Optimal. Leaf size=76 \[ -\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )\right )}{a d^2}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {e x}{a}+\frac {f x^2}{2 a} \]
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Rubi [A] time = 0.10, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4515, 3318, 4184, 3475} \[ -\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )\right )}{a d^2}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {e x}{a}+\frac {f x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 3318
Rule 3475
Rule 4184
Rule 4515
Rubi steps
\begin {align*} \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x) \, dx}{a}-\int \frac {e+f x}{a+a \sin (c+d x)} \, dx\\ &=\frac {e x}{a}+\frac {f x^2}{2 a}-\frac {\int (e+f x) \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a}\\ &=\frac {e x}{a}+\frac {f x^2}{2 a}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {f \int \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}\\ &=\frac {e x}{a}+\frac {f x^2}{2 a}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}\\ \end {align*}
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Mathematica [B] time = 0.57, size = 199, normalized size = 2.62 \[ \frac {\cos \left (\frac {d x}{2}\right ) \left (d^2 x (2 e+f x)-4 f \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+2 d^2 e x \sin \left (c+\frac {d x}{2}\right )+d^2 f x^2 \sin \left (c+\frac {d x}{2}\right )+2 d f x \cos \left (c+\frac {d x}{2}\right )-4 f \sin \left (c+\frac {d x}{2}\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 d e \sin \left (\frac {d x}{2}\right )-2 d f x \sin \left (\frac {d x}{2}\right )}{2 a d^2 \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 151, normalized size = 1.99 \[ \frac {d^{2} f x^{2} + 2 \, d e + 2 \, {\left (d^{2} e + d f\right )} x + {\left (d^{2} f x^{2} + 2 \, d e + 2 \, {\left (d^{2} e + d f\right )} x\right )} \cos \left (d x + c\right ) - 2 \, {\left (f \cos \left (d x + c\right ) + f \sin \left (d x + c\right ) + f\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (d^{2} f x^{2} - 2 \, d e + 2 \, {\left (d^{2} e - d f\right )} x\right )} \sin \left (d x + c\right )}{2 \, {\left (a d^{2} \cos \left (d x + c\right ) + a d^{2} \sin \left (d x + c\right ) + a d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.71, size = 772, normalized size = 10.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 446, normalized size = 5.87 \[ \frac {2 e \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {2 e}{a d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f x}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) d}+\frac {f x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) d}+\frac {f \,x^{2}}{2 a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f \,x^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) d}-\frac {f x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) d}+\frac {f \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}-\frac {2 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 273, normalized size = 3.59 \[ -\frac {4 \, c f {\left (\frac {1}{a d + \frac {a d \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d}\right )} - 4 \, e {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )} - \frac {{\left ({\left (d x + c\right )}^{2} \cos \left (d x + c\right )^{2} + {\left (d x + c\right )}^{2} \sin \left (d x + c\right )^{2} + 2 \, {\left (d x + c\right )}^{2} \sin \left (d x + c\right ) + {\left (d x + c\right )}^{2} + 4 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )\right )} f}{a d \cos \left (d x + c\right )^{2} + a d \sin \left (d x + c\right )^{2} + 2 \, a d \sin \left (d x + c\right ) + a d}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 80, normalized size = 1.05 \[ \frac {f\,x^2}{2\,a}-\frac {2\,f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{a\,d^2}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {x\,\left (d\,e+f\,2{}\mathrm {i}\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.00, size = 456, normalized size = 6.00 \[ \begin {cases} \frac {2 d^{2} e x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 d^{2} e x}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {d^{2} f x^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {d^{2} f x^{2}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {4 d e}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {2 d f x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 d f x}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {4 f \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {4 f \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 f \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 f \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} & \text {for}\: d \neq 0 \\\frac {\left (e x + \frac {f x^{2}}{2}\right ) \sin {\relax (c )}}{a \sin {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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