Optimal. Leaf size=175 \[ \frac {2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt {a+i a \tan (c+d x)}}+\frac {16 i (e \cos (c+d x))^{5/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {12 i (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {32 i (e \cos (c+d x))^{5/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d} \]
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Rubi [A]
time = 0.25, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3596, 3583,
3578, 3569} \begin {gather*} -\frac {12 i \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{35 a d}+\frac {2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt {a+i a \tan (c+d x)}}-\frac {32 i \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{5/2}}{35 a d}+\frac {16 i \sec ^2(c+d x) (e \cos (c+d x))^{5/2}}{35 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
Rule 3578
Rule 3583
Rule 3596
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{5/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx &=\left ((e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {1}{(e \sec (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}} \, dx\\ &=\frac {2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt {a+i a \tan (c+d x)}}+\frac {\left (6 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{5/2}} \, dx}{7 a}\\ &=\frac {2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt {a+i a \tan (c+d x)}}-\frac {12 i (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}{35 a d}+\frac {\left (24 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {1}{\sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx}{35 e^2}\\ &=\frac {2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt {a+i a \tan (c+d x)}}+\frac {16 i (e \cos (c+d x))^{5/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {12 i (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}{35 a d}+\frac {\left (16 (e \cos (c+d x))^{5/2} (e \sec (c+d x))^{5/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}} \, dx}{35 a e^2}\\ &=\frac {2 i (e \cos (c+d x))^{5/2}}{7 d \sqrt {a+i a \tan (c+d x)}}+\frac {16 i (e \cos (c+d x))^{5/2} \sec ^2(c+d x)}{35 d \sqrt {a+i a \tan (c+d x)}}-\frac {12 i (e \cos (c+d x))^{5/2} \sqrt {a+i a \tan (c+d x)}}{35 a d}-\frac {32 i (e \cos (c+d x))^{5/2} \sec ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{35 a d}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 80, normalized size = 0.46 \begin {gather*} -\frac {i e^3 (35 \cos (c+d x)+\cos (3 (c+d x))+70 i \sin (c+d x)+6 i \sin (3 (c+d x)))}{70 d \sqrt {e \cos (c+d x)} \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.05, size = 110, normalized size = 0.63
method | result | size |
default | \(\frac {2 \sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (5 i \left (\cos ^{4}\left (d x +c \right )\right )+5 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )+2 i \left (\cos ^{2}\left (d x +c \right )\right )+8 \sin \left (d x +c \right ) \cos \left (d x +c \right )-16 i\right )}{35 d \cos \left (d x +c \right )^{2} a}\) | \(110\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.60, size = 177, normalized size = 1.01 \begin {gather*} \frac {{\left (5 i \, \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) - 7 i \, \cos \left (\frac {5}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 35 i \, \cos \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) - 105 i \, \cos \left (\frac {1}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 5 \, \sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ) + 7 \, \sin \left (\frac {5}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 35 \, \sin \left (\frac {3}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right ) + 105 \, \sin \left (\frac {1}{7} \, \arctan \left (\sin \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right ), \cos \left (\frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )\right )\right )} e^{\frac {5}{2}}}{140 \, \sqrt {a} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 94, normalized size = 0.54 \begin {gather*} \frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (5 i \, e^{\frac {5}{2}} - 7 i \, e^{\left (6 i \, d x + 6 i \, c + \frac {5}{2}\right )} - 105 i \, e^{\left (4 i \, d x + 4 i \, c + \frac {5}{2}\right )} + 35 i \, e^{\left (2 i \, d x + 2 i \, c + \frac {5}{2}\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {7}{2} i \, d x - \frac {7}{2} i \, c\right )}}{140 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 5.24, size = 110, normalized size = 0.63 \begin {gather*} \frac {e^2\,\sqrt {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 )\,28{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,5{}\mathrm {i}+42\,\sin \left (2\,c+2\,d\,x\right )+5\,\sin \left (4\,c+4\,d\,x\right )-105{}\mathrm {i}\right )}{140\,a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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