Integrand size = 21, antiderivative size = 84 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {x}{b^2}+\frac {2 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2772, 2814, 2739, 632, 210} \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 a \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^2 d \sqrt {a^2-b^2}}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {x}{b^2} \]
[In]
[Out]
Rule 210
Rule 632
Rule 2739
Rule 2772
Rule 2814
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {\int \frac {\sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b} \\ & = -\frac {x}{b^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac {a \int \frac {1}{a+b \sin (c+d x)} \, dx}{b^2} \\ & = -\frac {x}{b^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^2 d} \\ & = -\frac {x}{b^2}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac {(4 a) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^2 d} \\ & = -\frac {x}{b^2}+\frac {2 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}-\frac {\cos (c+d x)}{b d (a+b \sin (c+d x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(414\) vs. \(2(84)=168\).
Time = 2.02 (sec) , antiderivative size = 414, normalized size of antiderivative = 4.93 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\cos (c+d x) \left (-2 a (a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{\sqrt {a+b} \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}}}\right ) \sqrt {1-\sin (c+d x)} (a+b \sin (c+d x))+\sqrt {a+b} \left (2 a \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{\sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}}}\right ) \sqrt {1-\sin (c+d x)} (a+b \sin (c+d x))+(-a+b) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \left (\sqrt {a-b} (a+b) \sqrt {1-\sin (c+d x)} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}+2 \sqrt {b} \text {arcsinh}\left (\frac {\sqrt {a-b} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{\sqrt {2} \sqrt {b}}\right ) (a+b \sin (c+d x))\right )\right )\right )}{(a-b)^{3/2} b (a+b)^{3/2} d \sqrt {1-\sin (c+d x)} \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}} (a+b \sin (c+d x))} \]
[In]
[Out]
Time = 0.75 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.44
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-b \right )}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{2}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2}}}{d}\) | \(121\) |
default | \(\frac {\frac {\frac {2 \left (-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-b \right )}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 a \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{b^{2}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2}}}{d}\) | \(121\) |
risch | \(-\frac {x}{b^{2}}-\frac {2 \left (i b +a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{b^{2} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}\) | \(201\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 388, normalized size of antiderivative = 4.62 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {2 \, {\left (a^{2} b - b^{3}\right )} d x \sin \left (d x + c\right ) + 2 \, {\left (a^{3} - a b^{2}\right )} d x + {\left (a b \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{3} - b^{5}\right )} d \sin \left (d x + c\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} d\right )}}, -\frac {{\left (a^{2} b - b^{3}\right )} d x \sin \left (d x + c\right ) + {\left (a^{3} - a b^{2}\right )} d x + {\left (a b \sin \left (d x + c\right ) + a^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + {\left (a^{2} b - b^{3}\right )} \cos \left (d x + c\right )}{{\left (a^{2} b^{3} - b^{5}\right )} d \sin \left (d x + c\right ) + {\left (a^{3} b^{2} - a b^{4}\right )} d}\right ] \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.50 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a}{\sqrt {a^{2} - b^{2}} b^{2}} - \frac {d x + c}{b^{2}} - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a b}}{d} \]
[In]
[Out]
Time = 5.10 (sec) , antiderivative size = 329, normalized size of antiderivative = 3.92 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {b^2\,\sin \left (c+d\,x\right )+\frac {\left (2\,a^3\,\mathrm {atan}\left (\frac {\left (-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}\,\left (a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )-a^2\,\sqrt {b^2-a^2}\,\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}}-\frac {b\,\left (a\,\sqrt {b^2-a^2}\,1{}\mathrm {i}+a\,\cos \left (c+d\,x\right )\,\sqrt {b^2-a^2}\,1{}\mathrm {i}-2\,a^2\,\mathrm {atan}\left (\frac {\left (-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}\,\left (a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+2\,b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )\,\sin \left (c+d\,x\right )+a\,\sin \left (c+d\,x\right )\,\sqrt {b^2-a^2}\,\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{\sqrt {b^2-a^2}}}{a\,b^2\,d\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \]
[In]
[Out]