Integrand size = 34, antiderivative size = 114 \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=\frac {\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac {2 a^3 B \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} d}-\frac {a B \sin (c+d x)}{b^2 d}+\frac {B \cos (c+d x) \sin (c+d x)}{2 b d} \]
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Time = 0.24 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {21, 2872, 3102, 2814, 2738, 211} \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=-\frac {2 a^3 B \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d \sqrt {a-b} \sqrt {a+b}}+\frac {B x \left (2 a^2+b^2\right )}{2 b^3}-\frac {a B \sin (c+d x)}{b^2 d}+\frac {B \sin (c+d x) \cos (c+d x)}{2 b d} \]
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Rule 21
Rule 211
Rule 2738
Rule 2814
Rule 2872
Rule 3102
Rubi steps \begin{align*} \text {integral}& = B \int \frac {\cos ^3(c+d x)}{a+b \cos (c+d x)} \, dx \\ & = \frac {B \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {B \int \frac {a+b \cos (c+d x)-2 a \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b} \\ & = -\frac {a B \sin (c+d x)}{b^2 d}+\frac {B \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {B \int \frac {a b+\left (2 a^2+b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2} \\ & = \frac {\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac {a B \sin (c+d x)}{b^2 d}+\frac {B \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (a^3 B\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^3} \\ & = \frac {\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac {a B \sin (c+d x)}{b^2 d}+\frac {B \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (2 a^3 B\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^3 d} \\ & = \frac {\left (2 a^2+b^2\right ) B x}{2 b^3}-\frac {2 a^3 B \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} d}-\frac {a B \sin (c+d x)}{b^2 d}+\frac {B \cos (c+d x) \sin (c+d x)}{2 b d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=\frac {B \left (2 \left (2 a^2+b^2\right ) (c+d x)+\frac {8 a^3 \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-4 a b \sin (c+d x)+b^2 \sin (2 (c+d x))\right )}{4 b^3 d} \]
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Time = 1.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {2 B \left (-\frac {a^{3} \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {\left (-a b -\frac {1}{2} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{b^{3}}\right )}{d}\) | \(139\) |
default | \(\frac {2 B \left (-\frac {a^{3} \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {\left (-a b -\frac {1}{2} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (2 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{b^{3}}\right )}{d}\) | \(139\) |
risch | \(\frac {B x \,a^{2}}{b^{3}}+\frac {B x}{2 b}+\frac {i B a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}-\frac {i B a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {a^{3} B \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {a^{3} B \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {B \sin \left (2 d x +2 c \right )}{4 b d}\) | \(227\) |
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Time = 0.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.07 \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=\left [-\frac {\sqrt {-a^{2} + b^{2}} B a^{3} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (2 \, B a^{4} - B a^{2} b^{2} - B b^{4}\right )} d x + {\left (2 \, B a^{3} b - 2 \, B a b^{3} - {\left (B a^{2} b^{2} - B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} d}, -\frac {2 \, \sqrt {a^{2} - b^{2}} B a^{3} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (2 \, B a^{4} - B a^{2} b^{2} - B b^{4}\right )} d x + {\left (2 \, B a^{3} b - 2 \, B a b^{3} - {\left (B a^{2} b^{2} - B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} d}\right ] \]
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Timed out. \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.62 \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )} B a^{3}}{\sqrt {a^{2} - b^{2}} b^{3}} - \frac {{\left (2 \, B a^{2} + B b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {2 \, {\left (2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, d} \]
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Time = 1.24 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.52 \[ \int \frac {\cos ^3(c+d x) (a B+b B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx=\frac {B\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b\,d}+\frac {B\,\sin \left (2\,c+2\,d\,x\right )}{4\,b\,d}+\frac {2\,B\,a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^3\,d}-\frac {B\,a\,\sin \left (c+d\,x\right )}{b^2\,d}-\frac {B\,a^3\,\mathrm {atan}\left (\frac {\left (a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,1{}\mathrm {i}}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}\right )\,2{}\mathrm {i}}{b^3\,d\,\sqrt {b^2-a^2}} \]
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