Optimal. Leaf size=113 \[ -\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))} \]
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Rubi [A]
time = 0.05, antiderivative size = 113, 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 = {2743, 2833, 12,
2738, 212} \begin {gather*} -\frac {45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac {5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (2 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 212
Rule 2738
Rule 2743
Rule 2833
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 \text {Subst}\left (\int \frac {1}{2-8 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}\\ &=-\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \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 = 0.09, size = 211, normalized size = 1.87 \begin {gather*} -\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {5}{512 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{1024 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {5}{512 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {45 \sin \left (\frac {1}{2} (c+d x)\right )}{1024 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 106, normalized size = 0.94
method | result | size |
norman | \(\frac {\frac {85 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d}-\frac {35 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d}}{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )^{2}}-\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2048 d}+\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048 d}\) | \(89\) |
derivativedivides | \(\frac {\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2048}-\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048}}{d}\) | \(106\) |
default | \(\frac {\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2048}-\frac {25}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {35}{2048 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {43 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048}}{d}\) | \(106\) |
risch | \(-\frac {i \left (215 \,{\mathrm e}^{3 i \left (d x +c \right )}-387 \,{\mathrm e}^{2 i \left (d x +c \right )}+325 \,{\mathrm e}^{i \left (d x +c \right )}-225\right )}{256 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{2}}-\frac {43 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}-\frac {4 i}{5}\right )}{2048 d}+\frac {43 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}+\frac {4 i}{5}\right )}{2048 d}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 137, normalized size = 1.21 \begin {gather*} -\frac {\frac {20 \, {\left (\frac {17 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {28 \, \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 {16 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1} - 43 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 43 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{2048 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 129, normalized size = 1.14 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 490 vs.
\(2 (102) = 204\).
time = 1.27, size = 490, normalized size = 4.34 \begin {gather*} \begin {cases} \frac {x}{\left (-3 + 5 \cos {\left (2 \operatorname {atan}{\left (\frac {1}{2} \right )} \right )}\right )^{3}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (\frac {1}{2} \right )} \vee c = - d x + 2 \operatorname {atan}{\left (\frac {1}{2} \right )} \\\frac {x}{\left (5 \cos {\left (c \right )} - 3\right )^{3}} & \text {for}\: d = 0 \\- \frac {688 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 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 )} + 2048 d} + \frac {344 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 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 )} + 2048 d} - \frac {43 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )}}{32768 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 )} + 2048 d} + \frac {688 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 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 )} + 2048 d} - \frac {344 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 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 )} + 2048 d} + \frac {43 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{32768 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 )} + 2048 d} - \frac {560 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 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 )} + 2048 d} + \frac {340 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{32768 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 )} + 2048 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 84, normalized size = 0.74 \begin {gather*} -\frac {\frac {20 \, {\left (28 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 17 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - 43 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 43 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{2048 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 75, normalized size = 0.66 \begin {gather*} \frac {43\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{1024\,d}+\frac {\frac {85\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8192}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2048}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {1}{16}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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