Optimal. Leaf size=81 \[ -\frac {4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3578, 3569}
\begin {gather*} -\frac {4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3578
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^{5/2}}{(e \sec (c+d x))^{7/2}} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}}+\frac {(2 a) \int \frac {(a+i a \tan (c+d x))^{3/2}}{(e \sec (c+d x))^{3/2}} \, dx}{7 e^2}\\ &=-\frac {4 i a (a+i a \tan (c+d x))^{3/2}}{21 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{7 d (e \sec (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 92, normalized size = 1.14 \begin {gather*} -\frac {2 a^2 (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x))) (5 i+2 \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}{21 d e^2 (e \sec (c+d x))^{3/2} (\cos (d x)+i \sin (d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.81, size = 105, normalized size = 1.30
method | result | size |
risch | \(-\frac {i a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{21 e^{3} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(88\) |
default | \(-\frac {2 \left (6 i \left (\cos ^{3}\left (d x +c \right )\right )-6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-i \cos \left (d x +c \right )-2 \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \left (\cos ^{4}\left (d x +c \right )\right ) a^{2}}{21 d \,e^{7}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 93, normalized size = 1.15 \begin {gather*} \frac {{\left (-7 i \, a^{2} \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) - 3 i \, a^{2} \cos \left (\frac {7}{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 ) + 7 \, a^{2} \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, a^{2} \sin \left (\frac {7}{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 )} \sqrt {a} e^{\left (-\frac {7}{2}\right )}}{21 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 88, normalized size = 1.09 \begin {gather*} \frac {{\left (-3 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} - 10 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} - 7 i \, a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c - \frac {7}{2}\right )}}{21 \, d \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.99, size = 112, normalized size = 1.38 \begin {gather*} -\frac {a^2\,\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 (\cos \left (2\,c+2\,d\,x\right )\,10{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,3{}\mathrm {i}-10\,\sin \left (2\,c+2\,d\,x\right )-3\,\sin \left (4\,c+4\,d\,x\right )+7{}\mathrm {i}\right )}{42\,d\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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