Integrand size = 33, antiderivative size = 154 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=-\frac {\left (6 a^2-b^2\right ) x}{2 b^4}+\frac {2 a \left (3 a^2-2 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} d}+\frac {3 a \sin (c+d x)}{b^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))} \]
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Time = 0.47 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3127, 3129, 3102, 2814, 2738, 211} \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {2 a \left (3 a^2-2 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d \sqrt {a-b} \sqrt {a+b}}-\frac {x \left (6 a^2-b^2\right )}{2 b^4}+\frac {3 a \sin (c+d x)}{b^3 d}+\frac {\sin (c+d x) \cos ^2(c+d x)}{b d (a+b \cos (c+d x))}-\frac {3 \sin (c+d x) \cos (c+d x)}{2 b^2 d} \]
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Rule 211
Rule 2738
Rule 2814
Rule 3102
Rule 3127
Rule 3129
Rubi steps \begin{align*} \text {integral}& = \frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-2 \left (a^2-b^2\right )+3 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )} \\ & = -\frac {3 \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {3 a \left (a^2-b^2\right )-b \left (a^2-b^2\right ) \cos (c+d x)-6 a \left (a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )} \\ & = \frac {3 a \sin (c+d x)}{b^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {3 a b \left (a^2-b^2\right )+\left (a^2-b^2\right ) \left (6 a^2-b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )} \\ & = -\frac {\left (6 a^2-b^2\right ) x}{2 b^4}+\frac {3 a \sin (c+d x)}{b^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}+\frac {\left (a \left (3 a^2-2 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^4} \\ & = -\frac {\left (6 a^2-b^2\right ) x}{2 b^4}+\frac {3 a \sin (c+d x)}{b^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}+\frac {\left (2 a \left (3 a^2-2 b^2\right )\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^4 d} \\ & = -\frac {\left (6 a^2-b^2\right ) x}{2 b^4}+\frac {2 a \left (3 a^2-2 b^2\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} d}+\frac {3 a \sin (c+d x)}{b^3 d}-\frac {3 \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {\cos ^2(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {2 \left (-6 a^2+b^2\right ) (c+d x)-\frac {8 a \left (3 a^2-2 b^2\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+8 a b \sin (c+d x)+\frac {4 a^2 b \sin (c+d x)}{a+b \cos (c+d x)}-b^2 \sin (2 (c+d x))}{4 b^4 d} \]
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Time = 1.95 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.27
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a^{2}-2 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-2 a b -\frac {1}{2} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (6 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{4}}}{d}\) | \(195\) |
default | \(\frac {\frac {2 a \left (\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (3 a^{2}-2 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}-\frac {2 \left (\frac {\left (-2 a b -\frac {1}{2} b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a b +\frac {1}{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (6 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{4}}}{d}\) | \(195\) |
risch | \(-\frac {3 x \,a^{2}}{b^{4}}+\frac {x}{2 b^{2}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{2} d}-\frac {i a \,{\mathrm e}^{i \left (d x +c \right )}}{b^{3} d}+\frac {i a \,{\mathrm e}^{-i \left (d x +c \right )}}{b^{3} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{2} d}+\frac {2 i a^{2} \left (a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{4} d \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{4}}+\frac {2 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+a \sqrt {-a^{2}+b^{2}}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{4}}-\frac {2 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+a \sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}\) | \(435\) |
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Time = 0.30 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.55 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\left [-\frac {{\left (6 \, a^{4} b - 7 \, a^{2} b^{3} + b^{5}\right )} d x \cos \left (d x + c\right ) + {\left (6 \, a^{5} - 7 \, a^{3} b^{2} + a b^{4}\right )} d x - {\left (3 \, a^{4} - 2 \, a^{2} b^{2} + {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (6 \, a^{4} b - 6 \, a^{2} b^{3} - {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6}\right )} d\right )}}, -\frac {{\left (6 \, a^{4} b - 7 \, a^{2} b^{3} + b^{5}\right )} d x \cos \left (d x + c\right ) + {\left (6 \, a^{5} - 7 \, a^{3} b^{2} + a b^{4}\right )} d x - 2 \, {\left (3 \, a^{4} - 2 \, a^{2} b^{2} + {\left (3 \, a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (6 \, a^{4} b - 6 \, a^{2} b^{3} - {\left (a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{4} - a b^{6}\right )} d\right )}}\right ] \]
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Timed out. \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.33 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.55 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {4 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} b^{3}} - \frac {{\left (6 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{4}} - \frac {4 \, {\left (3 \, a^{3} - 2 \, a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{4}} + \frac {2 \, {\left (4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{3}}}{2 \, d} \]
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Time = 2.33 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.31 \[ \int \frac {\cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (6\,a^2+b^2\right )}{b^3}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,a^2+3\,a\,b-b^2\right )}{b^3}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (-6\,a^2+3\,a\,b+b^2\right )}{b^3}}{d\,\left (\left (a-b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (3\,a-b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (3\,a+b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a+b\right )}-\frac {\ln \left (b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sqrt {b^2-a^2}\right )\,\left (3\,a^3\,\sqrt {b^2-a^2}-2\,a\,b^2\,\sqrt {b^2-a^2}\right )}{b^4\,d\,\left (a^2-b^2\right )}-\frac {a\,\ln \left (a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sqrt {b^2-a^2}\right )\,\sqrt {-\left (a+b\right )\,\left (a-b\right )}\,\left (3\,a^2-2\,b^2\right )}{d\,\left (b^6-a^2\,b^4\right )}-\frac {\mathrm {atan}\left (\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\frac {8\,a}{b}+\frac {24\,a^2}{b^2}-\frac {24\,a^3}{b^3}+\frac {144\,a^4}{b^4}-\frac {144\,a^5}{b^5}-8}-\frac {8\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a-8\,b+\frac {24\,a^2}{b}-\frac {24\,a^3}{b^2}+\frac {144\,a^4}{b^3}-\frac {144\,a^5}{b^4}}-\frac {24\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,b+24\,a^2-8\,b^2-\frac {24\,a^3}{b}+\frac {144\,a^4}{b^2}-\frac {144\,a^5}{b^3}}+\frac {24\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,b^2+24\,a^2\,b-24\,a^3-8\,b^3+\frac {144\,a^4}{b}-\frac {144\,a^5}{b^2}}-\frac {144\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a\,b^3-24\,a^3\,b+144\,a^4-8\,b^4+24\,a^2\,b^2-\frac {144\,a^5}{b}}+\frac {144\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{-144\,a^5+144\,a^4\,b-24\,a^3\,b^2+24\,a^2\,b^3+8\,a\,b^4-8\,b^5}\right )\,\left (a^2\,6{}\mathrm {i}-b^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{b^4\,d} \]
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