Optimal. Leaf size=138 \[ -\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 138, 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 = {2743, 2833, 12,
2738, 212} \begin {gather*} -\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (2 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{32768 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))^4} \, dx &=-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {1}{48} \int \frac {9+10 \cos (c+d x)}{(-3+5 \cos (c+d x))^3} \, dx\\ &=-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}+\frac {\int \frac {154+75 \cos (c+d x)}{(-3+5 \cos (c+d x))^2} \, dx}{1536}\\ &=-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac {\int \frac {837}{-3+5 \cos (c+d x)} \, dx}{24576}\\ &=-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac {279 \int \frac {1}{-3+5 \cos (c+d x)} \, dx}{8192}\\ &=-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac {279 \text {Subst}\left (\int \frac {1}{2-8 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d}\\ &=-\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac {995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(288\) vs. \(2(138)=276\).
time = 0.02, size = 288, normalized size = 2.09 \begin {gather*} \frac {467046 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-765855 \cos (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-2 \sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+2 \sin \left (\frac {1}{2} (c+d x)\right )\right )+226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))}{393216 d (-3+5 \cos (c+d x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 140, normalized size = 1.01
method | result | size |
norman | \(\frac {-\frac {745 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192 d}+\frac {265 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}-\frac {295 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512 d}}{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )^{3}}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768 d}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768 d}\) | \(105\) |
risch | \(-\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}-62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}-119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}-24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\) | \(129\) |
derivativedivides | \(\frac {-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\) | \(140\) |
default | \(\frac {-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}-\frac {125}{49152 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {25}{8192 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {295}{32768 \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {279 \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{32768}}{d}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 177, normalized size = 1.28 \begin {gather*} -\frac {\frac {20 \, {\left (\frac {447 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1696 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2832 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1} - 837 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) + 837 \, \log \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{98304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 170, normalized size = 1.23 \begin {gather*} \frac {837 \, {\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\right )} \log \left (-\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) - 837 \, {\left (125 \, \cos \left (d x + c\right )^{3} - 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) - 27\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 (4975 \, \cos \left (d x + c\right )^{2} - 4770 \, \cos \left (d x + c\right ) + 1583\right )} \sin \left (d x + c\right )}{196608 \, {\left (125 \, d \cos \left (d x + c\right )^{3} - 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) - 27 \, 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. 831 vs.
\(2 (126) = 252\).
time = 2.79, size = 831, normalized size = 6.02 \begin {gather*} \begin {cases} \frac {x}{\left (-3 + 5 \cos {\left (2 \operatorname {atan}{\left (\frac {1}{2} \right )} \right )}\right )^{4}} & \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 )^{4}} & \text {for}\: d = 0 \\- \frac {53568 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )} \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} + \frac {40176 \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 )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} - \frac {10044 \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 )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} + \frac {837 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1 \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} + \frac {53568 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} - \frac {40176 \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 )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} + \frac {10044 \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 )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} - \frac {837 \log {\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} - \frac {56640 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} + \frac {33920 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} - \frac {8940 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{6291456 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 4718592 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1179648 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 98304 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 97, normalized size = 0.70 \begin {gather*} -\frac {\frac {20 \, {\left (2832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1696 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 447 \, \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 )}^{3}} - 837 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 837 \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{98304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.00, size = 102, normalized size = 0.74 \begin {gather*} \frac {279\,\mathrm {atanh}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16384\,d}-\frac {\frac {295\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32768}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{49152}+\frac {745\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{524288}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16}-\frac {1}{64}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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