Optimal. Leaf size=29 \[ -\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d} \]
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Rubi [A]
time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3568, 32}
\begin {gather*} -\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=-\frac {i \text {Subst}\left (\int (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {2 i (a+i a \tan (c+d x))^{5/2}}{5 a d}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(69\) vs. \(2(29)=58\).
time = 0.40, size = 69, normalized size = 2.38 \begin {gather*} \frac {2 a \sec ^2(c+d x) (\cos (d x)-i \sin (d x)) (-i \cos (2 c+3 d x)+\sin (2 c+3 d x)) \sqrt {a+i a \tan (c+d x)}}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 24, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d a}\) | \(24\) |
default | \(-\frac {2 i \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}}{5 d a}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 21, normalized size = 0.72 \begin {gather*} -\frac {2 i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{5 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 59 vs. \(2 (21) = 42\).
time = 0.41, size = 59, normalized size = 2.03 \begin {gather*} -\frac {8 i \, \sqrt {2} a \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (5 i \, d x + 5 i \, c\right )}}{5 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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 \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \sec ^{2}{\left (c + d x \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 = 1.36, size = 153, normalized size = 5.28 \begin {gather*} -\frac {4\,a\,\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 )\,7{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,4{}\mathrm {i}+\cos \left (6\,c+6\,d\,x\right )\,1{}\mathrm {i}-5\,\sin \left (2\,c+2\,d\,x\right )-4\,\sin \left (4\,c+4\,d\,x\right )-\sin \left (6\,c+6\,d\,x\right )+4{}\mathrm {i}\right )}{5\,d\,\left (15\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (6\,c+6\,d\,x\right )+10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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