Optimal. Leaf size=46 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3486, 3767} \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {a \tan (c+d x)}{d}+\frac {i a \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3767
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x)) \, dx &=\frac {i a \sec ^4(c+d x)}{4 d}+a \int \sec ^4(c+d x) \, dx\\ &=\frac {i a \sec ^4(c+d x)}{4 d}-\frac {a \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {i a \sec ^4(c+d x)}{4 d}+\frac {a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 43, normalized size = 0.93 \[ \frac {a \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac {i a \sec ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 81, normalized size = 1.76 \[ \frac {24 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 16 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, a}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 48, normalized size = 1.04 \[ -\frac {-3 i \, a \tan \left (d x + c\right )^{4} - 4 \, a \tan \left (d x + c\right )^{3} - 6 i \, a \tan \left (d x + c\right )^{2} - 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 39, normalized size = 0.85 \[ \frac {\frac {i a}{4 \cos \left (d x +c \right )^{4}}-a \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 48, normalized size = 1.04 \[ \frac {3 i \, a \tan \left (d x + c\right )^{4} + 4 \, a \tan \left (d x + c\right )^{3} + 6 i \, a \tan \left (d x + c\right )^{2} + 12 \, a \tan \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.24, size = 48, normalized size = 1.04 \[ \frac {\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}+\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+\frac {1{}\mathrm {i}\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+a\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.29, size = 48, normalized size = 1.04 \[ \begin {cases} \frac {a \left (\frac {\tan ^{3}{\left (c + d x \right )}}{3} + \tan {\left (c + d x \right )}\right ) + \frac {i a \sec ^{4}{\left (c + d x \right )}}{4}}{d} & \text {for}\: d \neq 0 \\x \left (i a \tan {\relax (c )} + a\right ) \sec ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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