Optimal. Leaf size=113 \[ \frac {45 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)}-\frac {5 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}-\frac {43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
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Rubi [A] time = 0.08, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2664, 2754, 12, 2660, 616, 31} \[ \frac {45 \cos (c+d x)}{512 d (5 \sin (c+d x)+3)}-\frac {5 \cos (c+d x)}{32 d (5 \sin (c+d x)+3)^2}-\frac {43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 616
Rule 2660
Rule 2664
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(3+5 \sin (c+d x))^3} \, dx &=-\frac {5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac {1}{32} \int \frac {-6+5 \sin (c+d x)}{(3+5 \sin (c+d x))^2} \, dx\\ &=-\frac {5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac {1}{512} \int \frac {43}{3+5 \sin (c+d x)} \, dx\\ &=-\frac {5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac {43}{512} \int \frac {1}{3+5 \sin (c+d x)} \, dx\\ &=-\frac {5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac {43 \operatorname {Subst}\left (\int \frac {1}{3+10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{256 d}\\ &=-\frac {5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}+\frac {129 \operatorname {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {129 \operatorname {Subst}\left (\int \frac {1}{9+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}\\ &=-\frac {43 \log \left (3+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {43 \log \left (1+3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {5 \cos (c+d x)}{32 d (3+5 \sin (c+d x))^2}+\frac {45 \cos (c+d x)}{512 d (3+5 \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 180, normalized size = 1.59 \[ \frac {\sin \left (\frac {1}{2} (c+d x)\right ) \left (-\frac {180}{3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )}-\frac {60}{\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )}\right )+\frac {40}{\left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {40}{\left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-43 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+3 \cos \left (\frac {1}{2} (c+d x)\right )\right )+43 \log \left (3 \sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 133, normalized size = 1.18 \[ -\frac {43 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right ) - 34\right )} \log \left (4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) - 43 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right ) - 34\right )} \log \left (-4 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) + 5\right ) + 1800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 440 \, \cos \left (d x + c\right )}{4096 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 30 \, d \sin \left (d x + c\right ) - 34 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 107, normalized size = 0.95 \[ -\frac {\frac {40 \, {\left (75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 649 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 735 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 99\right )}}{{\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}^{2}} - 387 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 387 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \right |}\right )}{18432 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 114, normalized size = 1.01 \[ \frac {25}{128 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )^{2}}-\frac {15}{512 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}-\frac {43 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}{2048 d}-\frac {25}{1152 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {155}{4608 d \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {43 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2048 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 195, normalized size = 1.73 \[ \frac {\frac {40 \, {\left (\frac {735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {649 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {75 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 99\right )}}{\frac {60 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {118 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {60 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {9 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 9} + 387 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - 387 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 3\right )}{18432 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.73, size = 115, normalized size = 1.02 \[ \frac {-\frac {125\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6912}+\frac {3245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{20736}+\frac {1225\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6912}+\frac {55}{2304}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {118\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{9}+\frac {20\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+1\right )}-\frac {43\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {5}{4}\right )}{1024\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.54, size = 1227, normalized size = 10.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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