Optimal. Leaf size=179 \[ -\frac {2 i \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{7 d}+\frac {32 i a \sec ^4(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{35 d}+\frac {12 i a \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.38, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3515, 3497, 3502, 3488} \[ -\frac {2 i \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{7 d}+\frac {32 i a \sec ^4(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {16 i \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}{35 d}+\frac {12 i a \sec ^2(c+d x) (e \cos (c+d x))^{7/2}}{35 d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3488
Rule 3497
Rule 3502
Rule 3515
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx &=\left ((e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx\\ &=-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (6 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{7 e^2}\\ &=\frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}+\frac {\left (24 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{35 e^2}\\ &=\frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}+\frac {\left (16 a (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}\right ) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{35 e^4}\\ &=\frac {12 i a (e \cos (c+d x))^{7/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a (e \cos (c+d x))^{7/2} \sec ^4(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}{7 d}-\frac {16 i (e \cos (c+d x))^{7/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 d}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 80, normalized size = 0.45 \[ \frac {e^3 \sqrt {a+i a \tan (c+d x)} \sqrt {e \cos (c+d x)} (70 \sin (c+d x)+6 \sin (3 (c+d x))+35 i \cos (c+d x)+i \cos (3 (c+d x)))}{70 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 100, normalized size = 0.56 \[ \frac {\sqrt {2} \sqrt {\frac {1}{2}} {\left (-5 i \, e^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 35 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 105 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, e^{3}\right )} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {5}{2} i \, d x - \frac {5}{2} i \, c\right )}}{140 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.50, size = 97, normalized size = 0.54 \[ \frac {2 \left (i \left (\cos ^{3}\left (d x +c \right )\right )+6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+8 i \cos \left (d x +c \right )+16 \sin \left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}}{35 d \cos \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 202, normalized size = 1.13 \[ \frac {{\left (7 i \, e^{3} \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 i \, e^{3} \cos \left (\frac {7}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) - 35 i \, e^{3} \cos \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 105 i \, e^{3} \cos \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 7 \, e^{3} \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, e^{3} \sin \left (\frac {7}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 35 \, e^{3} \sin \left (\frac {3}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right ) + 105 \, e^{3} \sin \left (\frac {1}{5} \, \arctan \left (\sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ), \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right )\right )\right )\right )} \sqrt {a} \sqrt {e}}{140 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 96, normalized size = 0.54 \[ \frac {e^3\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (\sin \left (c+d\,x\right )+\frac {3\,\sin \left (3\,c+3\,d\,x\right )}{35}+\frac {\cos \left (c+d\,x\right )\,1{}\mathrm {i}}{2}+\frac {\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}}{70}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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