Optimal. Leaf size=81 \[ -\frac {4 i a \sqrt {a+i a \tan (c+d x)}}{5 d e^2 \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}} \]
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Rubi [A]
time = 0.10, 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 \sqrt {a+i a \tan (c+d x)}}{5 d e^2 \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}} \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))^{3/2}}{(e \sec (c+d x))^{5/2}} \, dx &=-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}}+\frac {(2 a) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx}{5 e^2}\\ &=-\frac {4 i a \sqrt {a+i a \tan (c+d x)}}{5 d e^2 \sqrt {e \sec (c+d x)}}-\frac {2 i (a+i a \tan (c+d x))^{3/2}}{5 d (e \sec (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 84, normalized size = 1.04 \begin {gather*} -\frac {2 a (\cos (d x)-i \sin (d x)) (\cos (c+2 d x)+i \sin (c+2 d x)) (3 i+2 \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}{5 d e (e \sec (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 86, normalized size = 1.06
method | result | size |
risch | \(-\frac {i a \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+5\right )}{5 e^{2} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(74\) |
default | \(-\frac {2 \left (i \left (\cos ^{2}\left (d x +c \right )\right )-\sin \left (d x +c \right ) \cos \left (d x +c \right )+2 i\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\cos ^{3}\left (d x +c \right )\right ) a}{5 d \,e^{5}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 58, normalized size = 0.72 \begin {gather*} \frac {{\left (-i \, a \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 i \, a \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, a \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a} e^{\left (-\frac {5}{2}\right )}}{5 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 73, normalized size = 0.90 \begin {gather*} \frac {{\left (-i \, a e^{\left (4 i \, d x + 4 i \, c\right )} - 6 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - 5 i \, a\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 {5}{2}\right )}}{5 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \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 = 4.71, size = 102, normalized size = 1.26 \begin {gather*} -\frac {a\,\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 (-\sin \left (c+d\,x\right )-\sin \left (3\,c+3\,d\,x\right )+\cos \left (c+d\,x\right )\,11{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,1{}\mathrm {i}\right )}{10\,d\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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