Optimal. Leaf size=121 \[ \frac {2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {16 i \sqrt {e \sec (c+d x)}}{15 d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i \sqrt {a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.15, antiderivative size = 121, 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 = {3583, 3578,
3569} \begin {gather*} \frac {16 i \sqrt {e \sec (c+d x)}}{15 d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i \sqrt {a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}+\frac {2 i}{5 d \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3578
Rule 3583
Rubi steps
\begin {align*} \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx &=\frac {2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {4 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{5 a}\\ &=\frac {2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {8 i \sqrt {a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}+\frac {8 \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{15 e^2}\\ &=\frac {2 i}{5 d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {16 i \sqrt {e \sec (c+d x)}}{15 d e^2 \sqrt {a+i a \tan (c+d x)}}-\frac {8 i \sqrt {a+i a \tan (c+d x)}}{15 a d (e \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 68, normalized size = 0.56 \begin {gather*} -\frac {i \sec ^2(c+d x) (-15+\cos (2 (c+d x))+4 i \sin (2 (c+d x)))}{15 d (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.87, size = 105, normalized size = 0.87
method | result | size |
default | \(\frac {2 \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \left (3 i \left (\cos ^{3}\left (d x +c \right )\right )+3 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 i \cos \left (d x +c \right )+8 \sin \left (d x +c \right )\right )}{15 d \,e^{3} a}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 129, normalized size = 1.07 \begin {gather*} \frac {{\left (3 i \, \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 i \, \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 ) + 30 i \, \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 ) + 3 \, \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \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 ) + 30 \, \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 )} e^{\left (-\frac {3}{2}\right )}}{30 \, \sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 83, normalized size = 0.69 \begin {gather*} \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-5 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 25 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 33 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-\frac {5}{2} i \, d x - \frac {5}{2} i \, c - \frac {3}{2}\right )}}{30 \, a 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 {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, 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 = 4.03, size = 86, normalized size = 0.71 \begin {gather*} \frac {\sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\left (4\,\sin \left (2\,c+2\,d\,x\right )-\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}+15{}\mathrm {i}\right )}{15\,d\,e^2\,\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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