Optimal. Leaf size=164 \[ \frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a \sqrt {e \sec (c+d x)}}{35 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3578, 3583,
3569} \begin {gather*} \frac {32 i a \sqrt {e \sec (c+d x)}}{35 d e^4 \sqrt {a+i a \tan (c+d x)}}+\frac {12 i a}{35 d e^2 \sqrt {a+i a \tan (c+d x)} (e \sec (c+d x))^{3/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3578
Rule 3583
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{7/2}} \, dx &=-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}+\frac {(6 a) \int \frac {1}{(e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx}{7 e^2}\\ &=\frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}+\frac {24 \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx}{35 e^2}\\ &=\frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}}+\frac {(16 a) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{35 e^4}\\ &=\frac {12 i a}{35 d e^2 (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}+\frac {32 i a \sqrt {e \sec (c+d x)}}{35 d e^4 \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)}}{7 d (e \sec (c+d x))^{7/2}}-\frac {16 i \sqrt {a+i a \tan (c+d x)}}{35 d e^2 (e \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 80, normalized size = 0.49 \begin {gather*} \frac {(35 i \cos (c+d x)+i \cos (3 (c+d x))+70 \sin (c+d x)+6 \sin (3 (c+d x))) \sqrt {a+i a \tan (c+d x)}}{70 d e^3 \sqrt {e \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.89, size = 102, normalized size = 0.62
method | result | size |
default | \(\frac {2 \left (i \left (\cos ^{3}\left (d x +c \right )\right )+6 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+8 i \cos \left (d x +c \right )+16 \sin \left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {e}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}}}{35 d \,e^{7}}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.58, size = 177, normalized size = 1.08 \begin {gather*} \frac {\sqrt {a} {\left (7 i \, \cos \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) - 5 i \, \cos \left (\frac {7}{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 ) - 35 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 ) + 105 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 ) + 7 \, \sin \left (\frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 5 \, \sin \left (\frac {7}{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 ) + 35 \, \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 ) + 105 \, \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 {7}{2}\right )}}{140 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 91, normalized size = 0.55 \begin {gather*} \frac {\sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-5 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 40 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 112 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-\frac {5}{2} i \, d x - \frac {5}{2} i \, c - \frac {7}{2}\right )}}{140 \, 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 = 5.16, size = 109, normalized size = 0.66 \begin {gather*} \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 (\cos \left (2\,c+2\,d\,x\right )\,36{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,1{}\mathrm {i}+76\,\sin \left (2\,c+2\,d\,x\right )+6\,\sin \left (4\,c+4\,d\,x\right )+35{}\mathrm {i}\right )}{140\,d\,e^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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