Optimal. Leaf size=59 \[ \frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f}+\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{a f^2} \]
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Rubi [A] time = 0.07, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3318, 4184, 3475} \[ \frac {(c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f}+\frac {2 d \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )\right )}{a f^2} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
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) \tan \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 {2 d \log \left (\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )\right )}{a f^2}+\frac {(c+d x) \tan \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 47, normalized size = 0.80 \[ \frac {f (c+d x) \tan \left (\frac {1}{4} (2 e+2 f x+\pi )\right )+2 d \log \left (\cos \left (\frac {1}{4} (2 e+2 f x+\pi )\right )\right )}{a f^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 101, normalized size = 1.71 \[ \frac {d f x + c f + {\left (d f x + c f\right )} \cos \left (f x + e\right ) + {\left (d \cos \left (f x + e\right ) - d \sin \left (f x + e\right ) + d\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + {\left (d f x + c f\right )} \sin \left (f x + e\right )}{a f^{2} \cos \left (f x + e\right ) - a f^{2} \sin \left (f x + e\right ) + a f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 697, normalized size = 11.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 123, normalized size = 2.08 \[ -\frac {2 c}{a f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d x}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) f}-\frac {d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right ) f}-\frac {d \ln \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a \,f^{2}}+\frac {2 d \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{a \,f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.18, size = 169, normalized size = 2.86 \[ \frac {\frac {{\left (2 \, {\left (f x + e\right )} \cos \left (f x + e\right ) + {\left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right )\right )} d}{a f \cos \left (f x + e\right )^{2} + a f \sin \left (f x + e\right )^{2} - 2 \, a f \sin \left (f x + e\right ) + a f} - \frac {2 \, d e}{a f - \frac {a f \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} + \frac {2 \, c}{a - \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 66, normalized size = 1.12 \[ \frac {2\,d\,\ln \left ({\mathrm {e}}^{e\,1{}\mathrm {i}}\,{\mathrm {e}}^{f\,x\,1{}\mathrm {i}}-\mathrm {i}\right )}{a\,f^2}+\frac {2\,\left (c+d\,x\right )}{a\,f\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}-\mathrm {i}\right )}-\frac {d\,x\,2{}\mathrm {i}}{a\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 272, normalized size = 4.61 \[ \begin {cases} - \frac {2 c f}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} - \frac {d f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} - \frac {d f x}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} + \frac {2 d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1 \right )} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} - \frac {2 d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 1 \right )}}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} - \frac {d \log {\left (\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} + \frac {d \log {\left (\tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 1 \right )}}{a f^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a f^{2}} & \text {for}\: f \neq 0 \\\frac {c x + \frac {d x^{2}}{2}}{- a \sin {\relax (e )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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