Optimal. Leaf size=90 \[ \frac {5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{64 d} \]
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Rubi [A] time = 0.05, antiderivative size = 90, 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} \[ \frac {5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}+\frac {3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {3 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{64 d} \]
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}{1-3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}+\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 {3 \log \left (1-3 \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 {5 \cos (c+d x)}{16 d (3-5 \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 130, normalized size = 1.44 \[ \frac {20 \sin \left (\frac {1}{2} (c+d x)\right ) \left (\frac {1}{3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {3}{\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )}\right )+9 \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 88, normalized size = 0.98 \[ -\frac {3 \, {\left (5 \, \sin \left (d x + c\right ) - 3\right )} \log \left (4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) - 3 \, {\left (5 \, \sin \left (d x + c\right ) - 3\right )} \log \left (-4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) + 40 \, \cos \left (d x + c\right )}{128 \, {\left (5 \, d \sin \left (d x + c\right ) - 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 81, normalized size = 0.90 \[ -\frac {\frac {40 \, {\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3\right )}}{3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3} - 9 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 9 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \right |}\right )}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 76, normalized size = 0.84 \[ -\frac {5}{48 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {3 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{64 d}-\frac {5}{16 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}-\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{64 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 115, normalized size = 1.28 \[ \frac {\frac {40 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 3\right )}}{\frac {10 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 3} + 9 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right ) - 9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 3\right )}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 64, normalized size = 0.71 \[ \frac {3\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {5}{4}\right )}{32\,d}-\frac {\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{72}-\frac {5}{24}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 462, normalized size = 5.13 \[ \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 )} - 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 {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 {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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