Optimal. Leaf size=27 \[ \frac {i (a-i a \tan (c+d x))^4}{4 a^7 d} \]
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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))^4}{4 a^7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3568
Rubi steps
\begin {align*} \int \frac {\sec ^8(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=-\frac {i \text {Subst}\left (\int (a-x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=\frac {i (a-i a \tan (c+d x))^4}{4 a^7 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. \(84\) vs. \(2(27)=54\).
time = 0.52, size = 84, normalized size = 3.11 \begin {gather*} \frac {\sec (c) \sec ^4(c+d x) (-3 i \cos (c)-2 i \cos (c+2 d x)-2 i \cos (3 c+2 d x)-3 \sin (c)+2 \sin (c+2 d x)-2 \sin (3 c+2 d x)+\sin (3 c+4 d x))}{4 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 21, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {i \left (\tan \left (d x +c \right )+i\right )^{4}}{4 d \,a^{3}}\) | \(21\) |
default | \(\frac {i \left (\tan \left (d x +c \right )+i\right )^{4}}{4 d \,a^{3}}\) | \(21\) |
risch | \(\frac {4 i}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
time = 0.29, size = 47, normalized size = 1.74 \begin {gather*} -\frac {-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} 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. 69 vs. \(2 (21) = 42\).
time = 0.36, size = 69, normalized size = 2.56 \begin {gather*} \frac {4 i}{a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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 ^{8}{\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. 47 vs. \(2 (21) = 42\).
time = 0.75, size = 47, normalized size = 1.74 \begin {gather*} -\frac {-i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 4 \, \tan \left (d x + c\right )}{4 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.34, size = 77, normalized size = 2.85 \begin {gather*} -\frac {\sin \left (c+d\,x\right )\,\left (-4\,{\cos \left (c+d\,x\right )}^3+{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,6{}\mathrm {i}+4\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^2-{\sin \left (c+d\,x\right )}^3\,1{}\mathrm {i}\right )}{4\,a^3\,d\,{\cos \left (c+d\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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