Optimal. Leaf size=84 \[ \frac{\cos ^3(c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{\sin ^3(c+d x) \cos (c+d x)}{4 a^3 d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{15 x}{8 a^3} \]
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Rubi [A] time = 0.165566, antiderivative size = 105, normalized size of antiderivative = 1.25, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2859, 2679, 2682, 2635, 8} \[ -\frac{5 \cos ^3(c+d x)}{4 a^3 d}-\frac{3 \cos ^5(c+d x)}{4 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{15 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{15 x}{8 a^3}-\frac{\cos ^7(c+d x)}{d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2859
Rule 2679
Rule 2682
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=-\frac{\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac{3 \int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{a}\\ &=-\frac{\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac{3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{15 \int \frac{\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{4 a^2}\\ &=-\frac{5 \cos ^3(c+d x)}{4 a^3 d}-\frac{\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac{3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{15 \int \cos ^2(c+d x) \, dx}{4 a^3}\\ &=-\frac{5 \cos ^3(c+d x)}{4 a^3 d}-\frac{15 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac{3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac{15 \int 1 \, dx}{8 a^3}\\ &=-\frac{15 x}{8 a^3}-\frac{5 \cos ^3(c+d x)}{4 a^3 d}-\frac{15 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac{\cos ^7(c+d x)}{d (a+a \sin (c+d x))^3}-\frac{3 \cos ^5(c+d x)}{4 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.4017, size = 255, normalized size = 3.04 \[ -\frac{120 d x \sin \left (\frac{c}{2}\right )-104 \sin \left (\frac{c}{2}+d x\right )+104 \sin \left (\frac{3 c}{2}+d x\right )-32 \sin \left (\frac{3 c}{2}+2 d x\right )-32 \sin \left (\frac{5 c}{2}+2 d x\right )+8 \sin \left (\frac{5 c}{2}+3 d x\right )-8 \sin \left (\frac{7 c}{2}+3 d x\right )+\sin \left (\frac{7 c}{2}+4 d x\right )+\sin \left (\frac{9 c}{2}+4 d x\right )+\cos \left (\frac{c}{2}\right ) (120 d x+1)+104 \cos \left (\frac{c}{2}+d x\right )+104 \cos \left (\frac{3 c}{2}+d x\right )-32 \cos \left (\frac{3 c}{2}+2 d x\right )+32 \cos \left (\frac{5 c}{2}+2 d x\right )-8 \cos \left (\frac{5 c}{2}+3 d x\right )-8 \cos \left (\frac{7 c}{2}+3 d x\right )+\cos \left (\frac{7 c}{2}+4 d x\right )-\cos \left (\frac{9 c}{2}+4 d x\right )-\sin \left (\frac{c}{2}\right )}{64 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.109, size = 279, normalized size = 3.3 \begin{align*} -{\frac{15}{4\,d{a}^{3}} \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}}-2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{23}{4\,d{a}^{3}} \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}}-18\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}+{\frac{23}{4\,d{a}^{3}} \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}}-22\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}+{\frac{15}{4\,d{a}^{3}}\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}}-6\,{\frac{1}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{4}}}-{\frac{15}{4\,d{a}^{3}}\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 = 1.53987, size = 360, normalized size = 4.29 \begin{align*} \frac{\frac{\frac{15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{88 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{23 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{72 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{23 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{8 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 24}{a^{3} + \frac{4 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac{15 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.06535, size = 150, normalized size = 1.79 \begin{align*} \frac{8 \, \cos \left (d x + c\right )^{3} - 15 \, d x -{\left (2 \, \cos \left (d x + c\right )^{3} - 17 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 32 \, \cos \left (d x + c\right )}{8 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37456, size = 171, normalized size = 2.04 \begin{align*} -\frac{\frac{15 \,{\left (d x + c\right )}}{a^{3}} + \frac{2 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 8 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 23 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 72 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 23 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 88 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 24\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{4} a^{3}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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