Integrand size = 24, antiderivative size = 58 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {a A \cos ^3(c+d x)}{5 d (a+a \sin (c+d x))^4}-\frac {A \cos ^3(c+d x)}{15 d (a+a \sin (c+d x))^3} \]
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Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2815, 2751, 2750} \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {A \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^3}-\frac {a A \cos ^3(c+d x)}{5 d (a \sin (c+d x)+a)^4} \]
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Rule 2750
Rule 2751
Rule 2815
Rubi steps \begin{align*} \text {integral}& = (a A) \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx \\ & = -\frac {a A \cos ^3(c+d x)}{5 d (a+a \sin (c+d x))^4}+\frac {1}{5} A \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx \\ & = -\frac {a A \cos ^3(c+d x)}{5 d (a+a \sin (c+d x))^4}-\frac {A \cos ^3(c+d x)}{15 d (a+a \sin (c+d x))^3} \\ \end{align*}
Time = 1.55 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.59 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {A \left (-15 \cos \left (c+\frac {d x}{2}\right )+5 \cos \left (c+\frac {3 d x}{2}\right )+5 \sin \left (\frac {d x}{2}\right )+\sin \left (2 c+\frac {5 d x}{2}\right )\right )}{30 a^3 d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
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Result contains complex when optimal does not.
Time = 0.56 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.07
method | result | size |
risch | \(\frac {2 i A \left (-5 i {\mathrm e}^{2 i \left (d x +c \right )}+15 \,{\mathrm e}^{3 i \left (d x +c \right )}-i-5 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{15 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{5}}\) | \(62\) |
parallelrisch | \(-\frac {2 A \left (15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+25 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )}{15 d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(75\) |
derivativedivides | \(\frac {2 A \left (\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {14}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {8}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\right )}{d \,a^{3}}\) | \(86\) |
default | \(\frac {2 A \left (\frac {3}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {14}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {8}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\right )}{d \,a^{3}}\) | \(86\) |
norman | \(\frac {-\frac {8 A}{15 a d}-\frac {58 A \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}-\frac {8 A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {16 A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}-\frac {2 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 A \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\) | \(161\) |
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Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (54) = 108\).
Time = 0.25 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.66 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {A \cos \left (d x + c\right )^{3} - 2 \, A \cos \left (d x + c\right )^{2} + 3 \, A \cos \left (d x + c\right ) - {\left (A \cos \left (d x + c\right )^{2} + 3 \, A \cos \left (d x + c\right ) + 6 \, A\right )} \sin \left (d x + c\right ) + 6 \, A}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (53) = 106\).
Time = 2.32 (sec) , antiderivative size = 573, normalized size of antiderivative = 9.88 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=\begin {cases} - \frac {30 A \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} - \frac {30 A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} - \frac {50 A \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} - \frac {10 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} - \frac {8 A}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \left (- A \sin {\left (c \right )} + A\right )}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 387, normalized size of antiderivative = 6.67 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (\frac {A {\left (\frac {20 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} - \frac {3 \, A {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}\right )}}{15 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.36 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2 \, {\left (15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A\right )}}{15 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}} \]
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Time = 12.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.31 \[ \int \frac {A-A \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {2\,A\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+25\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}{15\,a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^5} \]
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