Integrand size = 15, antiderivative size = 89 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=-\frac {b^3 \arctan \left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac {(2 a+3 b) \cot ^3(x)}{3 (a+b)^2}-\frac {\cot ^5(x)}{5 (a+b)} \] Output:
-b^3*arctan((a+b)^(1/2)*cot(x)/a^(1/2))/a^(1/2)/(a+b)^(7/2)-(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-cot(x)^5/(5*a+5*b)
Time = 0.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=\frac {b^3 \arctan \left (\frac {\sqrt {a} \tan (x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\cot (x) \left (8 a^2+26 a b+33 b^2+\left (4 a^2+13 a b+9 b^2\right ) \csc ^2(x)+3 (a+b)^2 \csc ^4(x)\right )}{15 (a+b)^3} \] Input:
Integrate[Csc[x]^6/(a + b*Cos[x]^2),x]
Output:
(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)
Time = 0.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3670, 300, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\cos \left (x+\frac {\pi }{2}\right )^6 \left (a+b \sin \left (x+\frac {\pi }{2}\right )^2\right )}dx\) |
\(\Big \downarrow \) 3670 |
\(\displaystyle -\int \frac {\left (\cot ^2(x)+1\right )^3}{(a+b) \cot ^2(x)+a}d\cot (x)\) |
\(\Big \downarrow \) 300 |
\(\displaystyle -\int \left (\frac {\cot ^4(x)}{a+b}+\frac {(2 a+3 b) \cot ^2(x)}{(a+b)^2}+\frac {a^2+3 b a+3 b^2}{(a+b)^3}+\frac {b^3}{(a+b)^3 \left ((a+b) \cot ^2(x)+a\right )}\right )d\cot (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (a^2+3 a b+3 b^2\right ) \cot (x)}{(a+b)^3}-\frac {b^3 \arctan \left (\frac {\sqrt {a+b} \cot (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^{7/2}}-\frac {\cot ^5(x)}{5 (a+b)}-\frac {(2 a+3 b) \cot ^3(x)}{3 (a+b)^2}\) |
Input:
Int[Csc[x]^6/(a + b*Cos[x]^2),x]
Output:
-((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))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Su bst[Int[(a + (a + b)*ff^2*x^2)^p/(1 + ff^2*x^2)^(m/2 + p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]
Time = 0.77 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {b^{3} \arctan \left (\frac {a \tan \left (x \right )}{\sqrt {\left (a +b \right ) a}}\right )}{\left (a +b \right )^{3} \sqrt {\left (a +b \right ) a}}-\frac {1}{5 \left (a +b \right ) \tan \left (x \right )^{5}}-\frac {2 a +3 b}{3 \left (a +b \right )^{2} \tan \left (x \right )^{3}}-\frac {a^{2}+3 a b +3 b^{2}}{\left (a +b \right )^{3} \tan \left (x \right )}\) | \(83\) |
risch | \(-\frac {2 i \left (15 b^{2} {\mathrm e}^{8 i x}-30 a b \,{\mathrm e}^{6 i x}-90 b^{2} {\mathrm e}^{6 i x}+80 a^{2} {\mathrm e}^{4 i x}+230 a b \,{\mathrm e}^{4 i x}+240 b^{2} {\mathrm e}^{4 i x}-40 a^{2} {\mathrm e}^{2 i x}-130 a b \,{\mathrm e}^{2 i x}-150 \,{\mathrm e}^{2 i x} b^{2}+8 a^{2}+26 a b +33 b^{2}\right )}{15 \left (a +b \right )^{3} \left ({\mathrm e}^{2 i x}-1\right )^{5}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}+\frac {-2 i a^{2}-2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{\sqrt {-a^{2}-a b}\, b}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}-\frac {b^{3} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i a^{2}+2 i a b +2 a \sqrt {-a^{2}-a b}+b \sqrt {-a^{2}-a b}}{b \sqrt {-a^{2}-a b}}\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{3}}\) | \(293\) |
Input:
int(csc(x)^6/(a+b*cos(x)^2),x,method=_RETURNVERBOSE)
Output:
1/(a+b)^3*b^3/((a+b)*a)^(1/2)*arctan(a*tan(x)/((a+b)*a)^(1/2))-1/5/(a+b)/t an(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)
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (77) = 154\).
