Optimal. Leaf size=34 \[ \frac {\tan (a+b x) \sec (a+b x)}{2 b}-\frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 34, 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 = {2611, 3770} \[ \frac {\tan (a+b x) \sec (a+b x)}{2 b}-\frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3770
Rubi steps
\begin {align*} \int \sec (a+b x) \tan ^2(a+b x) \, dx &=\frac {\sec (a+b x) \tan (a+b x)}{2 b}-\frac {1}{2} \int \sec (a+b x) \, dx\\ &=-\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\sec (a+b x) \tan (a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \[ \frac {\tan (a+b x) \sec (a+b x)}{2 b}-\frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 61, normalized size = 1.79 \[ -\frac {\cos \left (b x + a\right )^{2} \log \left (\sin \left (b x + a\right ) + 1\right ) - \cos \left (b x + a\right )^{2} \log \left (-\sin \left (b x + a\right ) + 1\right ) - 2 \, \sin \left (b x + a\right )}{4 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 48, normalized size = 1.41 \[ -\frac {\frac {2 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} + \log \left ({\left | \sin \left (b x + a\right ) + 1 \right |}\right ) - \log \left ({\left | \sin \left (b x + a\right ) - 1 \right |}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 53, normalized size = 1.56 \[ \frac {\sin ^{3}\left (b x +a \right )}{2 b \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{2 b}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 46, normalized size = 1.35 \[ -\frac {\frac {2 \, \sin \left (b x + a\right )}{\sin \left (b x + a\right )^{2} - 1} + \log \left (\sin \left (b x + a\right ) + 1\right ) - \log \left (\sin \left (b x + a\right ) - 1\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 69, normalized size = 2.03 \[ \frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^3+\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}{b\,\left ({\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2+1\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{b} \]
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 \[ \int \sin ^{2}{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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