Optimal. Leaf size=10 \[ \frac {\tan (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3767, 8} \[ \frac {\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rubi steps
\begin {align*} \int \sec ^2(a+b x) \, dx &=-\frac {\operatorname {Subst}(\int 1 \, dx,x,-\tan (a+b x))}{b}\\ &=\frac {\tan (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 10, normalized size = 1.00 \[ \frac {\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 18, normalized size = 1.80 \[ \frac {\sin \left (b x + a\right )}{b \cos \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 10, normalized size = 1.00 \[ \frac {\tan \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 11, normalized size = 1.10 \[ \frac {\tan \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 10, normalized size = 1.00 \[ \frac {\tan \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 10, normalized size = 1.00 \[ \frac {\mathrm {tan}\left (a+b\,x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 58, normalized size = 5.80 \[ \begin {cases} \tilde {\infty } x & \text {for}\: \left (a = - \frac {\pi }{2} \vee a = - b x - \frac {\pi }{2}\right ) \wedge \left (a = - b x - \frac {\pi }{2} \vee b = 0\right ) \\\frac {x}{\cos ^{2}{\relax (a )}} & \text {for}\: b = 0 \\- \frac {2 \tan {\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} - b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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