Time = 0.19 (sec) , antiderivative size = 610, normalized size of antiderivative = 6.85 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx =\text {Too large to display} \] Input:
integrate(csc(x)^6/(a+b*cos(x)^2),x, algorithm="fricas")
Output:
[-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*co s(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)*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))]
\[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=\int \frac {\csc ^{6}{\left (x \right )}}{a + b \cos ^{2}{\left (x \right )}}\, dx \] Input:
integrate(csc(x)**6/(a+b*cos(x)**2),x)
Output:
Integral(csc(x)**6/(a + b*cos(x)**2), x)
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.43 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=\frac {b^{3} \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {15 \, {\left (a^{2} + 3 \, a b + 3 \, b^{2}\right )} \tan \left (x\right )^{4} + 5 \, {\left (2 \, a^{2} + 5 \, a b + 3 \, b^{2}\right )} \tan \left (x\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (x\right )^{5}} \] Input:
integrate(csc(x)^6/(a+b*cos(x)^2),x, algorithm="maxima")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (77) = 154\).
Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.75 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=\frac {{\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (x\right )}{\sqrt {a^{2} + a b}}\right )\right )} b^{3}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a^{2} + a b}} - \frac {15 \, a^{2} \tan \left (x\right )^{4} + 45 \, a b \tan \left (x\right )^{4} + 45 \, b^{2} \tan \left (x\right )^{4} + 10 \, a^{2} \tan \left (x\right )^{2} + 25 \, a b \tan \left (x\right )^{2} + 15 \, b^{2} \tan \left (x\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (x\right )^{5}} \] Input:
integrate(csc(x)^6/(a+b*cos(x)^2),x, algorithm="giac")
Output:
(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)
Time = 1.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=\frac {b^3\,\mathrm {atan}\left (\frac {\sqrt {a}\,\mathrm {tan}\left (x\right )\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}{{\left (a+b\right )}^{7/2}}\right )}{\sqrt {a}\,{\left (a+b\right )}^{7/2}}-\frac {\frac {1}{5\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (x\right )}^2\,\left (2\,a+3\,b\right )}{3\,{\left (a+b\right )}^2}+\frac {{\mathrm {tan}\left (x\right )}^4\,\left (a^2+3\,a\,b+3\,b^2\right )}{{\left (a+b\right )}^3}}{{\mathrm {tan}\left (x\right )}^5} \] Input:
int(1/(sin(x)^6*(a + b*cos(x)^2)),x)
Output:
(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
Time = 0.18 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.67 \[ \int \frac {\csc ^6(x)}{a+b \cos ^2(x)} \, dx=\frac {15 \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {\sqrt {a +b}\, \tan \left (\frac {x}{2}\right )-\sqrt {b}}{\sqrt {a}}\right ) \sin \left (x \right )^{5} b^{3}+15 \sqrt {a}\, \sqrt {a +b}\, \mathit {atan} \left (\frac {\sqrt {a +b}\, \tan \left (\frac {x}{2}\right )+\sqrt {b}}{\sqrt {a}}\right ) \sin \left (x \right )^{5} b^{3}-8 \cos \left (x \right ) \sin \left (x \right )^{4} a^{4}-34 \cos \left (x \right ) \sin \left (x \right )^{4} a^{3} b -59 \cos \left (x \right ) \sin \left (x \right )^{4} a^{2} b^{2}-33 \cos \left (x \right ) \sin \left (x \right )^{4} a \,b^{3}-4 \cos \left (x \right ) \sin \left (x \right )^{2} a^{4}-17 \cos \left (x \right ) \sin \left (x \right )^{2} a^{3} b -22 \cos \left (x \right ) \sin \left (x \right )^{2} a^{2} b^{2}-9 \cos \left (x \right ) \sin \left (x \right )^{2} a \,b^{3}-3 \cos \left (x \right ) a^{4}-9 \cos \left (x \right ) a^{3} b -9 \cos \left (x \right ) a^{2} b^{2}-3 \cos \left (x \right ) a \,b^{3}}{15 \sin \left (x \right )^{5} a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )} \] Input:
int(csc(x)^6/(a+b*cos(x)^2),x)
Output:
(15*sqrt(a)*sqrt(a + b)*atan((sqrt(a + b)*tan(x/2) - sqrt(b))/sqrt(a))*sin (x)**5*b**3 + 15*sqrt(a)*sqrt(a + b)*atan((sqrt(a + b)*tan(x/2) + sqrt(b)) /sqrt(a))*sin(x)**5*b**3 - 8*cos(x)*sin(x)**4*a**4 - 34*cos(x)*sin(x)**4*a **3*b - 59*cos(x)*sin(x)**4*a**2*b**2 - 33*cos(x)*sin(x)**4*a*b**3 - 4*cos (x)*sin(x)**2*a**4 - 17*cos(x)*sin(x)**2*a**3*b - 22*cos(x)*sin(x)**2*a**2 *b**2 - 9*cos(x)*sin(x)**2*a*b**3 - 3*cos(x)*a**4 - 9*cos(x)*a**3*b - 9*co s(x)*a**2*b**2 - 3*cos(x)*a*b**3)/(15*sin(x)**5*a*(a**4 + 4*a**3*b + 6*a** 2*b**2 + 4*a*b**3 + b**4))