Optimal. Leaf size=56 \[ \frac {5 x}{64}+\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{32 d}+\frac {3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2743, 12, 2737}
\begin {gather*} \frac {5 \text {ArcTan}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{32 d}+\frac {3 \cos (c+d x)}{16 d (3 \sin (c+d x)+5)}+\frac {5 x}{64} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2737
Rule 2743
Rubi steps
\begin {align*} \int \frac {1}{(-5-3 \sin (c+d x))^2} \, dx &=\frac {3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}-\frac {1}{16} \int \frac {5}{-5-3 \sin (c+d x)} \, dx\\ &=\frac {3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}-\frac {5}{16} \int \frac {1}{-5-3 \sin (c+d x)} \, dx\\ &=\frac {5 x}{64}+\frac {5 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{32 d}+\frac {3 \cos (c+d x)}{16 d (5+3 \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 91, normalized size = 1.62 \begin {gather*} \frac {25 \tan ^{-1}\left (\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right )+\frac {6 (-5+5 \cos (c+d x)-3 \sin (c+d x))}{5+3 \sin (c+d x)}}{160 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 63, normalized size = 1.12
method | result | size |
derivativedivides | \(\frac {\frac {\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{200}+\frac {3}{40}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+1}+\frac {5 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{32}}{d}\) | \(63\) |
default | \(\frac {\frac {\frac {9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{200}+\frac {3}{40}}{\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{5}+1}+\frac {5 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{32}}{d}\) | \(63\) |
risch | \(\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}+3 i}{8 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-3+10 i {\mathrm e}^{i \left (d x +c \right )}\right )}+\frac {5 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3 i\right )}{64 d}-\frac {5 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i}{3}\right )}{64 d}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 93, normalized size = 1.66 \begin {gather*} \frac {\frac {12 \, {\left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 5\right )}}{\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5} + 25 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right )}{160 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 59, normalized size = 1.05 \begin {gather*} \frac {5 \, {\left (3 \, \sin \left (d x + c\right ) + 5\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right ) + 12 \, \cos \left (d x + c\right )}{64 \, {\left (3 \, d \sin \left (d x + c\right ) + 5 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.93, size = 389, normalized size = 6.95 \begin {gather*} \begin {cases} \frac {x}{\left (-5 + 3 \sin {\left (2 \operatorname {atan}{\left (\frac {3}{5} - \frac {4 i}{5} \right )} \right )}\right )^{2}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (\frac {3}{5} - \frac {4 i}{5} \right )} \\\frac {x}{\left (-5 + 3 \sin {\left (2 \operatorname {atan}{\left (\frac {3}{5} + \frac {4 i}{5} \right )} \right )}\right )^{2}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (\frac {3}{5} + \frac {4 i}{5} \right )} \\\frac {x}{\left (- 3 \sin {\left (c \right )} - 5\right )^{2}} & \text {for}\: d = 0 \\\frac {125 \left (\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} + \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} + \frac {150 \left (\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} + \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} + \frac {125 \left (\operatorname {atan}{\left (\frac {5 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{4} + \frac {3}{4} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} + \frac {36 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} + \frac {60}{800 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 960 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 800 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.52, size = 95, normalized size = 1.70 \begin {gather*} \frac {25 \, d x + 25 \, c + \frac {24 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}}{5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5} + 50 \, \arctan \left (-\frac {3 \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) - 9}\right )}{320 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 83, normalized size = 1.48 \begin {gather*} \frac {5\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3}{4}\right )}{32\,d}-\frac {5\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{32\,d}+\frac {\frac {9\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{200}+\frac {3}{40}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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