Optimal. Leaf size=140 \[ \frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \sin (c+d x)}{48 d (3+5 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (3+5 \cos (c+d x))^2}+\frac {995 \sin (c+d x)}{24576 d (3+5 \cos (c+d x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 140, 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 (5 \cos (c+d x)+3)}-\frac {25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac {5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \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}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d}\\ &=\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\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. \(296\) vs. \(2(140)=280\).
time = 0.20, size = 296, normalized size = 2.11 \begin {gather*} \frac {467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+765855 \cos (c+d x) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+376650 \cos (2 (c+d x)) \left (\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-467046 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-104625 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\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 = 0.10, size = 124, normalized size = 0.89
method | result | size |
norman | \(\frac {-\frac {295 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{512 d}+\frac {265 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}-\frac {745 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8192 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-4\right )^{3}}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768 d}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768 d}\) | \(99\) |
derivativedivides | \(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\) | \(124\) |
default | \(\frac {-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}-\frac {125}{6144 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {175}{4096 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {745}{16384 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\) | \(124\) |
risch | \(\frac {i \left (20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+62775 \,{\mathrm e}^{4 i \left (d x +c \right )}+111042 \,{\mathrm e}^{3 i \left (d x +c \right )}+119310 \,{\mathrm e}^{2 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}+24875\right )}{12288 d \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )^{3}}+\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 174, normalized size = 1.24 \begin {gather*} -\frac {\frac {20 \, {\left (\frac {2832 \, \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 {447 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} + 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 837 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{98304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 170, normalized size = 1.21 \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. 813 vs.
\(2 (126) = 252\).
time = 2.76, size = 813, normalized size = 5.81 \begin {gather*} \begin {cases} \frac {x}{\left (5 \cos {\left (2 \operatorname {atan}{\left (2 \right )} \right )} + 3\right )^{4}} & \text {for}\: c = - d x - 2 \operatorname {atan}{\left (2 \right )} \vee c = - d x + 2 \operatorname {atan}{\left (2 \right )} \\\frac {x}{\left (5 \cos {\left (c \right )} + 3\right )^{4}} & \text {for}\: d = 0 \\\frac {837 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {10044 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {40176 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {53568 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} - 2 \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {837 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {10044 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {40176 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )} \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {53568 \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {8940 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} + \frac {33920 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} - \frac {56640 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{98304 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 1179648 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 4718592 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 6291456 d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 91, normalized size = 0.65 \begin {gather*} -\frac {\frac {20 \, {\left (447 \, \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} + 2832 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{3}} + 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) - 837 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{98304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.97, size = 102, normalized size = 0.73 \begin {gather*} -\frac {279\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{16384\,d}-\frac {\frac {745\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192}-\frac {265\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{768}+\frac {295\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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