Optimal. Leaf size=27 \[ -\frac {i (a+i a \tan (c+d x))^3}{3 a d} \]
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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 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))^2 \, dx &=-\frac {i \text {Subst}\left (\int (a+x)^2 \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac {i (a+i a \tan (c+d x))^3}{3 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. \(68\) vs. \(2(27)=54\).
time = 0.32, size = 68, normalized size = 2.52 \begin {gather*} \frac {a^2 \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 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 50 vs. \(2 (23 ) = 46\).
time = 0.27, size = 51, normalized size = 1.89
method | result | size |
risch | \(\frac {8 i a^{2} \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}+3 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(47\) |
derivativedivides | \(\frac {-\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {i a^{2}}{\cos \left (d x +c \right )^{2}}+a^{2} \tan \left (d x +c \right )}{d}\) | \(51\) |
default | \(\frac {-\frac {a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}+\frac {i a^{2}}{\cos \left (d x +c \right )^{2}}+a^{2} \tan \left (d x +c \right )}{d}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 21, normalized size = 0.78 \begin {gather*} -\frac {i \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{3 \, 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. 75 vs. \(2 (21) = 42\).
time = 0.33, size = 75, normalized size = 2.78 \begin {gather*} -\frac {8 \, {\left (-3 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 3 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{2}\right )}}{3 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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*} - a^{2} \left (\int \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \left (- 2 i \tan {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int \left (- \sec ^{2}{\left (c + d x \right )}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 42, normalized size = 1.56 \begin {gather*} -\frac {a^{2} \tan \left (d x + c\right )^{3} - 3 i \, a^{2} \tan \left (d x + c\right )^{2} - 3 \, a^{2} \tan \left (d x + c\right )}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.22, size = 35, normalized size = 1.30 \begin {gather*} \frac {a^2\,\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\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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