Integrand size = 28, antiderivative size = 187 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{33 d e^8}+\frac {2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}+\frac {2 a^4 \sin (c+d x)}{33 d e^7 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}} \]
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Time = 0.27 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3577, 3854, 3856, 2720} \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{33 d e^8}+\frac {2 a^4 \sin (c+d x)}{33 d e^7 \sqrt {e \sec (c+d x)}}+\frac {2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}}-\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}} \]
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Rule 2720
Rule 3577
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}+\frac {a^2 \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{11/2}} \, dx}{5 e^2} \\ & = -\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}}+\frac {\left (7 a^4\right ) \int \frac {1}{(e \sec (c+d x))^{7/2}} \, dx}{55 e^4} \\ & = \frac {2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}-\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}}+\frac {a^4 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{11 e^6} \\ & = \frac {2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}+\frac {2 a^4 \sin (c+d x)}{33 d e^7 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}}+\frac {a^4 \int \sqrt {e \sec (c+d x)} \, dx}{33 e^8} \\ & = \frac {2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}+\frac {2 a^4 \sin (c+d x)}{33 d e^7 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}}+\frac {\left (a^4 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{33 e^8} \\ & = \frac {2 a^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{33 d e^8}+\frac {2 a^4 \sin (c+d x)}{55 d e^5 (e \sec (c+d x))^{5/2}}+\frac {2 a^4 \sin (c+d x)}{33 d e^7 \sqrt {e \sec (c+d x)}}-\frac {4 i a (a+i a \tan (c+d x))^3}{15 d (e \sec (c+d x))^{15/2}}-\frac {4 i \left (a^4+i a^4 \tan (c+d x)\right )}{55 d e^2 (e \sec (c+d x))^{11/2}} \\ \end{align*}
Time = 3.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.83 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=-\frac {i a^4 \sqrt {e \sec (c+d x)} \left (64+112 \cos (2 (c+d x))+48 \cos (4 (c+d x))-54 i \sin (2 (c+d x))-37 i \sin (4 (c+d x))+40 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (i \cos (4 (c+d x))+\sin (4 (c+d x)))\right ) (\cos (4 (c+2 d x))+i \sin (4 (c+2 d x)))}{660 d e^8 (\cos (d x)+i \sin (d x))^4} \]
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Time = 33.74 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {2 a^{4} \left (-88 i \left (\cos ^{7}\left (d x +c \right )\right )+88 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+60 i \left (\cos ^{5}\left (d x +c \right )\right )-16 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+5 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+5 i \sec \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+3 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+5 \sin \left (d x +c \right )\right )}{165 e^{7} d \sqrt {e \sec \left (d x +c \right )}}\) | \(217\) |
risch | \(-\frac {i {\mathrm e}^{i \left (d x +c \right )} \left (11 \,{\mathrm e}^{6 i \left (d x +c \right )}+47 \,{\mathrm e}^{4 i \left (d x +c \right )}+81 \,{\mathrm e}^{2 i \left (d x +c \right )}+85\right ) a^{4} \sqrt {2}}{1320 d \,e^{7} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}+\frac {2 \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right ) a^{4} \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{33 d \sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}\, e^{7} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(273\) |
parts | \(-\frac {2 a^{4} \left (-77 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-91 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+195 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}}+195 i \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}}-117 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-195 \sin \left (d x +c \right )\right )}{1155 d \sqrt {e \sec \left (d x +c \right )}\, e^{7}}+\frac {2 a^{4} \left (77 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-119 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )-20 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}}+12 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-20 i \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}}+20 \sin \left (d x +c \right )\right )}{1155 d \sqrt {e \sec \left (d x +c \right )}\, e^{7}}-\frac {8 i a^{4} \left (11 \left (\cos ^{7}\left (d x +c \right )\right )-15 \left (\cos ^{5}\left (d x +c \right )\right )\right )}{165 d \sqrt {e \sec \left (d x +c \right )}\, e^{7}}-\frac {8 i a^{4}}{15 d \left (e \sec \left (d x +c \right )\right )^{\frac {15}{2}}}+\frac {4 a^{4} \left (77 \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )-14 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+30 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}}+30 i \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}}-18 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-30 \sin \left (d x +c \right )\right )}{385 d \sqrt {e \sec \left (d x +c \right )}\, e^{7}}\) | \(646\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.66 \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\frac {-80 i \, \sqrt {2} a^{4} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right ) + \sqrt {2} {\left (-11 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 58 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 128 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 166 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} - 85 i \, a^{4}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{1320 \, d e^{8}} \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {15}{2}}} \,d x } \]
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\[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4}}{\left (e \sec \left (d x + c\right )\right )^{\frac {15}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+i a \tan (c+d x))^4}{(e \sec (c+d x))^{15/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^4}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{15/2}} \,d x \]
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