Optimal. Leaf size=35 \[ a x+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3473, 8} \[ a x+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \left (a+b \tan ^4(c+d x)\right ) \, dx &=a x+b \int \tan ^4(c+d x) \, dx\\ &=a x+\frac {b \tan ^3(c+d x)}{3 d}-b \int \tan ^2(c+d x) \, dx\\ &=a x-\frac {b \tan (c+d x)}{d}+\frac {b \tan ^3(c+d x)}{3 d}+b \int 1 \, dx\\ &=a x+b x-\frac {b \tan (c+d x)}{d}+\frac {b \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 1.26 \[ a x+\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 32, normalized size = 0.91 \[ \frac {b \tan \left (d x + c\right )^{3} + 3 \, {\left (a + b\right )} d x - 3 \, b \tan \left (d x + c\right )}{3 \, d} \]
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 = 0.02, size = 43, normalized size = 1.23 \[ a x +\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}-\frac {b \tan \left (d x +c \right )}{d}+\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 34, normalized size = 0.97 \[ a x + \frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} b}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.59, size = 31, normalized size = 0.89 \[ \frac {\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-b\,\mathrm {tan}\left (c+d\,x\right )+d\,x\,\left (a+b\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 32, normalized size = 0.91 \[ a x + b \left (\begin {cases} x + \frac {\tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {\tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \tan ^{4}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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