Optimal. Leaf size=69 \[ -\frac {15}{578 d (3 \tan (c+d x)+5)}-\frac {3}{68 d (3 \tan (c+d x)+5)^2}+\frac {99 \log (3 \sin (c+d x)+5 \cos (c+d x))}{19652 d}-\frac {5 x}{19652} \]
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Rubi [A] time = 0.09, antiderivative size = 69, 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 = {3483, 3529, 3531, 3530} \[ -\frac {15}{578 d (3 \tan (c+d x)+5)}-\frac {3}{68 d (3 \tan (c+d x)+5)^2}+\frac {99 \log (3 \sin (c+d x)+5 \cos (c+d x))}{19652 d}-\frac {5 x}{19652} \]
Antiderivative was successfully verified.
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Rule 3483
Rule 3529
Rule 3530
Rule 3531
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \tan (c+d x))^3} \, dx &=-\frac {3}{68 d (5+3 \tan (c+d x))^2}+\frac {1}{34} \int \frac {5-3 \tan (c+d x)}{(5+3 \tan (c+d x))^2} \, dx\\ &=-\frac {3}{68 d (5+3 \tan (c+d x))^2}-\frac {15}{578 d (5+3 \tan (c+d x))}+\frac {\int \frac {16-30 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx}{1156}\\ &=-\frac {5 x}{19652}-\frac {3}{68 d (5+3 \tan (c+d x))^2}-\frac {15}{578 d (5+3 \tan (c+d x))}+\frac {99 \int \frac {3-5 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx}{19652}\\ &=-\frac {5 x}{19652}+\frac {99 \log (5 \cos (c+d x)+3 \sin (c+d x))}{19652 d}-\frac {3}{68 d (5+3 \tan (c+d x))^2}-\frac {15}{578 d (5+3 \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.69, size = 86, normalized size = 1.25 \[ \frac {\left (\frac {1}{39304}+\frac {i}{39304}\right ) \left ((-47+52 i) \log (-\tan (c+d x)+i)-(52-47 i) \log (\tan (c+d x)+i)+(3-3 i) \left (33 \log (3 \tan (c+d x)+5)-\frac {17 (30 \tan (c+d x)+67)}{(3 \tan (c+d x)+5)^2}\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 120, normalized size = 1.74 \[ -\frac {18 \, {\left (5 \, d x - 87\right )} \tan \left (d x + c\right )^{2} + 250 \, d x - 99 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25\right )} \log \left (\frac {9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, {\left (5 \, d x - 36\right )} \tan \left (d x + c\right ) + 2484}{39304 \, {\left (9 \, d \tan \left (d x + c\right )^{2} + 30 \, d \tan \left (d x + c\right ) + 25 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 74, normalized size = 1.07 \[ -\frac {10 \, d x + 10 \, c + \frac {3 \, {\left (891 \, \tan \left (d x + c\right )^{2} + 3990 \, \tan \left (d x + c\right ) + 4753\right )}}{{\left (3 \, \tan \left (d x + c\right ) + 5\right )}^{2}} + 99 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 198 \, \log \left ({\left | 3 \, \tan \left (d x + c\right ) + 5 \right |}\right )}{39304 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 80, normalized size = 1.16 \[ -\frac {99 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{39304 d}-\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{19652 d}-\frac {3}{68 d \left (5+3 \tan \left (d x +c \right )\right )^{2}}-\frac {15}{578 d \left (5+3 \tan \left (d x +c \right )\right )}+\frac {99 \ln \left (5+3 \tan \left (d x +c \right )\right )}{19652 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 73, normalized size = 1.06 \[ -\frac {10 \, d x + 10 \, c + \frac {102 \, {\left (30 \, \tan \left (d x + c\right ) + 67\right )}}{9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25} + 99 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 198 \, \log \left (3 \, \tan \left (d x + c\right ) + 5\right )}{39304 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.09, size = 84, normalized size = 1.22 \[ \frac {99\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {5}{3}\right )}{19652\,d}-\frac {\frac {5\,\mathrm {tan}\left (c+d\,x\right )}{578}+\frac {67}{3468}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2+\frac {10\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {25}{9}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {99}{39304}+\frac {5}{39304}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-\frac {99}{39304}-\frac {5}{39304}{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.79, size = 442, normalized size = 6.41 \[ \begin {cases} - \frac {90 d x \tan ^{2}{\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {300 d x \tan {\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {250 d x}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} + \frac {1782 \log {\left (\tan {\left (c + d x \right )} + \frac {5}{3} \right )} \tan ^{2}{\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} + \frac {5940 \log {\left (\tan {\left (c + d x \right )} + \frac {5}{3} \right )} \tan {\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} + \frac {4950 \log {\left (\tan {\left (c + d x \right )} + \frac {5}{3} \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {891 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {2970 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {2475 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {3060 \tan {\left (c + d x \right )}}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} - \frac {6834}{353736 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 982600 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (3 \tan {\relax (c )} + 5\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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