Optimal. Leaf size=122 \[ \frac {8 i a}{15 d e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{15 d e^2 \sqrt {e \sec (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3497, 3502, 3488} \[ \frac {8 i a}{15 d e^2 \sqrt {a+i a \tan (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{15 d e^2 \sqrt {e \sec (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3488
Rule 3497
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx &=-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}+\frac {(4 a) \int \frac {1}{\sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx}{5 e^2}\\ &=\frac {8 i a}{15 d e^2 \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}+\frac {8 \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx}{15 e^2}\\ &=\frac {8 i a}{15 d e^2 \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{5 d (e \sec (c+d x))^{5/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{15 d e^2 \sqrt {e \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 63, normalized size = 0.52 \[ \frac {i \sqrt {a+i a \tan (c+d x)} (-4 i \sin (2 (c+d x))+\cos (2 (c+d x))-15)}{15 d e^2 \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 86, normalized size = 0.70 \[ \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-3 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 33 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 25 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i\right )} e^{\left (-\frac {3}{2} i \, d x - \frac {3}{2} i \, c\right )}}{30 \, d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {i \, a \tan \left (d x + c\right ) + a}}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.30, size = 85, normalized size = 0.70 \[ \frac {2 \left (i \left (\cos ^{2}\left (d x +c \right )\right )+4 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d \,e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 130, normalized size = 1.07 \[ \frac {\sqrt {a} {\left (5 i \, \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - 3 i \, \cos \left (\frac {5}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right ) - 30 i \, \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, \sin \left (\frac {5}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right ) + 30 \, \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ), \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right )\right )\right )\right )}}{30 \, d e^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 101, normalized size = 0.83 \[ \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (2\,c+2\,d\,x\right )+1+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,c+2\,d\,x\right )+1}}\,\left (4\,\sin \left (c+d\,x\right )+4\,\sin \left (3\,c+3\,d\,x\right )-\cos \left (c+d\,x\right )\,29{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}\right )}{30\,d\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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