Integrand size = 15, antiderivative size = 57 \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=\frac {2 A \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {B \log (a+b \cos (x))}{b} \]
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=-\frac {2 A \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {B \log (a+b \cos (x))}{b} \]
(-2*A*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - (B* Log[a + b*Cos[x]])/b
Time = 0.32 (sec) , antiderivative size = 57, 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.
\(\displaystyle \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+B \sin (x)}{a+b \cos (x)}dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle \int \left (\frac {A}{a+b \cos (x)}+\frac {B \sin (x)}{a+b \cos (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 A \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b}}-\frac {B \log (a+b \cos (x))}{b}\) |
(2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - (B*Log[a + b*Cos[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]
Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(47)=94\).
Time = 0.35 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.68
method | result | size |
default | \(\frac {\frac {2 \left (-B a +B b \right ) \ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{2 a -2 b}+\frac {2 A b \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}+\frac {B \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b}\) | \(96\) |
risch | \(-\frac {i B x}{b}+\frac {2 i B x \,a^{2} b}{a^{2} b^{2}-b^{4}}-\frac {2 i B x \,b^{3}}{a^{2} b^{2}-b^{4}}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {A a b -i \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B \,a^{2}}{\left (a^{2}-b^{2}\right ) b}+\frac {b \ln \left ({\mathrm e}^{i x}+\frac {A a b -i \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B}{a^{2}-b^{2}}+\frac {\ln \left ({\mathrm e}^{i x}+\frac {A a b -i \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{\left (a^{2}-b^{2}\right ) b}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {A a b +i \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B \,a^{2}}{\left (a^{2}-b^{2}\right ) b}+\frac {b \ln \left ({\mathrm e}^{i x}+\frac {A a b +i \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) B}{a^{2}-b^{2}}-\frac {\ln \left ({\mathrm e}^{i x}+\frac {A a b +i \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{A \,b^{2}}\right ) \sqrt {-A^{2} a^{2} b^{2}+A^{2} b^{4}}}{\left (a^{2}-b^{2}\right ) b}\) | \(456\) |
2/b*(1/2*(-B*a+B*b)/(a-b)*ln(a*tan(1/2*x)^2-b*tan(1/2*x)^2+a+b)+A*b/((a+b) *(a-b))^(1/2)*arctan((a-b)*tan(1/2*x)/((a+b)*(a-b))^(1/2)))+B/b*ln(1+tan(1 /2*x)^2)
Time = 0.26 (sec) , antiderivative size = 227, normalized size of antiderivative = 3.98 \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} A b \log \left (\frac {2 \, a b \cos \left (x\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right ) + {\left (B a^{2} - B b^{2}\right )} \log \left (b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}, \frac {2 \, \sqrt {a^{2} - b^{2}} A b \arctan \left (-\frac {a \cos \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x\right )}\right ) - {\left (B a^{2} - B b^{2}\right )} \log \left (b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}\right ] \]
[-1/2*(sqrt(-a^2 + b^2)*A*b*log((2*a*b*cos(x) + (2*a^2 - b^2)*cos(x)^2 + 2 *sqrt(-a^2 + b^2)*(a*cos(x) + b)*sin(x) - a^2 + 2*b^2)/(b^2*cos(x)^2 + 2*a *b*cos(x) + a^2)) + (B*a^2 - B*b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2) )/(a^2*b - b^3), 1/2*(2*sqrt(a^2 - b^2)*A*b*arctan(-(a*cos(x) + b)/(sqrt(a ^2 - b^2)*sin(x))) - (B*a^2 - B*b^2)*log(b^2*cos(x)^2 + 2*a*b*cos(x) + a^2 ))/(a^2*b - b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 688 vs. \(2 (48) = 96\).
Time = 10.69 (sec) , antiderivative size = 688, normalized size of antiderivative = 12.07 \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=\begin {cases} \tilde {\infty } \left (- A \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} + A \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} - B \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} - B \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} + B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {A \tan {\left (\frac {x}{2} \right )}}{b} + \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{b} & \text {for}\: a = b \\\frac {A}{b \tan {\left (\frac {x}{2} \right )}} + \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{b} - \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} \right )}}{b} & \text {for}\: a = - b \\\frac {A x - B \cos {\left (x \right )}}{a} & \text {for}\: b = 0 \\\frac {A b \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} - \frac {A b \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} - \frac {B a \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} - \frac {B a \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} + \frac {B a \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} + \frac {B b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} + \frac {B b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {x}{2} \right )} \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} - \frac {B b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{a b \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-A*log(tan(x/2) - 1) + A*log(tan(x/2) + 1) - B*log(tan(x/2 ) - 1) - B*log(tan(x/2) + 1) + B*log(tan(x/2)**2 + 1)), Eq(a, 0) & Eq(b, 0 )), (A*tan(x/2)/b + B*log(tan(x/2)**2 + 1)/b, Eq(a, b)), (A/(b*tan(x/2)) + B*log(tan(x/2)**2 + 1)/b - 2*B*log(tan(x/2))/b, Eq(a, -b)), ((A*x - B*cos (x))/a, Eq(b, 0)), (A*b*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b *sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) - A*b*l og(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) - B*a*sqrt(-a/(a - b) - b/(a - b) )*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/( a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) - B*a*sqrt(-a/(a - b) - b/(a - b))*log(sqrt(-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))) + B*a*sqrt(-a/(a - b) - b/ (a - b))*log(tan(x/2)**2 + 1)/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqr t(-a/(a - b) - b/(a - b))) + B*b*sqrt(-a/(a - b) - b/(a - b))*log(-sqrt(-a /(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2 *sqrt(-a/(a - b) - b/(a - b))) + B*b*sqrt(-a/(a - b) - b/(a - b))*log(sqrt (-a/(a - b) - b/(a - b)) + tan(x/2))/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b **2*sqrt(-a/(a - b) - b/(a - b))) - B*b*sqrt(-a/(a - b) - b/(a - b))*log(t an(x/2)**2 + 1)/(a*b*sqrt(-a/(a - b) - b/(a - b)) - b**2*sqrt(-a/(a - b) - b/(a - b))), True))
Exception generated. \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (47) = 94\).
