Integrand size = 23, antiderivative size = 23 \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Int}\left (\frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.77 (sec) , antiderivative size = 273, normalized size of antiderivative = 11.87 \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\frac {-i \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+12 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-6 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]+\frac {12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}}{18 a d} \]
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Time = 2.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 7.61
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a}}{d}\) | \(175\) |
default | \(\frac {\frac {-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a}}{a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{9 a}}{d}\) | \(175\) |
risch | \(\text {Expression too large to display}\) | \(1243\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.24 (sec) , antiderivative size = 36403, normalized size of antiderivative = 1582.74 \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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Not integrable
Time = 125.45 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )}}{\left (a + b \sin ^{3}{\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 4.41 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.13 \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}} \,d x } \]
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Time = 15.24 (sec) , antiderivative size = 1648, normalized size of antiderivative = 71.65 \[ \int \frac {\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx=\text {Too large to display} \]
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