Optimal. Leaf size=26 \[ \frac {\tan ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3767} \[ \frac {\tan ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \sec ^4(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (a+b x)\right )}{b}\\ &=\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 23, normalized size = 0.88 \[ \frac {\frac {1}{3} \tan ^3(a+b x)+\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 31, normalized size = 1.19 \[ \frac {{\left (2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 22, normalized size = 0.85 \[ \frac {\tan \left (b x + a\right )^{3} + 3 \, \tan \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 24, normalized size = 0.92 \[ -\frac {\left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (b x +a \right )\right )}{3}\right ) \tan \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 22, normalized size = 0.85 \[ \frac {\tan \left (b x + a\right )^{3} + 3 \, \tan \left (b x + a\right )}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 21, normalized size = 0.81 \[ \frac {\mathrm {tan}\left (a+b\,x\right )\,\left ({\mathrm {tan}\left (a+b\,x\right )}^2+3\right )}{3\,b} \]
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 \[ \int \sec ^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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