Optimal. Leaf size=15 \[ \frac {\sec ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2606, 30} \[ \frac {\sec ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rubi steps
\begin {align*} \int \sec ^2(a+b x) \tan (a+b x) \, dx &=\frac {\operatorname {Subst}(\int x \, dx,x,\sec (a+b x))}{b}\\ &=\frac {\sec ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\sec ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 13, normalized size = 0.87 \[ \frac {1}{2 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 13, normalized size = 0.87 \[ \frac {1}{2 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 14, normalized size = 0.93 \[ \frac {\sec ^{2}\left (b x +a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 17, normalized size = 1.13 \[ -\frac {1}{2 \, {\left (\sin \left (b x + a\right )^{2} - 1\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 13, normalized size = 0.87 \[ \frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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