Optimal. Leaf size=81 \[ \frac {59 x}{2048}+\frac {59 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{1024 d}+\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 2833, 12,
2736} \begin {gather*} \frac {59 \text {ArcTan}\left (\frac {\cos (c+d x)}{\sin (c+d x)+3}\right )}{1024 d}+\frac {45 \cos (c+d x)}{512 d (3 \sin (c+d x)+5)}+\frac {3 \cos (c+d x)}{32 d (3 \sin (c+d x)+5)^2}+\frac {59 x}{2048} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2736
Rule 2743
Rule 2833
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \sin (c+d x))^3} \, dx &=\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}-\frac {1}{32} \int \frac {-10+3 \sin (c+d x)}{(5+3 \sin (c+d x))^2} \, dx\\ &=\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))}+\frac {1}{512} \int \frac {59}{5+3 \sin (c+d x)} \, dx\\ &=\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))}+\frac {59}{512} \int \frac {1}{5+3 \sin (c+d x)} \, dx\\ &=\frac {59 x}{2048}+\frac {59 \tan ^{-1}\left (\frac {\cos (c+d x)}{3+\sin (c+d x)}\right )}{1024 d}+\frac {3 \cos (c+d x)}{32 d (5+3 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (5+3 \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 113, normalized size = 1.40 \begin {gather*} \frac {59 \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 {546 \cos (c+d x)+9 (-59+9 \cos (2 (c+d x))-60 \sin (c+d x)+15 \sin (2 (c+d x)))}{(5+3 \sin (c+d x))^2}}{1024 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 91, normalized size = 1.12
method | result | size |
derivativedivides | \(\frac {\frac {\frac {963 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1280}+\frac {11739 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6400}+\frac {2313 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1280}+\frac {273}{256}}{\left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{2}}+\frac {59 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{1024}}{d}\) | \(91\) |
default | \(\frac {\frac {\frac {963 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1280}+\frac {11739 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6400}+\frac {2313 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1280}+\frac {273}{256}}{\left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{2}}+\frac {59 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}+\frac {3}{4}\right )}{1024}}{d}\) | \(91\) |
risch | \(\frac {\frac {885 i {\mathrm e}^{2 i \left (d x +c \right )}}{256}+\frac {177 \,{\mathrm e}^{3 i \left (d x +c \right )}}{256}-\frac {723 \,{\mathrm e}^{i \left (d x +c \right )}}{256}-\frac {135 i}{256}}{\left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-3+10 i {\mathrm e}^{i \left (d x +c \right )}\right )^{2} d}+\frac {59 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3 i\right )}{2048 d}-\frac {59 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i}{3}\right )}{2048 d}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 173 vs.
\(2 (73) = 146\).
time = 0.58, size = 173, normalized size = 2.14 \begin {gather*} \frac {\frac {12 \, {\left (\frac {3855 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3913 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1605 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 2275\right )}}{\frac {60 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {86 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {60 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {25 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 25} + 1475 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {3}{4}\right )}{25600 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 94, normalized size = 1.16 \begin {gather*} \frac {59 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right ) - 34\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) + 3}{4 \, \cos \left (d x + c\right )}\right ) - 540 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 1092 \, \cos \left (d x + c\right )}{2048 \, {\left (9 \, d \cos \left (d x + c\right )^{2} - 30 \, d \sin \left (d x + c\right ) - 34 \, 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 = 2.26, size = 918, normalized size = 11.33 \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 )^{3}} & \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 )^{3}} & \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 )^{3}} & \text {for}\: d = 0 \\\frac {36875 \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 ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {88500 \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 ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {126850 \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 )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {88500 \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 )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {36875 \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 )}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {19260 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {46956 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {46260 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} + \frac {27300}{640000 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2201600 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1536000 d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 640000 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.65, size = 121, normalized size = 1.49 \begin {gather*} \frac {1475 \, d x + 1475 \, c + \frac {24 \, {\left (1605 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3913 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3855 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2275\right )}}{{\left (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\right )}^{2}} + 2950 \, \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 )}{51200 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.60, size = 111, normalized size = 1.37 \begin {gather*} \frac {59\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3}{4}\right )}{1024\,d}-\frac {59\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{1024\,d}+\frac {\frac {963\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1280}+\frac {11739\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6400}+\frac {2313\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{1280}+\frac {273}{256}}{d\,{\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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