Optimal. Leaf size=88 \[ -\frac {5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d} \]
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Rubi [A] time = 0.04, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2664, 12, 2660, 616, 31} \[ -\frac {5 \cos (c+d x)}{16 d (5 \sin (c+d x)+3)}+\frac {3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{64 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 616
Rule 2660
Rule 2664
Rubi steps
\begin {align*} \int \frac {1}{(-3-5 \sin (c+d x))^2} \, dx &=-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}+\frac {1}{16} \int \frac {3}{-3-5 \sin (c+d x)} \, dx\\ &=-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}+\frac {3}{16} \int \frac {1}{-3-5 \sin (c+d x)} \, dx\\ &=-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-10 x-3 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}\\ &=-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}-\frac {9 \operatorname {Subst}\left (\int \frac {1}{-9-3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\frac {9 \operatorname {Subst}\left (\int \frac {1}{-1-3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}\\ &=\frac {3 \log \left (3+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (1+3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {5 \cos (c+d x)}{16 d (3+5 \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 126, normalized size = 1.43 \[ \frac {20 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {3}{3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}+\frac {1}{\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )}\right )+9 \left (\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 88, normalized size = 1.00 \[ \frac {3 \, {\left (5 \, \sin \left (d x + c\right ) + 3\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 3 \, {\left (5 \, \sin \left (d x + c\right ) + 3\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 40 \, \cos \left (d x + c\right )}{128 \, {\left (5 \, d \sin \left (d x + c\right ) + 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 81, normalized size = 0.92 \[ -\frac {\frac {40 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}}{3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3} + 9 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \right |}\right )}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 76, normalized size = 0.86 \[ -\frac {5}{16 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{64 d}-\frac {5}{48 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {3 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{64 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 115, normalized size = 1.31 \[ -\frac {\frac {40 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}}{\frac {10 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 3} + 9 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 64, normalized size = 0.73 \[ \frac {3\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {5}{4}\right )}{32\,d}-\frac {\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{72}+\frac {5}{24}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.65, size = 468, normalized size = 5.32 \[ \begin {cases} \frac {x}{\left (-3 + 5 \sin {\left (2 \operatorname {atan}{\left (\frac {1}{3} \right )} \right )}\right )^{2}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (\frac {1}{3} \right )} \\\frac {x}{\left (-3 + 5 \sin {\left (2 \operatorname {atan}{\relax (3 )} \right )}\right )^{2}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\relax (3 )} \\\frac {x}{\left (- 5 \sin {\relax (c )} - 3\right )^{2}} & \text {for}\: d = 0 \\- \frac {27 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + \frac {1}{3} \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {90 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + \frac {1}{3} \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {27 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + \frac {1}{3} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} + \frac {27 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} + \frac {90 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} + \frac {27 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 3 \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {200 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} - \frac {120}{576 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1920 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 576 d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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