Integrand size = 28, antiderivative size = 116 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3581, 3854, 3856, 2720} \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=\frac {4 i e^2}{7 d \left (a^2+i a^2 \tan (c+d x)\right ) (e \sec (c+d x))^{3/2}}+\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d} \]
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Rule 2720
Rule 3581
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (3 e^2\right ) \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a^2} \\ & = \frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\int \sqrt {e \sec (c+d x)} \, dx}{7 a^2} \\ & = \frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2} \\ & = \frac {2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{7 a^2 d}+\frac {2 e \sin (c+d x)}{7 a^2 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2}{7 d (e \sec (c+d x))^{3/2} \left (a^2+i a^2 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.48 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\sec ^2(c+d x) \sqrt {e \sec (c+d x)} \left (2 i+2 i \cos (2 (c+d x))+2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))-\sin (2 (c+d x))\right )}{7 a^2 d (-i+\tan (c+d x))^2} \]
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Time = 5.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.53
method | result | size |
default | \(-\frac {2 \sqrt {e \sec \left (d x +c \right )}\, \left (i \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-2 i \left (\cos ^{4}\left (d x +c \right )\right )+i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-2 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-\sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{7 a^{2} d}\) | \(177\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (3 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - 4 i \, \sqrt {2} \sqrt {e} e^{\left (4 i \, d x + 4 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{14 \, a^{2} d} \]
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\[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {\sqrt {e \sec {\left (c + d x \right )}}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
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Exception generated. \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=\int { \frac {\sqrt {e \sec \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e \sec (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx=\int \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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