Integrand size = 21, antiderivative size = 94 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {3 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]
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Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2650, 2716, 2721, 2719} \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=-\frac {3 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]
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Rule 2650
Rule 2716
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = -\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}+\frac {3}{2} \int \frac {1}{(d \cos (a+b x))^{3/2}} \, dx \\ & = -\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {3 \int \sqrt {d \cos (a+b x)} \, dx}{2 d^2} \\ & = -\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {\left (3 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{2 d^2 \sqrt {\cos (a+b x)}} \\ & = -\frac {\csc (a+b x)}{b d \sqrt {d \cos (a+b x)}}-\frac {3 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b d^2 \sqrt {\cos (a+b x)}}+\frac {3 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.69 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {-\cos (a+b x) \cot (a+b x)-3 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+2 \sin (a+b x)}{b d \sqrt {d \cos (a+b x)}} \]
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Time = 0.50 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.22
method | result | size |
default | \(-\frac {\sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, {\left (-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}^{\frac {3}{2}} \left (6 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}+12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1\right )}{2 d^{3} \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{5} {\left (2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}^{2} \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(209\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.39 \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\frac {-3 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {d} \cos \left (b x + a\right ) \sin \left (b x + a\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) - 2 \, \sqrt {d \cos \left (b x + a\right )} {\left (3 \, \cos \left (b x + a\right )^{2} - 2\right )}}{2 \, b d^{2} \cos \left (b x + a\right ) \sin \left (b x + a\right )} \]
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\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {\csc ^{2}{\left (a + b x \right )}}{\left (d \cos {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\left (d \cos \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\left (d\,\cos \left (a+b\,x\right )\right )}^{3/2}} \,d x \]
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