Integrand size = 28, antiderivative size = 124 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}} \]
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Time = 0.17 (sec) , antiderivative size = 124, 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 = {3578, 3577, 3856, 2719} \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}} \]
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Rule 2719
Rule 3577
Rule 3578
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}+\frac {a \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{5/2}} \, dx}{3 e^2} \\ & = -\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}+\frac {a^3 \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{15 e^4} \\ & = -\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}}+\frac {a^3 \int \sqrt {\cos (c+d x)} \, dx}{15 e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {2 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d e^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^3}{9 d (e \sec (c+d x))^{9/2}}-\frac {4 i \left (a^3+i a^3 \tan (c+d x)\right )}{15 d e^2 (e \sec (c+d x))^{5/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.95 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.95 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=-\frac {a^3 e^{-2 i (c+d x)} \left (11+16 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+4 \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (-i+\tan (c+d x))^3}{90 d e^2 (e \sec (c+d x))^{5/2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (132 ) = 264\).
Time = 14.16 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73
method | result | size |
risch | \(-\frac {i \left (5 \,{\mathrm e}^{4 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}+12\right ) a^{3} \sqrt {2}}{90 d \,e^{4} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i E\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a^{3} \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{15 d \,e^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(339\) |
default | \(\frac {2 i a^{3} \left (-20 i \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-20 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-20 \left (\cos ^{5}\left (d x +c \right )\right )+3 \cos \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}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )-3 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}}\, \cos \left (d x +c \right )-i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-20 \left (\cos ^{4}\left (d x +c \right )\right )+6 \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}}\, E\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )-6 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}}-i \cos \left (d x +c \right ) \sin \left (d x +c \right )+9 \left (\cos ^{3}\left (d x +c \right )\right )+3 \sec \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\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}}-3 \sec \left (d x +c \right ) \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 ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}-3 i \sin \left (d x +c \right )+9 \left (\cos ^{2}\left (d x +c \right )\right )\right )}{45 e^{4} d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}}\) | \(497\) |
parts | \(\text {Expression too large to display}\) | \(976\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\frac {12 i \, \sqrt {2} a^{3} \sqrt {e} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-5 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 16 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 11 i \, a^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{90 \, d e^{5}} \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{9/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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