Optimal. Leaf size=94 \[ -\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{3 x}{2 a} \]
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Rubi [A] time = 0.0923199, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2767, 2748, 2635, 8, 2633} \[ -\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{d (a \cos (c+d x)+a)}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{3 x}{2 a} \]
Antiderivative was successfully verified.
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Rule 2767
Rule 2748
Rule 2635
Rule 8
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x)}{a+a \cos (c+d x)} \, dx &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{\int \cos ^2(c+d x) (3 a-4 a \cos (c+d x)) \, dx}{a^2}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{3 \int \cos ^2(c+d x) \, dx}{a}+\frac{4 \int \cos ^3(c+d x) \, dx}{a}\\ &=-\frac{3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{3 \int 1 \, dx}{2 a}-\frac{4 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac{3 x}{2 a}+\frac{4 \sin (c+d x)}{a d}-\frac{3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{4 \sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.271445, size = 143, normalized size = 1.52 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (21 \sin \left (c+\frac{d x}{2}\right )+18 \sin \left (c+\frac{3 d x}{2}\right )+18 \sin \left (2 c+\frac{3 d x}{2}\right )-2 \sin \left (2 c+\frac{5 d x}{2}\right )-2 \sin \left (3 c+\frac{5 d x}{2}\right )+\sin \left (3 c+\frac{7 d x}{2}\right )+\sin \left (4 c+\frac{7 d x}{2}\right )-36 d x \cos \left (c+\frac{d x}{2}\right )+69 \sin \left (\frac{d x}{2}\right )-36 d x \cos \left (\frac{d x}{2}\right )\right )}{48 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 136, normalized size = 1.5 \begin{align*}{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+5\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{16}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}+3\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-3\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.72912, size = 238, normalized size = 2.53 \begin{align*} \frac{\frac{\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac{3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac{9 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{3 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61676, size = 180, normalized size = 1.91 \begin{align*} -\frac{9 \, d x \cos \left (d x + c\right ) + 9 \, d x -{\left (2 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} + 7 \, \cos \left (d x + c\right ) + 16\right )} \sin \left (d x + c\right )}{6 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.5499, size = 570, normalized size = 6.06 \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 = 1.3614, size = 119, normalized size = 1.27 \begin{align*} -\frac{\frac{9 \,{\left (d x + c\right )}}{a} - \frac{6 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 16 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{3} a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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