Integrand size = 15, antiderivative size = 54 \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\frac {2 A \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {B \log (a+b \sin (x))}{b} \]
Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\frac {2 A \arctan \left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {B \log (a+b \sin (x))}{b} \]
Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4901, 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.
\(\displaystyle \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \cos (x)}{a+b \sin (x)}dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle \int \left (\frac {A}{a+b \sin (x)}+\frac {B \cos (x)}{a+b \sin (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 A \arctan \left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {B \log (a+b \sin (x))}{b}\) |
3.1.1.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
Time = 0.76 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98
method | result | size |
parts | \(\frac {2 A \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}+\frac {B \ln \left (a +b \sin \left (x \right )\right )}{b}\) | \(53\) |
default | \(-\frac {B \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b}+\frac {B \ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )+\frac {2 A b \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b}\) | \(83\) |
risch | \(\frac {i x B}{b}-\frac {2 i B x \,a^{2} b}{a^{2} b^{2}-b^{4}}+\frac {2 i B x \,b^{3}}{a^{2} b^{2}-b^{4}}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a A b +\sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B \,a^{2}}{\left (a^{2}-b^{2}\right ) b}-\frac {b \ln \left ({\mathrm e}^{i x}+\frac {i a A b +\sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B}{a^{2}-b^{2}}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a A b +\sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{\left (a^{2}-b^{2}\right ) b}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a A b -\sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B \,a^{2}}{\left (a^{2}-b^{2}\right ) b}-\frac {b \ln \left ({\mathrm e}^{i x}+\frac {i a A b -\sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B}{a^{2}-b^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {i a A b -\sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{\left (a^{2}-b^{2}\right ) b}\) | \(456\) |
Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 4.56 \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} A b \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (-b^{2} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}, -\frac {2 \, \sqrt {a^{2} - b^{2}} A b \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (-b^{2} \cos \left (x\right )^{2} + 2 \, a b \sin \left (x\right ) + a^{2} + b^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}\right ] \]
[-1/2*(sqrt(-a^2 + b^2)*A*b*log(((2*a^2 - b^2)*cos(x)^2 - 2*a*b*sin(x) - a ^2 - b^2 + 2*(a*cos(x)*sin(x) + b*cos(x))*sqrt(-a^2 + b^2))/(b^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (B*a^2 - B*b^2)*log(-b^2*cos(x)^2 + 2*a*b*s in(x) + a^2 + b^2))/(a^2*b - b^3), -1/2*(2*sqrt(a^2 - b^2)*A*b*arctan(-(a* sin(x) + b)/(sqrt(a^2 - b^2)*cos(x))) - (B*a^2 - B*b^2)*log(-b^2*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2))/(a^2*b - b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (44) = 88\).
Time = 14.07 (sec) , antiderivative size = 552, normalized size of antiderivative = 10.22 \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\begin {cases} \tilde {\infty } \left (A \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} - B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tan {\left (\frac {x}{2} \right )} \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {A \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} - B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tan {\left (\frac {x}{2} \right )} \right )}}{b} & \text {for}\: a = 0 \\\frac {2 A}{b \tan {\left (\frac {x}{2} \right )} - b} + \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} \tan {\left (\frac {x}{2} \right )}}{b \tan {\left (\frac {x}{2} \right )} - b} - \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{b \tan {\left (\frac {x}{2} \right )} - b} - \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan {\left (\frac {x}{2} \right )}}{b \tan {\left (\frac {x}{2} \right )} - b} + \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{b \tan {\left (\frac {x}{2} \right )} - b} & \text {for}\: a = - b \\- \frac {2 A}{b \tan {\left (\frac {x}{2} \right )} + b} + \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan {\left (\frac {x}{2} \right )}}{b \tan {\left (\frac {x}{2} \right )} + b} + \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{b \tan {\left (\frac {x}{2} \right )} + b} - \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan {\left (\frac {x}{2} \right )}}{b \tan {\left (\frac {x}{2} \right )} + b} - \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{b \tan {\left (\frac {x}{2} \right )} + b} & \text {for}\: a = b \\\frac {A x + B \sin {\left (x \right )}}{a} & \text {for}\: b = 0 \\- \frac {A b \sqrt {- a^{2} + b^{2}} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} + \frac {A b \sqrt {- a^{2} + b^{2}} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} - \frac {B a^{2} \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{a^{2} b - b^{3}} + \frac {B a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} + \frac {B a^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} + \frac {B b^{2} \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{a^{2} b - b^{3}} - \frac {B b^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} - \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} - \frac {B b^{2} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {b}{a} + \frac {\sqrt {- a^{2} + b^{2}}}{a} \right )}}{a^{2} b - b^{3}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(A*log(tan(x/2)) - B*log(tan(x/2)**2 + 1) + B*log(tan(x/2)) ), Eq(a, 0) & Eq(b, 0)), ((A*log(tan(x/2)) - B*log(tan(x/2)**2 + 1) + B*lo g(tan(x/2)))/b, Eq(a, 0)), (2*A/(b*tan(x/2) - b) + 2*B*log(tan(x/2) - 1)*t an(x/2)/(b*tan(x/2) - b) - 2*B*log(tan(x/2) - 1)/(b*tan(x/2) - b) - B*log( tan(x/2)**2 + 1)*tan(x/2)/(b*tan(x/2) - b) + B*log(tan(x/2)**2 + 1)/(b*tan (x/2) - b), Eq(a, -b)), (-2*A/(b*tan(x/2) + b) + 2*B*log(tan(x/2) + 1)*tan (x/2)/(b*tan(x/2) + b) + 2*B*log(tan(x/2) + 1)/(b*tan(x/2) + b) - B*log(ta n(x/2)**2 + 1)*tan(x/2)/(b*tan(x/2) + b) - B*log(tan(x/2)**2 + 1)/(b*tan(x /2) + b), Eq(a, b)), ((A*x + B*sin(x))/a, Eq(b, 0)), (-A*b*sqrt(-a**2 + b* *2)*log(tan(x/2) + b/a - sqrt(-a**2 + b**2)/a)/(a**2*b - b**3) + A*b*sqrt( -a**2 + b**2)*log(tan(x/2) + b/a + sqrt(-a**2 + b**2)/a)/(a**2*b - b**3) - B*a**2*log(tan(x/2)**2 + 1)/(a**2*b - b**3) + B*a**2*log(tan(x/2) + b/a - sqrt(-a**2 + b**2)/a)/(a**2*b - b**3) + B*a**2*log(tan(x/2) + b/a + sqrt( -a**2 + b**2)/a)/(a**2*b - b**3) + B*b**2*log(tan(x/2)**2 + 1)/(a**2*b - b **3) - B*b**2*log(tan(x/2) + b/a - sqrt(-a**2 + b**2)/a)/(a**2*b - b**3) - B*b**2*log(tan(x/2) + b/a + sqrt(-a**2 + b**2)/a)/(a**2*b - b**3), True))
Exception generated. \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.32 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.63 \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} A}{\sqrt {a^{2} - b^{2}}} + \frac {B \log \left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}{b} - \frac {B \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}{b} \]
2*(pi*floor(1/2*x/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*x) + b)/sqrt(a^2 - b^2)))*A/sqrt(a^2 - b^2) + B*log(a*tan(1/2*x)^2 + 2*b*tan(1/2*x) + a)/b - B*log(tan(1/2*x)^2 + 1)/b
Time = 18.26 (sec) , antiderivative size = 1779, normalized size of antiderivative = 32.94 \[ \int \frac {A+B \cos (x)}{a+b \sin (x)} \, dx=\text {Too large to display} \]
(log((a + b*sin(x))/(cos(x) + 1))*(2*B*b^3 - 2*B*a^2*b))/(2*(b^4 - a^2*b^2 )) - (2*A*atan((tan(x/2)*(a^2 - b^2)^(3/2)*((((((A*(((2*B*b^3 - 2*B*a^2*b) *(96*a*b^4 - 64*a^3*b^2))/(2*(b^4 - a^2*b^2)) + 64*A*a*b^3 + 32*B*a*b^3))/ (a^2 - b^2)^(1/2) + (A*(2*B*b^3 - 2*B*a^2*b)*(96*a*b^4 - 64*a^3*b^2))/(2*( b^4 - a^2*b^2)*(a^2 - b^2)^(1/2)))*(2*B*b^3 - 2*B*a^2*b))/(2*(b^4 - a^2*b^ 2)) - (A^3*(96*a*b^4 - 64*a^3*b^2))/(a^2 - b^2)^(3/2) + (A*(64*B^2*a^3 - 3 2*A^2*a*b^2 - 96*B^2*a*b^2 + ((2*B*b^3 - 2*B*a^2*b)*(((2*B*b^3 - 2*B*a^2*b )*(96*a*b^4 - 64*a^3*b^2))/(2*(b^4 - a^2*b^2)) + 64*A*a*b^3 + 32*B*a*b^3)) /(2*(b^4 - a^2*b^2)) + 128*A*B*a*b^2))/(a^2 - b^2)^(1/2))*(2*A^2*b^4 + 4*B ^2*a^4 + 8*B^2*b^4 - A^2*a^2*b^2 - 12*B^2*a^2*b^2 + 8*A*B*b^4 - 8*A*B*a^2* b^2))/(a^3*(a^2 - b^2)^(1/2)*(A^2*b^2 + 4*B^2*a^2 - 4*B^2*b^2)^2) - (2*b*( A + 2*B)*(A*b^2 - 2*B*a^2 + 2*B*b^2)*((A*((A*(((2*B*b^3 - 2*B*a^2*b)*(96*a *b^4 - 64*a^3*b^2))/(2*(b^4 - a^2*b^2)) + 64*A*a*b^3 + 32*B*a*b^3))/(a^2 - b^2)^(1/2) + (A*(2*B*b^3 - 2*B*a^2*b)*(96*a*b^4 - 64*a^3*b^2))/(2*(b^4 - a^2*b^2)*(a^2 - b^2)^(1/2))))/(a^2 - b^2)^(1/2) + 32*B^3*a*b - ((2*B*b^3 - 2*B*a^2*b)*(64*B^2*a^3 - 32*A^2*a*b^2 - 96*B^2*a*b^2 + ((2*B*b^3 - 2*B*a^ 2*b)*(((2*B*b^3 - 2*B*a^2*b)*(96*a*b^4 - 64*a^3*b^2))/(2*(b^4 - a^2*b^2)) + 64*A*a*b^3 + 32*B*a*b^3))/(2*(b^4 - a^2*b^2)) + 128*A*B*a*b^2))/(2*(b^4 - a^2*b^2)) - 64*A*B^2*a*b + 32*A^2*B*a*b + (A^2*(2*B*b^3 - 2*B*a^2*b)*(96 *a*b^4 - 64*a^3*b^2))/(2*(b^4 - a^2*b^2)*(a^2 - b^2))))/(a^3*(A^2*b^2 +...