Optimal. Leaf size=28 \[ \frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}+x \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 8} \[ \frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}+x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \tan ^4(c+d x) \, dx &=\frac {\tan ^3(c+d x)}{3 d}-\int \tan ^2(c+d x) \, dx\\ &=-\frac {\tan (c+d x)}{d}+\frac {\tan ^3(c+d x)}{3 d}+\int 1 \, dx\\ &=x-\frac {\tan (c+d x)}{d}+\frac {\tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 1.36 \[ \frac {\tan ^{-1}(\tan (c+d x))}{d}+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 26, normalized size = 0.93 \[ \frac {\tan \left (d x + c\right )^{3} + 3 \, d x - 3 \, \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 1.25 \[ \frac {\tan ^{3}\left (d x +c \right )}{3 d}-\frac {\tan \left (d x +c \right )}{d}+\frac {d x +c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 29, normalized size = 1.04 \[ \frac {\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.50, size = 24, normalized size = 0.86 \[ x-\frac {\mathrm {tan}\left (c+d\,x\right )-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 27, normalized size = 0.96 \[ \begin {cases} x + \frac {\tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {\tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \tan ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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