Optimal. Leaf size=60 \[ \frac{2 d \log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )\right )}{a f^2}-\frac{(c+d x) \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{a f} \]
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Rubi [A] time = 0.0636577, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3318, 4184, 3475} \[ \frac{2 d \log \left (\sin \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )\right )}{a f^2}-\frac{(c+d x) \cot \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{a f} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int \frac{c+d x}{a+a \sin (e+f x)} \, dx &=\frac{\int (c+d x) \csc ^2\left (\frac{1}{2} \left (e+\frac{\pi }{2}\right )+\frac{f x}{2}\right ) \, dx}{2 a}\\ &=-\frac{(c+d x) \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{d \int \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{a f}\\ &=-\frac{(c+d x) \cot \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{2 d \log \left (\sin \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )\right )}{a f^2}\\ \end{align*}
Mathematica [A] time = 0.149789, size = 51, normalized size = 0.85 \[ \frac{f (c+d x) \tan \left (\frac{1}{4} (2 e+2 f x-\pi )\right )+2 d \log \left (\cos \left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )}{a f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.043, size = 122, normalized size = 2. \begin{align*} -2\,{\frac{c}{af \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }}+{\frac{dx}{af}\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}-{\frac{dx}{af} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) ^{-1}}+2\,{\frac{d\ln \left ( \tan \left ( 1/2\,fx+e/2 \right ) +1 \right ) }{a{f}^{2}}}-{\frac{d}{a{f}^{2}}\ln \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.985481, size = 228, normalized size = 3.8 \begin{align*} -\frac{\frac{{\left (2 \,{\left (f x + e\right )} \cos \left (f x + e\right ) -{\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right )\right )} d}{a f \cos \left (f x + e\right )^{2} + a f \sin \left (f x + e\right )^{2} + 2 \, a f \sin \left (f x + e\right ) + a f} - \frac{2 \, d e}{a f + \frac{a f \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} + \frac{2 \, c}{a + \frac{a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77034, size = 251, normalized size = 4.18 \begin{align*} -\frac{d f x + c f +{\left (d f x + c f\right )} \cos \left (f x + e\right ) -{\left (d \cos \left (f x + e\right ) + d \sin \left (f x + e\right ) + d\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) -{\left (d f x + c f\right )} \sin \left (f x + e\right )}{a f^{2} \cos \left (f x + e\right ) + a f^{2} \sin \left (f x + e\right ) + a f^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.12534, size = 272, normalized size = 4.53 \begin{align*} \begin{cases} - \frac{2 c f}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} + \frac{d f x \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} - \frac{d f x}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} + \frac{2 d \log{\left (\tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 1 \right )} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} + \frac{2 d \log{\left (\tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 1 \right )}}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} - \frac{d \log{\left (\tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 1 \right )} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )}}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} - \frac{d \log{\left (\tan ^{2}{\left (\frac{e}{2} + \frac{f x}{2} \right )} + 1 \right )}}{a f^{2} \tan{\left (\frac{e}{2} + \frac{f x}{2} \right )} + a f^{2}} & \text{for}\: f \neq 0 \\\frac{c x + \frac{d x^{2}}{2}}{a \sin{\left (e \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24122, size = 940, normalized size = 15.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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