Integrand size = 21, antiderivative size = 41 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}} \]
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Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {16, 2721, 2719} \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{b^2 d \sqrt {\cos (c+d x)}} \]
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Rule 16
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {b \cos (c+d x)} \, dx}{b^2} \\ & = \frac {\sqrt {b \cos (c+d x)} \int \sqrt {\cos (c+d x)} \, dx}{b^2 \sqrt {\cos (c+d x)}} \\ & = \frac {2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {2 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(63)=126\).
Time = 2.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.51
method | result | size |
default | \(\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{b \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(144\) |
risch | \(-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {2}\, {\mathrm e}^{-i \left (d x +c \right )}}{d b \sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b \,{\mathrm e}^{-i \left (d x +c \right )}}}-\frac {i \left (-\frac {2 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}{b \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i E\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {b \,{\mathrm e}^{3 i \left (d x +c \right )}+b \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b \,{\mathrm e}^{i \left (d x +c \right )}}\, {\mathrm e}^{-i \left (d x +c \right )}}{d b \sqrt {\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) b \,{\mathrm e}^{-i \left (d x +c \right )}}}\) | \(325\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.61 \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\frac {i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - i \, \sqrt {2} \sqrt {b} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{b^{2} d} \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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