Optimal. Leaf size=58 \[ \frac {4 x}{a^3}+\frac {4 i \log (\cos (c+d x))}{a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {i \tan ^2(c+d x)}{2 a^3 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45}
\begin {gather*} \frac {i \tan ^2(c+d x)}{2 a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {4 i \log (\cos (c+d x))}{a^3 d}+\frac {4 x}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {i \text {Subst}\left (\int \frac {(a-x)^2}{a+x} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac {i \text {Subst}\left (\int \left (-3 a+x+\frac {4 a^2}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=\frac {4 x}{a^3}+\frac {4 i \log (\cos (c+d x))}{a^3 d}-\frac {3 \tan (c+d x)}{a^3 d}+\frac {i \tan ^2(c+d x)}{2 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 113, normalized size = 1.95 \begin {gather*} \frac {\sec (c) \sec ^2(c+d x) (2 d x \cos (3 c+2 d x)+2 \cos (c+2 d x) (d x+i \log (\cos (c+d x)))+\cos (c) (i+4 d x+4 i \log (\cos (c+d x)))+2 i \cos (3 c+2 d x) \log (\cos (c+d x))+3 \sin (c)-3 \sin (c+2 d x))}{2 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 41, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {-3 \tan \left (d x +c \right )+\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}-4 i \ln \left (\tan \left (d x +c \right )-i\right )}{d \,a^{3}}\) | \(41\) |
default | \(\frac {-3 \tan \left (d x +c \right )+\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}-4 i \ln \left (\tan \left (d x +c \right )-i\right )}{d \,a^{3}}\) | \(41\) |
risch | \(\frac {8 x}{a^{3}}+\frac {8 c}{a^{3} d}-\frac {2 i \left (2 \,{\mathrm e}^{2 i \left (d x +c \right )}+3\right )}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {4 i \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 45, normalized size = 0.78 \begin {gather*} \frac {\frac {i \, \tan \left (d x + c\right )^{2} - 6 \, \tan \left (d x + c\right )}{a^{3}} - \frac {8 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{3}}}{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. 113 vs. \(2 (52) = 104\).
time = 0.37, size = 113, normalized size = 1.95 \begin {gather*} \frac {2 \, {\left (4 \, d x e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d x + 2 \, {\left (4 \, d x - i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 2 \, {\left (-i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3 i\right )}}{a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} d} \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 {i \int \frac {\sec ^{6}{\left (c + d x \right )}}{\tan ^{3}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} + i}\, dx}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 128 vs. \(2 (52) = 104\).
time = 0.87, size = 128, normalized size = 2.21 \begin {gather*} \frac {2 \, {\left (\frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3}} - \frac {4 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}{a^{3}} + \frac {2 i \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{3}} + \frac {-3 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 i}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}}\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.37, size = 41, normalized size = 0.71 \begin {gather*} -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,8{}\mathrm {i}+6\,\mathrm {tan}\left (c+d\,x\right )-{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}}{2\,a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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