Optimal. Leaf size=34 \[ -\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\sec (a+b x) \tan (a+b x)}{2 b} \]
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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 = {2691, 3855}
\begin {gather*} \frac {\tan (a+b x) \sec (a+b x)}{2 b}-\frac {\tanh ^{-1}(\sin (a+b x))}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2691
Rule 3855
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 \begin {gather*} -\frac {\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac {\sec (a+b x) \tan (a+b x)}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 48, normalized size = 1.41
method | result | size |
derivativedivides | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{2 \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{2}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2}}{b}\) | \(48\) |
default | \(\frac {\frac {\sin ^{3}\left (b x +a \right )}{2 \cos \left (b x +a \right )^{2}}+\frac {\sin \left (b x +a \right )}{2}-\frac {\ln \left (\sec \left (b x +a \right )+\tan \left (b x +a \right )\right )}{2}}{b}\) | \(48\) |
risch | \(-\frac {i \left ({\mathrm e}^{3 i \left (b x +a \right )}-{\mathrm e}^{i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}{2 b}\) | \(78\) |
norman | \(\frac {\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}+\frac {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )}{b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{2}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{2 b}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{2 b}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 46, normalized size = 1.35 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (30) = 60\).
time = 0.41, size = 61, normalized size = 1.79 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \int \sin ^{2}{\left (a + b x \right )} \sec ^{3}{\left (a + b x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.67, size = 48, normalized size = 1.41 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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