Optimal. Leaf size=29 \[ \frac{\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac{\sin (a+b x)}{b} \]
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Rubi [A] time = 0.015958, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4576, 2637, 3770} \[ \frac{\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac{\sin (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4576
Rule 2637
Rule 3770
Rubi steps
\begin{align*} \int \sin (a+b x) \tan (c+b x) \, dx &=\cos (a-c) \int \sec (c+b x) \, dx-\int \cos (a+b x) \, dx\\ &=\frac{\tanh ^{-1}(\sin (c+b x)) \cos (a-c)}{b}-\frac{\sin (a+b x)}{b}\\ \end{align*}
Mathematica [C] time = 0.0486601, size = 94, normalized size = 3.24 \[ -\frac{2 i \cos (a-c) \tan ^{-1}\left (\frac{(\sin (c)+i \cos (c)) \left (\sin (c) \cos \left (\frac{b x}{2}\right )+\cos (c) \sin \left (\frac{b x}{2}\right )\right )}{\cos (c) \cos \left (\frac{b x}{2}\right )-i \sin (c) \cos \left (\frac{b x}{2}\right )}\right )}{b}-\frac{\sin (a) \cos (b x)}{b}-\frac{\cos (a) \sin (b x)}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.066, size = 99, normalized size = 3.4 \begin{align*}{\frac{{\frac{i}{2}}{{\rm e}^{i \left ( bx+a \right ) }}}{b}}-{\frac{{\frac{i}{2}}{{\rm e}^{-i \left ( bx+a \right ) }}}{b}}-{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}-i{{\rm e}^{i \left ( a-c \right ) }} \right ) \cos \left ( a-c \right ) }{b}}+{\frac{\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+i{{\rm e}^{i \left ( a-c \right ) }} \right ) \cos \left ( a-c \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.91388, size = 177, normalized size = 6.1 \begin{align*} -\frac{\cos \left (-a + c\right ) \log \left (\frac{\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} - 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} + 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}{\cos \left (b x + 2 \, c\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (b x + 2 \, c\right ) + \sin \left (b x + 2 \, c\right )^{2} - 2 \, \cos \left (b x + 2 \, c\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}}\right ) + 2 \, \sin \left (b x + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.516937, size = 509, normalized size = 17.55 \begin{align*} \frac{\sqrt{2} \sqrt{\cos \left (-2 \, a + 2 \, c\right ) + 1} \log \left (\frac{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \frac{2 \, \sqrt{2}{\left ({\left (\cos \left (-2 \, a + 2 \, c\right ) + 1\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right )\right )}}{\sqrt{\cos \left (-2 \, a + 2 \, c\right ) + 1}} - \cos \left (-2 \, a + 2 \, c\right ) - 3}{2 \, \cos \left (b x + a\right )^{2} \cos \left (-2 \, a + 2 \, c\right ) - 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) \sin \left (-2 \, a + 2 \, c\right ) - \cos \left (-2 \, a + 2 \, c\right ) + 1}\right ) - 4 \, \sin \left (b x + a\right )}{4 \, b} \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 \sin{\left (a + b x \right )} \tan{\left (b x + c \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x + a\right ) \tan \left (b x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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