Optimal. Leaf size=27 \[ \frac {i (a-i a \tan (c+d x))^3}{3 a^5 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32}
\begin {gather*} \frac {i (a-i a \tan (c+d x))^3}{3 a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^2 \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=\frac {i (a-i a \tan (c+d x))^3}{3 a^5 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. \(68\) vs. \(2(27)=54\).
time = 0.29, size = 68, normalized size = 2.52 \begin {gather*} \frac {\sec (c) \sec ^3(c+d x) (-3 i \cos (d x)-3 i \cos (2 c+d x)+3 \sin (d x)-3 \sin (2 c+d x)+2 \sin (2 c+3 d x))}{6 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 20, normalized size = 0.74
method | result | size |
derivativedivides | \(-\frac {\left (\tan \left (d x +c \right )+i\right )^{3}}{3 d \,a^{2}}\) | \(20\) |
default | \(-\frac {\left (\tan \left (d x +c \right )+i\right )^{3}}{3 d \,a^{2}}\) | \(20\) |
risch | \(\frac {8 i}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 35, normalized size = 1.30 \begin {gather*} -\frac {\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} 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. 54 vs. \(2 (21) = 42\).
time = 0.36, size = 54, normalized size = 2.00 \begin {gather*} \frac {8 i}{3 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} 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*} - \frac {\int \frac {\sec ^{6}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 35, normalized size = 1.30 \begin {gather*} -\frac {\tan \left (d x + c\right )^{3} + 3 i \, \tan \left (d x + c\right )^{2} - 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.34, size = 33, normalized size = 1.22 \begin {gather*} -\frac {\mathrm {tan}\left (c+d\,x\right )\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}-3\right )}{3\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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