Optimal. Leaf size=19 \[ a x+\frac {b \tan (c+d x)}{d}-b x \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3473, 8} \[ a x+\frac {b \tan (c+d x)}{d}-b x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \left (a+b \tan ^2(c+d x)\right ) \, dx &=a x+b \int \tan ^2(c+d x) \, dx\\ &=a x+\frac {b \tan (c+d x)}{d}-b \int 1 \, dx\\ &=a x-b x+\frac {b \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.47 \[ a x-\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 21, normalized size = 1.11 \[ \frac {{\left (a - b\right )} d x + b \tan \left (d x + c\right )}{d} \]
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 = 0.03, size = 29, normalized size = 1.53 \[ a x +\frac {b \tan \left (d x +c \right )}{d}-\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 23, normalized size = 1.21 \[ a x - \frac {{\left (d x + c - \tan \left (d x + c\right )\right )} b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.44, size = 21, normalized size = 1.11 \[ \frac {b\,\mathrm {tan}\left (c+d\,x\right )+d\,x\,\left (a-b\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 1.05 \[ a x + b \left (\begin {cases} - x + \frac {\tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \tan ^{2}{\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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