Optimal. Leaf size=115 \[ -\frac {45 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)}+\frac {5 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
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Rubi [A] time = 0.08, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2664, 2754, 12, 2659, 206} \[ -\frac {45 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)}+\frac {5 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2659
Rule 2664
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(3+5 \cos (c+d x))^3} \, dx &=\frac {5 \sin (c+d x)}{32 d (3+5 \cos (c+d x))^2}+\frac {1}{32} \int \frac {-6+5 \cos (c+d x)}{(3+5 \cos (c+d x))^2} \, dx\\ &=\frac {5 \sin (c+d x)}{32 d (3+5 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (3+5 \cos (c+d x))}+\frac {1}{512} \int \frac {43}{3+5 \cos (c+d x)} \, dx\\ &=\frac {5 \sin (c+d x)}{32 d (3+5 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (3+5 \cos (c+d x))}+\frac {43}{512} \int \frac {1}{3+5 \cos (c+d x)} \, dx\\ &=\frac {5 \sin (c+d x)}{32 d (3+5 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (3+5 \cos (c+d x))}+\frac {43 \operatorname {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}\\ &=-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {5 \sin (c+d x)}{32 d (3+5 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (3+5 \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 217, normalized size = 1.89 \[ -\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{2048 d \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{2048 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {5}{512 d \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {5}{512 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {43 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 129, normalized size = 1.12 \[ \frac {43 \, {\left (25 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 43 \, {\left (25 \, \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right ) + 9\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 40 \, {\left (45 \, \cos \left (d x + c\right ) + 11\right )} \sin \left (d x + c\right )}{4096 \, {\left (25 \, d \cos \left (d x + c\right )^{2} + 30 \, d \cos \left (d x + c\right ) + 9 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.62, size = 78, normalized size = 0.68 \[ \frac {\frac {20 \, {\left (17 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 28 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{2}} + 43 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - 43 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{2048 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 108, normalized size = 0.94 \[ -\frac {25}{512 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {85}{1024 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{2048 d}+\frac {25}{512 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {85}{1024 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{2048 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.82, size = 135, normalized size = 1.17 \[ \frac {\frac {20 \, {\left (\frac {28 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {17 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{\frac {8 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {\sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 16} + 43 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 43 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{2048 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 76, normalized size = 0.66 \[ \frac {43\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{1024\,d}-\frac {\frac {35\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}-\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{512}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+16\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.46, size = 474, normalized size = 4.12 \[ \begin {cases} \frac {x}{\left (5 \cos {\left (2 \operatorname {atan}{\relax (2 )} \right )} + 3\right )^{3}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\relax (2 )} \vee c = - d x + 2 \operatorname {atan}{\relax (2 )} \\\frac {x}{\left (5 \cos {\relax (c )} + 3\right )^{3}} & \text {for}\: d = 0 \\- \frac {43 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} + \frac {344 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} - \frac {688 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} + \frac {43 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} - \frac {344 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} + \frac {688 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} + \frac {340 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} - \frac {560 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2048 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 16384 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 32768 d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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