Optimal. Leaf size=75 \[ -\frac {\cos (c+d x)}{a^3 d}+\frac {3}{d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {3 \log (\cos (c+d x)+1)}{a^3 d}-\frac {1}{2 a d (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.12, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2833, 12, 43} \[ -\frac {\cos (c+d x)}{a^3 d}+\frac {3}{d \left (a^3 \cos (c+d x)+a^3\right )}+\frac {3 \log (\cos (c+d x)+1)}{a^3 d}-\frac {1}{2 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin (c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{a^3 (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-\frac {a^3}{(a-x)^3}+\frac {3 a^2}{(a-x)^2}-\frac {3 a}{a-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\cos (c+d x)}{a^3 d}-\frac {1}{2 a d (a+a \cos (c+d x))^2}+\frac {3}{d \left (a^3+a^3 \cos (c+d x)\right )}+\frac {3 \log (1+\cos (c+d x))}{a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 103, normalized size = 1.37 \[ \frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (-2 \cos (3 (c+d x))+72 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\cos (2 (c+d x)) \left (24 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5\right )+\cos (c+d x) \left (96 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+22\right )+21\right )}{4 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 96, normalized size = 1.28 \[ -\frac {2 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - 6 \, {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, \cos \left (d x + c\right ) - 5}{2 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 63, normalized size = 0.84 \[ -\frac {\cos \left (d x + c\right )}{a^{3} d} + \frac {3 \, \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a^{3} d} + \frac {6 \, \cos \left (d x + c\right ) + 5}{2 \, a^{3} d {\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 86, normalized size = 1.15 \[ -\frac {1}{d \,a^{3} \sec \left (d x +c \right )}-\frac {3 \ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}-\frac {1}{2 a^{3} d \left (1+\sec \left (d x +c \right )\right )^{2}}-\frac {2}{d \,a^{3} \left (1+\sec \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sec \left (d x +c \right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 71, normalized size = 0.95 \[ \frac {\frac {6 \, \cos \left (d x + c\right ) + 5}{a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \cos \left (d x + c\right ) + a^{3}} - \frac {2 \, \cos \left (d x + c\right )}{a^{3}} + \frac {6 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 59, normalized size = 0.79 \[ \frac {3\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{a^3\,d}-\frac {\cos \left (c+d\,x\right )}{a^3\,d}+\frac {3\,\cos \left (c+d\,x\right )+\frac {5}{2}}{a^3\,d\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sin {\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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