Optimal. Leaf size=31 \[ \frac{\tan (c+d x) \log (\sin (c+d x))}{d \sqrt{b \tan ^2(c+d x)}} \]
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Rubi [A] time = 0.0160449, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3658, 3475} \[ \frac{\tan (c+d x) \log (\sin (c+d x))}{d \sqrt{b \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3475
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \tan ^2(c+d x)}} \, dx &=\frac{\tan (c+d x) \int \cot (c+d x) \, dx}{\sqrt{b \tan ^2(c+d x)}}\\ &=\frac{\log (\sin (c+d x)) \tan (c+d x)}{d \sqrt{b \tan ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0835878, size = 39, normalized size = 1.26 \[ \frac{\tan (c+d x) (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d \sqrt{b \tan ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 45, normalized size = 1.5 \begin{align*} -{\frac{\tan \left ( dx+c \right ) \left ( \ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) -2\,\ln \left ( \tan \left ( dx+c \right ) \right ) \right ) }{2\,d}{\frac{1}{\sqrt{b \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61233, size = 45, normalized size = 1.45 \begin{align*} -\frac{\frac{\log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt{b}} - \frac{2 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt{b}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31158, size = 119, normalized size = 3.84 \begin{align*} \frac{\sqrt{b \tan \left (d x + c\right )^{2}} \log \left (\frac{\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, b d \tan \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \tan ^{2}{\left (c + d x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.61994, size = 109, normalized size = 3.52 \begin{align*} \frac{\frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{\sqrt{b} \mathrm{sgn}\left (\tan \left (d x + c\right )\right )} - \frac{2 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{\sqrt{b} \mathrm{sgn}\left (\tan \left (d x + c\right )\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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