^{2} + a b} \cos \left (x\right ) \sin \left (x\right )}\right ) \sin \left (x\right ) - 6 \,{\left (a^{3} + 3 \, a^{2} b + 2 \, a b^{2}\right )} \cos \left (x\right )}{6 \,{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} -{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(4*(2*a^3 + 7*a^2*b + 5*a*b^2)*cos(x)^3 + 3*(b^2*cos(x)^2 - b^2)*sqrt(-a^2 - a*b)*log(((8*a^2 + 8*a*b +
b^2)*cos(x)^4 - 2*(4*a^2 + 3*a*b)*cos(x)^2 + 4*((2*a + b)*cos(x)^3 - a*cos(x))*sqrt(-a^2 - a*b)*sin(x) + a^2)/
(b^2*cos(x)^4 + 2*a*b*cos(x)^2 + a^2))*sin(x) - 12*(a^3 + 3*a^2*b + 2*a*b^2)*cos(x))/((a^4 + 3*a^3*b + 3*a^2*b
^2 + a*b^3 - (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(x)^2)*sin(x)), 1/6*(2*(2*a^3 + 7*a^2*b + 5*a*b^2)*cos(x)^
3 + 3*(b^2*cos(x)^2 - b^2)*sqrt(a^2 + a*b)*arctan(1/2*((2*a + b)*cos(x)^2 - a)/(sqrt(a^2 + a*b)*cos(x)*sin(x))
)*sin(x) - 6*(a^3 + 3*a^2*b + 2*a*b^2)*cos(x))/((a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 - (a^4 + 3*a^3*b + 3*a^2*b^
2 + a*b^3)*cos(x)^2)*sin(x))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc ^{4}{\left (x \right )}}{a + b \cos ^{2}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**4/(a + b*cos(x)**2), x)

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Giac [A]  time = 1.21711, size = 122, normalized size = 2. \begin{align*} \frac{{\left (\pi \left \lfloor \frac{x}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (x\right )}{\sqrt{a^{2} + a b}}\right )\right )} b^{2}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a^{2} + a b}} - \frac{3 \, a \tan \left (x\right )^{2} + 6 \, b \tan \left (x\right )^{2} + a + b}{3 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \tan \left (x\right )^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(pi*floor(x/pi + 1/2)*sgn(a) + arctan(a*tan(x)/sqrt(a^2 + a*b)))*b^2/((a^2 + 2*a*b + b^2)*sqrt(a^2 + a*b)) - 1
/3*(3*a*tan(x)^2 + 6*b*tan(x)^2 + a + b)/((a^2 + 2*a*b + b^2)*tan(x)^3)