Optimal. Leaf size=73 \[ \frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a} \]
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Rubi [A] time = 0.15003, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3872, 2839, 2564, 30, 2568, 2635, 8} \[ \frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2839
Rule 2564
Rule 30
Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sin ^4(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac{\int \cos (c+d x) \sin ^2(c+d x) \, dx}{a}-\frac{\int \cos ^2(c+d x) \sin ^2(c+d x) \, dx}{a}\\ &=\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac{\int \cos ^2(c+d x) \, dx}{4 a}+\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\sin (c+d x)\right )}{a d}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d}-\frac{\int 1 \, dx}{8 a}\\ &=-\frac{x}{8 a}-\frac{\cos (c+d x) \sin (c+d x)}{8 a d}+\frac{\cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac{\sin ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.578878, size = 83, normalized size = 1.14 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (24 \sin (c+d x)-8 \sin (3 (c+d x))+3 \left (\sin (4 (c+d x))+4 c-4 \tan \left (\frac{c}{2}\right )-4 d x\right )\right )}{48 a d (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.072, size = 154, normalized size = 2.1 \begin{align*} -{\frac{1}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{53}{12\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}+{\frac{11}{12\,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 ) ^{-4}}+{\frac{1}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-4}}-{\frac{1}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.50441, size = 265, normalized size = 3.63 \begin{align*} \frac{\frac{\frac{3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{11 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{53 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac{4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{3 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67449, size = 128, normalized size = 1.75 \begin{align*} -\frac{3 \, d x -{\left (6 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right )}{24 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin ^{4}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3091, size = 117, normalized size = 1.6 \begin{align*} -\frac{\frac{3 \,{\left (d x + c\right )}}{a} + \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 53 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 11 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, \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 )}^{4} a}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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