3.1.23 \(\int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx\) [23]

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

[Out]

-(a^2+3*a*b+3*b^2)*cot(x)/(a+b)^3-1/3*(2*a+3*b)*cot(x)^3/(a+b)^2-1/5*cot(x)^5/(a+b)-b^3*arctan(cot(x)*(a+b)^(1
/2)/a^(1/2))/(a+b)^(7/2)/a^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 89, 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 = {3270, 398, 211} \begin {gather*} -\frac {\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac {b^3 \text {ArcTan}\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} \end {gather*}

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 211

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

Rule 398

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 3270

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]

Rubi steps

\begin {align*} \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx &=-\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{a+(a+b) x^2} \, dx,x,\cot (x)\right )\\ &=-\text {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 \text {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*}

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Mathematica [A]
time = 0.43, size = 90, normalized size = 1.01 \begin {gather*} \frac {b^3 \text {ArcTan}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\cot (x) \left (8 a^2+26 a b+33 b^2+\left (4 a^2+13 a b+9 b^2\right ) \csc ^2(x)+3 (a+b)^2 \csc ^4(x)\right )}{15 (a+b)^3} \end {gather*}

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 = 0.20, size = 83, normalized size = 0.93

method	result	size
default	\(-\frac {1}{5 \left (a +b \right ) \tan \left (x \right )^{5}}-\frac {2 a +3 b}{3 \left (a +b \right )^{2} \tan \left (x \right )^{3}}-\frac {a^{2}+3 a b +3 b^{2}}{\left (a +b \right )^{3} \tan \left (x \right )}+\frac {b^{3} \arctan \left (\frac {a \tan \left (x \right )}{\sqrt {\left (a +b \right ) a}}\right )}{\left (a +b \right )^{3} \sqrt {\left (a +b \right ) a}}\)	\(83\)
risch	\(-\frac {2 i \left (15 b^{2} {\mathrm e}^{8 i x}-30 a b \,{\mathrm e}^{6 i x}-90 b^{2} {\mathrm e}^{6 i x}+80 a^{2} {\mathrm e}^{4 i x}+230 a b \,{\mathrm e}^{4 i x}+240 b^{2} {\mathrm e}^{4 i x}-40 a^{2} {\mathrm e}^{2 i x}-130 b \,{\mathrm e}^{2 i x} a -150 b^{2} {\mathrm e}^{2 i x}+8 a^{2}+26 a b +33 b^{2}\right )}{15 \left (a +b \right )^{3} \left ({\mathrm e}^{2 i x}-1\right )^{5}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}+\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, size = 127, normalized size = 1.43 \begin {gather*} \frac {b^{3} \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {15 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \tan \left (x\right )^{4} + 5 \, {\left (2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right )} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*arctan(a*tan(x)/sqrt((a + b)*a))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sqrt((a + b)*a)) - 1/15*(15*(a^2 + 3*a*b
+ 3*b^2)*tan(x)^4 + 5*(2*a^2 + 5*a*b + 3*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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (77) = 154\).
time = 0.45, size = 610, normalized size = 6.85 \begin {gather*} \left [-\frac {4 \, {\left (8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right )} \cos \left (x\right )^{5} - 20 \, {\left (4 \, a^{4} + 17 \, a^{3} b + 28 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \left (x\right )^{3} + 15 \, {\left (b^{3} \cos \left (x\right )^{4} - 2 \, b^{3} \cos \left (x\right )^{2} + b^{3}\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 ) + 60 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (x\right )}{60 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}, -\frac {2 \, {\left (8 \, a^{4} + 34 \, a^{3} b + 59 \, a^{2} b^{2} + 33 \, a b^{3}\right )} \cos \left (x\right )^{5} - 10 \, {\left (4 \, a^{4} + 17 \, a^{3} b + 28 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \left (x\right )^{3} + 15 \, {\left (b^{3} \cos \left (x\right )^{4} - 2 \, b^{3} \cos \left (x\right )^{2} + b^{3}\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 ) + 30 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (x\right )}{30 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4} + {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{4} - 2 \, {\left (a^{5} + 4 \, a^{4} b + 6 \, a^{3} b^{2} + 4 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}\right ] \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{6}{\left (x \right )}}{a + b \cos ^{2}{\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (77) = 154\).
time = 0.43, size = 156, normalized size = 1.75 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 2.37, size = 101, normalized size = 1.13 \begin {gather*} \frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\left (x\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{{\left (a+b\right )}^{7/2}}\right )}{\sqrt {a}\,{\left (a+b\right )}^{7/2}}-\frac {\frac {1}{5\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (x\right )}^2\,\left (2\,a+3\,b\right )}{3\,{\left (a+b\right )}^2}+\frac {{\mathrm {tan}\left (x\right )}^4\,\left (a^2+3\,a\,b+3\,b^2\right )}{{\left (a+b\right )}^3}}{{\mathrm {tan}\left (x\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^6*(a + b*cos(x)^2)),x)

[Out]

(b^3*atan((a^(1/2)*tan(x)*(3*a*b^2 + 3*a^2*b + a^3 + b^3))/(a + b)^(7/2)))/(a^(1/2)*(a + b)^(7/2)) - (1/(5*(a
+ b)) + (tan(x)^2*(2*a + 3*b))/(3*(a + b)^2) + (tan(x)^4*(3*a*b + a^2 + 3*b^2))/(a + b)^3)/tan(x)^5

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