Optimal. Leaf size=96 \[ \frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3486, 3769, 3771, 2639} \[ \frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3486
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {a+i a \tan (c+d x)}{(e \sec (c+d x))^{5/2}} \, dx &=-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+a \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx\\ &=-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}+\frac {(3 a) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 e^2}\\ &=-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac {2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}+\frac {(3 a) \int \sqrt {\cos (c+d x)} \, dx}{5 e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}\\ &=-\frac {2 i a}{5 d (e \sec (c+d x))^{5/2}}+\frac {6 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d e^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {2 a \sin (c+d x)}{5 d e (e \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.80, size = 99, normalized size = 1.03 \[ -\frac {a (\tan (c+d x)-i) \left (-2 \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-e^{2 i (c+d x)}\right )-3 i \sin (2 (c+d x))+2 \cos (2 (c+d x))+2\right )}{5 d e^2 \sqrt {e \sec (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ \frac {\sqrt {2} {\left (-i \, a e^{\left (5 i \, d x + 5 i \, c\right )} + i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, a e^{\left (3 i \, d x + 3 i \, c\right )} - 4 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 7 i \, a e^{\left (i \, d x + i \, c\right )} - 5 i \, a\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 10 \, {\left (d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - d e^{3} e^{\left (i \, d x + i \, c\right )}\right )} {\rm integral}\left (\frac {\sqrt {2} {\left (-3 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 6 i \, a e^{\left (i \, d x + i \, c\right )} - 3 i \, a\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}}{5 \, {\left (d e^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 2 \, d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + d e^{3} e^{\left (i \, d x + i \, c\right )}\right )}}, x\right )}{10 \, {\left (d e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - d e^{3} e^{\left (i \, d x + i \, c\right )}\right )}} \]
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 \[ \int \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.82, size = 341, normalized size = 3.55 \[ \frac {2 a \left (3 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 i \cos \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )+3 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-3 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \EllipticE \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right )-i \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-\left (\cos ^{4}\left (d x +c \right )\right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+3 \cos \left (d x +c \right )\right )}{5 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3} \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {i \, a \tan \left (d x + c\right ) + a}{\left (e \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ i a \left (\int \left (- \frac {i}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\right )\, dx + \int \frac {\tan {\left (c + d x \right )}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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