Integrand size = 28, antiderivative size = 139 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {6 i a^3 \sqrt {e \sec (c+d x)}}{d}+\frac {6 a^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2}{5 d}+\frac {6 i \sqrt {e \sec (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{5 d} \]
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Time = 0.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3579, 3567, 3856, 2720} \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {6 i a^3 \sqrt {e \sec (c+d x)}}{d}+\frac {6 i \left (a^3+i a^3 \tan (c+d x)\right ) \sqrt {e \sec (c+d x)}}{5 d}+\frac {6 a^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{d}+\frac {2 i a (a+i a \tan (c+d x))^2 \sqrt {e \sec (c+d x)}}{5 d} \]
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Rule 2720
Rule 3567
Rule 3579
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2}{5 d}+\frac {1}{5} (9 a) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx \\ & = \frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2}{5 d}+\frac {6 i \sqrt {e \sec (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\left (3 a^2\right ) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx \\ & = \frac {6 i a^3 \sqrt {e \sec (c+d x)}}{d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2}{5 d}+\frac {6 i \sqrt {e \sec (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\left (3 a^3\right ) \int \sqrt {e \sec (c+d x)} \, dx \\ & = \frac {6 i a^3 \sqrt {e \sec (c+d x)}}{d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2}{5 d}+\frac {6 i \sqrt {e \sec (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{5 d}+\left (3 a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {6 i a^3 \sqrt {e \sec (c+d x)}}{d}+\frac {6 a^3 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{d}+\frac {2 i a \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2}{5 d}+\frac {6 i \sqrt {e \sec (c+d x)} \left (a^3+i a^3 \tan (c+d x)\right )}{5 d} \\ \end{align*}
Time = 2.37 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.57 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 \sec ^2(c+d x) \sqrt {e \sec (c+d x)} \left (18 i+20 i \cos (2 (c+d x))+30 \cos ^{\frac {5}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-5 \sin (2 (c+d x))\right )}{5 d} \]
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Time = 14.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {2 a^{3} \sqrt {e \sec \left (d x +c \right )}\, \left (15 i \cos \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+15 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+20 i-5 \tan \left (d x +c \right )-i \left (\sec ^{2}\left (d x +c \right )\right )\right )}{5 d}\) | \(157\) |
parts | \(-\frac {2 i a^{3} \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )}{d}-\frac {i a^{3} \sqrt {e \sec \left (d x +c \right )}\, \left (5 \cos \left (d x +c \right ) \ln \left (\frac {2 \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-\cos \left (d x +c \right )+1}{\cos \left (d x +c \right )+1}\right )-5 \cos \left (d x +c \right ) \ln \left (\frac {4 \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+4 \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-2 \cos \left (d x +c \right )+2}{\cos \left (d x +c \right )+1}\right )-20 \cos \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-20 \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+4 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}+4 \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\right )}{10 d \sqrt {-\frac {\cos \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cos \left (d x +c \right )+1\right )}+\frac {6 i a^{3} \sqrt {e \sec \left (d x +c \right )}}{d}-\frac {2 a^{3} \sqrt {e \sec \left (d x +c \right )}\, \left (2 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 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}}+\tan \left (d x +c \right )\right )}{d}\) | \(560\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (-25 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 36 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 15 i \, a^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 15 \, \sqrt {2} {\left (i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=- i a^{3} \left (\int i \sqrt {e \sec {\left (c + d x \right )}}\, dx + \int \left (- 3 \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )}\right )\, dx + \int \sqrt {e \sec {\left (c + d x \right )}} \tan ^{3}{\left (c + d x \right )}\, dx + \int \left (- 3 i \sqrt {e \sec {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}\right )\, dx\right ) \]
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\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\int { \sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Exception generated. \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^3 \, dx=\int \sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]
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