Time = 0.32 (sec) , antiderivative size = 361, normalized size of antiderivative = 6.33 \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=-\frac {B {\left (a + b\right )} {\left (a - b\right )}^{2} \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{2 \, {\left (a - b\right )}}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (\sqrt {a^{2} - b^{2}} A b {\left | a - b \right |} + \sqrt {a^{2} - b^{2}} A {\left | a - b \right |} {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, x\right )}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (A b - A {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, x\right )}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{b^{2} - a {\left | b \right |}} - \frac {{\left (B a - B b\right )} \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + \frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{2 \, {\left (a - b\right )}}\right )}{b^{2} - a {\left | b \right |}} \]
-B*(a + b)*(a - b)^2*log(tan(1/2*x)^2 + 1/2*(2*a + sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))/((a^2 - 2*a*b + b^2)*b^2 + (a^3 - 2*a^2*b + a*b^2)*abs (b)) + (sqrt(a^2 - b^2)*A*b*abs(a - b) + sqrt(a^2 - b^2)*A*abs(a - b)*abs( b))*(pi*floor(1/2*x/pi + 1/2) + arctan(2*sqrt(1/2)*tan(1/2*x)/sqrt((2*a + sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))))/((a^2 - 2*a*b + b^2)*b^2 + (a ^3 - 2*a^2*b + a*b^2)*abs(b)) + (A*b - A*abs(b))*(pi*floor(1/2*x/pi + 1/2) + arctan(2*sqrt(1/2)*tan(1/2*x)/sqrt((2*a - sqrt(-4*(a + b)*(a - b) + 4*a ^2))/(a - b))))/(b^2 - a*abs(b)) - (B*a - B*b)*log(tan(1/2*x)^2 + 1/2*(2*a - sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))/(b^2 - a*abs(b))
Time = 6.17 (sec) , antiderivative size = 1537, normalized size of antiderivative = 26.96 \[ \int \frac {A+B \sin (x)}{a+b \cos (x)} \, dx=\text {Too large to display} \]
(B*log(1/(cos(x) + 1)))/b - (log((a + b*cos(x))/(cos(x) + 1))*(2*B*b^3 - 2 *B*a^2*b))/(2*(b^4 - a^2*b^2)) - (2*A*atan(((a^2 - b^2)*((A*(64*A*B*b^3 + ((2*B*b^3 - 2*B*a^2*b)*(32*A*b^4 + 32*A*a^2*b^2 - 64*A*a*b^3))/(2*(b^4 - a ^2*b^2)) - 128*A*B*a*b^2 + 64*A*B*a^2*b))/(a^2 - b^2)^(1/2) + (A*(2*B*b^3 - 2*B*a^2*b)*(32*A*b^4 + 32*A*a^2*b^2 - 64*A*a*b^3))/(2*(b^4 - a^2*b^2)*(a ^2 - b^2)^(1/2)))*(A^2*b^2 - 4*B^2*a^2 + 4*B^2*b^2))/((32*A*a - 32*A*b)*(a - b)*(A^2*b^2 + 4*B^2*a^2 - 4*B^2*b^2)^2) - (tan(x/2)*(a^2 - b^2)^(3/2)*( (((A^3*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(a^2 - b^2)^(3/2) + (((A*(64 *B*b^4 + 64*B*a^2*b^2 - 128*B*a*b^3 - ((2*B*b^3 - 2*B*a^2*b)*(64*a*b^4 - 1 28*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(a^2 - b^2)^(1/2) - (A*(2* B*b^3 - 2*B*a^2*b)*(64*a*b^4 - 128*a^2*b^3 + 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*(32 *A^2*b^3 + 64*B^2*a^3 - 32*A^2*a*b^2 + 64*B^2*a*b^2 - 128*B^2*a^2*b + ((2* B*b^3 - 2*B*a^2*b)*(64*B*b^4 + 64*B*a^2*b^2 - 128*B*a*b^3 - ((2*B*b^3 - 2* B*a^2*b)*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(2*( b^4 - a^2*b^2))))/(a^2 - b^2)^(1/2))*(A^2*b^2 - 4*B^2*a^2 + 4*B^2*b^2))/(( a^2 - b^2)^(1/2)*(a - b)*(A^2*b^2 + 4*B^2*a^2 - 4*B^2*b^2)^2) - (4*A*B*b*( 64*B^3*a^2 + 64*B^3*b^2 - 32*A^2*B*b^2 + (A*((A*(64*B*b^4 + 64*B*a^2*b^2 - 128*B*a*b^3 - ((2*B*b^3 - 2*B*a^2*b)*(64*a*b^4 - 128*a^2*b^3 + 64*a^3*b^2 ))/(2*(b^4 - a^2*b^2))))/(a^2 - b^2)^(1/2) - (A*(2*B*b^3 - 2*B*a^2*b)*(...