Integrand size = 16, antiderivative size = 68 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2} \]
-a^2*arctanh((b*cos(x)-a*sin(x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(3/2)-b*cos(x) /(a^2+b^2)-a*sin(x)/(a^2+b^2)
Time = 0.25 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {2 a^2 \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b \cos (x)+a \sin (x)}{a^2+b^2} \]
(2*a^2*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b* Cos[x] + a*Sin[x])/(a^2 + b^2)
Time = 0.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3578, 3042, 3118, 3553, 219}
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.
\(\displaystyle \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (x)^2}{a \cos (x)+b \sin (x)}dx\) |
\(\Big \downarrow \) 3578 |
\(\displaystyle \frac {b \int \sin (x)dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b \int \sin (x)dx}{a^2+b^2}+\frac {a^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {a^2 \int \frac {1}{a \cos (x)+b \sin (x)}dx}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2}-\frac {b \cos (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 3553 |
\(\displaystyle -\frac {a^2 \int \frac {1}{a^2+b^2-(b \cos (x)-a \sin (x))^2}d(b \cos (x)-a \sin (x))}{a^2+b^2}-\frac {a \sin (x)}{a^2+b^2}-\frac {b \cos (x)}{a^2+b^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {a \sin (x)}{a^2+b^2}-\frac {b \cos (x)}{a^2+b^2}\) |
-((a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - (b*Cos[x])/(a^2 + b^2) - (a*Sin[x])/(a^2 + b^2)
3.1.9.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a^2/(a^2 + b^2) Int[Sin[c + d*x]^(m - 2)/(a *Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Simp[b/(a^2 + b^2) Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ [m, 1]
Time = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {8 a^{2} \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{2}+4 b^{2}\right ) \sqrt {a^{2}+b^{2}}}+\frac {-2 a \tan \left (\frac {x}{2}\right )-2 b}{\left (a^{2}+b^{2}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}\) | \(84\) |
risch | \(\frac {i {\mathrm e}^{i x}}{-2 i b +2 a}-\frac {i {\mathrm e}^{-i x}}{2 \left (i b +a \right )}-\frac {a^{2} \ln \left ({\mathrm e}^{i x}-\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(146\) |
8*a^2/(4*a^2+4*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2+ b^2)^(1/2))+2/(a^2+b^2)*(-a*tan(1/2*x)-b)/(1+tan(1/2*x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (64) = 128\).
Time = 0.27 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.12 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} a^{2} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cos \left (x\right ) - 2 \, {\left (a^{3} + a b^{2}\right )} \sin \left (x\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]
1/2*(sqrt(a^2 + b^2)*a^2*log(-(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin (x) + (a^2 - b^2)*cos(x)^2 + b^2)) - 2*(a^2*b + b^3)*cos(x) - 2*(a^3 + a*b ^2)*sin(x))/(a^4 + 2*a^2*b^2 + b^4)
Result contains complex when optimal does not.
Time = 76.32 (sec) , antiderivative size = 706, normalized size of antiderivative = 10.38 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=\begin {cases} \tilde {\infty } \cos {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {\cos {\left (x \right )}}{b} & \text {for}\: a = 0 \\- \frac {\sin ^{2}{\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {2 i \sin {\left (x \right )} \cos {\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {2 \cos ^{2}{\left (x \right )}}{3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} & \text {for}\: a = - i b \\- \frac {\sin ^{2}{\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} + \frac {2 i \sin {\left (x \right )} \cos {\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} - \frac {2 \cos ^{2}{\left (x \right )}}{- 3 i b \sin {\left (x \right )} + 3 b \cos {\left (x \right )}} & \text {for}\: a = i b \\- \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} - \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} + \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} + \frac {a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {b}{a} + \frac {\sqrt {a^{2} + b^{2}}}{a} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {2 a \sqrt {a^{2} + b^{2}} \tan {\left (\frac {x}{2} \right )}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} - \frac {2 b \sqrt {a^{2} + b^{2}}}{a^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + a^{2} \sqrt {a^{2} + b^{2}} + b^{2} \sqrt {a^{2} + b^{2}} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2} \sqrt {a^{2} + b^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*cos(x), Eq(a, 0) & Eq(b, 0)), (-cos(x)/b, Eq(a, 0)), (-sin( x)**2/(3*I*b*sin(x) + 3*b*cos(x)) - 2*I*sin(x)*cos(x)/(3*I*b*sin(x) + 3*b* cos(x)) - 2*cos(x)**2/(3*I*b*sin(x) + 3*b*cos(x)), Eq(a, -I*b)), (-sin(x)* *2/(-3*I*b*sin(x) + 3*b*cos(x)) + 2*I*sin(x)*cos(x)/(-3*I*b*sin(x) + 3*b*c os(x)) - 2*cos(x)**2/(-3*I*b*sin(x) + 3*b*cos(x)), Eq(a, I*b)), (-a**2*log (tan(x/2) - b/a - sqrt(a**2 + b**2)/a)*tan(x/2)**2/(a**2*sqrt(a**2 + b**2) *tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) - a**2*log(tan(x/2) - b/a - sqrt(a**2 + b**2)/a )/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt (a**2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) + a**2*log(tan(x/2) - b/a + sqrt(a**2 + b**2)/a)*tan(x/2)**2/(a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2)*tan(x/2)**2 + b**2*sqrt (a**2 + b**2)) + a**2*log(tan(x/2) - b/a + sqrt(a**2 + b**2)/a)/(a**2*sqrt (a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a**2 + b**2 )*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) - 2*a*sqrt(a**2 + b**2)*tan(x/2)/( a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a* *2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)) - 2*b*sqrt(a**2 + b**2)/( a**2*sqrt(a**2 + b**2)*tan(x/2)**2 + a**2*sqrt(a**2 + b**2) + b**2*sqrt(a* *2 + b**2)*tan(x/2)**2 + b**2*sqrt(a**2 + b**2)), True))
Time = 0.31 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{2} \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (b + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}}{a^{2} + b^{2} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}} \]
-a^2*log((b - a*sin(x)/(cos(x) + 1) + sqrt(a^2 + b^2))/(b - a*sin(x)/(cos( x) + 1) - sqrt(a^2 + b^2)))/(a^2 + b^2)^(3/2) - 2*(b + a*sin(x)/(cos(x) + 1))/(a^2 + b^2 + (a^2 + b^2)*sin(x)^2/(cos(x) + 1)^2)
Time = 0.31 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.38 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {a^{2} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a \tan \left (\frac {1}{2} \, x\right ) + b\right )}}{{\left (a^{2} + b^{2}\right )} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}} \]
-a^2*log(abs(2*a*tan(1/2*x) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*x) - 2*b + 2*sqrt(a^2 + b^2)))/(a^2 + b^2)^(3/2) - 2*(a*tan(1/2*x) + b)/((a^2 + b^2)*(tan(1/2*x)^2 + 1))
Time = 20.94 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.38 \[ \int \frac {\sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx=-\frac {\frac {2\,b}{a^2+b^2}+\frac {2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )}{a^2+b^2}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1}-\frac {2\,a^2\,\mathrm {atanh}\left (\frac {2\,a^2\,b+2\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2+b^2\right )}{2\,{\left (a^2+b^2\right )}^{3/2}}\right )}{{\left (a^2+b^2\right )}^{3/2}} \]