3.23 \(\int \frac{\csc ^6(x)}{a+b \cos ^2(x)} \, dx\)

Optimal. Leaf size=89 \[ -\frac{\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\cot ^5(x)}{5 (a+b)}-\frac{(2 a+3 b) \cot ^3(x)}{3 (a+b)^2} \]

[Out]

-((b^3*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Cot[x])/(a + b)
^3 - ((2*a + 3*b)*Cot[x]^3)/(3*(a + b)^2) - Cot[x]^5/(5*(a + b))

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Rubi [A]  time = 0.101601, antiderivative size = 89, 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} \[ -\frac{\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\cot ^5(x)}{5 (a+b)}-\frac{(2 a+3 b) \cot ^3(x)}{3 (a+b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^6/(a + b*Cos[x]^2),x]

[Out]

-((b^3*ArcTan[(Sqrt[a + b]*Cot[x])/Sqrt[a]])/(Sqrt[a]*(a + b)^(7/2))) - ((a^2 + 3*a*b + 3*b^2)*Cot[x])/(a + b)
^3 - ((2*a + 3*b)*Cot[x]^3)/(3*(a + b)^2) - Cot[x]^5/(5*(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 ^6(x)}{a+b \cos ^2(x)} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{a^2+3 a b+3 b^2}{(a+b)^3}+\frac{(2 a+3 b) x^2}{(a+b)^2}+\frac{x^4}{a+b}+\frac{b^3}{(a+b)^3 \left (a+(a+b) x^2\right )}\right ) \, dx,x,\cot (x)\right )\\ &=-\frac{\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac{(2 a+3 b) \cot ^3(x)}{3 (a+b)^2}-\frac{\cot ^5(x)}{5 (a+b)}-\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\cot (x)\right )}{(a+b)^3}\\ &=-\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac{(2 a+3 b) \cot ^3(x)}{3 (a+b)^2}-\frac{\cot ^5(x)}{5 (a+b)}\\ \end{align*}

Mathematica [A]  time = 0.38139, size = 90, normalized size = 1.01 \[ \frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right )}{\sqrt{a} (a+b)^{7/2}}-\frac{\cot (x) \left (\left (4 a^2+13 a b+9 b^2\right ) \csc ^2(x)+8 a^2+3 (a+b)^2 \csc ^4(x)+26 a b+33 b^2\right )}{15 (a+b)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^6/(a + b*Cos[x]^2),x]

[Out]

(b^3*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2)) - (Cot[x]*(8*a^2 + 26*a*b + 33*b^2 + (4*a^2
+ 13*a*b + 9*b^2)*Csc[x]^2 + 3*(a + b)^2*Csc[x]^4))/(15*(a + b)^3)

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Maple [A]  time = 0.042, size = 106, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}}{ \left ( a+b \right ) ^{3}}\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 ( 5\,a+5\,b \right ) \left ( \tan \left ( x \right ) \right ) ^{5}}}-{\frac{2\,a}{3\, \left ( a+b \right ) ^{2} \left ( \tan \left ( x \right ) \right ) ^{3}}}-{\frac{b}{ \left ( a+b \right ) ^{2} \left ( \tan \left ( x \right ) \right ) ^{3}}}-{\frac{{a}^{2}}{ \left ( a+b \right ) ^{3}\tan \left ( x \right ) }}-3\,{\frac{ab}{ \left ( a+b \right ) ^{3}\tan \left ( x \right ) }}-3\,{\frac{{b}^{2}}{ \left ( a+b \right ) ^{3}\tan \left ( x \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^6/(a+b*cos(x)^2),x)

[Out]

b^3/(a+b)^3/((a+b)*a)^(1/2)*arctan(a*tan(x)/((a+b)*a)^(1/2))-1/5/(a+b)/tan(x)^5-2/3/(a+b)^2/tan(x)^3*a-1/(a+b)
^2/tan(x)^3*b-1/(a+b)^3/tan(x)*a^2-3/(a+b)^3/tan(x)*a*b-3/(a+b)^3/tan(x)*b^2

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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)^6/(a+b*cos(x)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.04315, size = 1446, normalized size = 16.25 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^6/(a+b*cos(x)^2),x, algorithm="fricas")

[Out]

[-1/60*(4*(8*a^4 + 34*a^3*b + 59*a^2*b^2 + 33*a*b^3)*cos(x)^5 - 20*(4*a^4 + 17*a^3*b + 28*a^2*b^2 + 15*a*b^3)*
cos(x)^3 + 15*(b^3*cos(x)^4 - 2*b^3*cos(x)^2 + b^3)*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) + 60*(a^4 + 4*a^3*b + 6*a^2*b^2 + 3*a*b^3)*cos(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*
a^2*b^3 + a*b^4 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^4 - 2*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*
a^2*b^3 + a*b^4)*cos(x)^2)*sin(x)), -1/30*(2*(8*a^4 + 34*a^3*b + 59*a^2*b^2 + 33*a*b^3)*cos(x)^5 - 10*(4*a^4 +
17*a^3*b + 28*a^2*b^2 + 15*a*b^3)*cos(x)^3 + 15*(b^3*cos(x)^4 - 2*b^3*cos(x)^2 + b^3)*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) + 30*(a^4 + 4*a^3*b + 6*a^2*b^2 + 3*a*b^3
)*cos(x))/((a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4 + (a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*co
s(x)^4 - 2*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*cos(x)^2)*sin(x))]

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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(csc(x)**6/(a+b*cos(x)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.21034, size = 211, normalized size = 2.37 \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^{3}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt{a^{2} + a b}} - \frac{15 \, a^{2} \tan \left (x\right )^{4} + 45 \, a b \tan \left (x\right )^{4} + 45 \, b^{2} \tan \left (x\right )^{4} + 10 \, a^{2} \tan \left (x\right )^{2} + 25 \, a b \tan \left (x\right )^{2} + 15 \, b^{2} \tan \left (x\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{15 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (x\right )^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^6/(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^3/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt(a^2
+ a*b)) - 1/15*(15*a^2*tan(x)^4 + 45*a*b*tan(x)^4 + 45*b^2*tan(x)^4 + 10*a^2*tan(x)^2 + 25*a*b*tan(x)^2 + 15*
b^2*tan(x)^2 + 3*a^2 + 6*a*b + 3*b^2)/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*tan(x)^5)