Optimal. Leaf size=70 \[ -\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x}{a^2}-\frac{\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.109522, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2859, 2682, 2635, 8} \[ -\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{x}{a^2}-\frac{\cos ^5(c+d x)}{d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2859
Rule 2682
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^2} \, dx &=-\frac{\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac{2 \int \frac{\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{a}\\ &=-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac{2 \int \cos ^2(c+d x) \, dx}{a^2}\\ &=-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac{\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}-\frac{\int 1 \, dx}{a^2}\\ &=-\frac{x}{a^2}-\frac{2 \cos ^3(c+d x)}{3 a^2 d}-\frac{\cos (c+d x) \sin (c+d x)}{a^2 d}-\frac{\cos ^5(c+d x)}{d (a+a \sin (c+d x))^2}\\ \end{align*}
Mathematica [B] time = 0.747531, size = 204, normalized size = 2.91 \[ \frac{-24 d x \sin \left (\frac{c}{2}\right )+21 \sin \left (\frac{c}{2}+d x\right )-21 \sin \left (\frac{3 c}{2}+d x\right )+6 \sin \left (\frac{3 c}{2}+2 d x\right )+6 \sin \left (\frac{5 c}{2}+2 d x\right )-\sin \left (\frac{5 c}{2}+3 d x\right )+\sin \left (\frac{7 c}{2}+3 d x\right )-2 \cos \left (\frac{c}{2}\right ) (12 d x+1)-21 \cos \left (\frac{c}{2}+d x\right )-21 \cos \left (\frac{3 c}{2}+d x\right )+6 \cos \left (\frac{3 c}{2}+2 d x\right )-6 \cos \left (\frac{5 c}{2}+2 d x\right )+\cos \left (\frac{5 c}{2}+3 d x\right )+\cos \left (\frac{7 c}{2}+3 d x\right )+2 \sin \left (\frac{c}{2}\right )}{24 a^2 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.089, size = 177, normalized size = 2.5 \begin{align*} -2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-8\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{2} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-{\frac{10}{3\,d{a}^{2}} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}-2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.67655, size = 248, normalized size = 3.54 \begin{align*} \frac{2 \,{\left (\frac{\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5}{a^{2} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09594, size = 115, normalized size = 1.64 \begin{align*} \frac{\cos \left (d x + c\right )^{3} - 3 \, d x + 3 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, \cos \left (d x + c\right )}{3 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 44.2726, size = 694, normalized size = 9.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33507, size = 119, normalized size = 1.7 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}}{a^{2}} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 12 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3} a^{2}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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