Optimal. Leaf size=88 \[ -\frac {99}{19652 d (3 \tan (c+d x)+5)}-\frac {15}{1156 d (3 \tan (c+d x)+5)^2}-\frac {1}{34 d (3 \tan (c+d x)+5)^3}+\frac {60 \log (3 \sin (c+d x)+5 \cos (c+d x))}{83521 d}-\frac {161 x}{334084} \]
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Rubi [A] time = 0.12, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3483, 3529, 3531, 3530} \[ -\frac {99}{19652 d (3 \tan (c+d x)+5)}-\frac {15}{1156 d (3 \tan (c+d x)+5)^2}-\frac {1}{34 d (3 \tan (c+d x)+5)^3}+\frac {60 \log (3 \sin (c+d x)+5 \cos (c+d x))}{83521 d}-\frac {161 x}{334084} \]
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))^4} \, dx &=-\frac {1}{34 d (5+3 \tan (c+d x))^3}+\frac {1}{34} \int \frac {5-3 \tan (c+d x)}{(5+3 \tan (c+d x))^3} \, dx\\ &=-\frac {1}{34 d (5+3 \tan (c+d x))^3}-\frac {15}{1156 d (5+3 \tan (c+d x))^2}+\frac {\int \frac {16-30 \tan (c+d x)}{(5+3 \tan (c+d x))^2} \, dx}{1156}\\ &=-\frac {1}{34 d (5+3 \tan (c+d x))^3}-\frac {15}{1156 d (5+3 \tan (c+d x))^2}-\frac {99}{19652 d (5+3 \tan (c+d x))}+\frac {\int \frac {-10-198 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx}{39304}\\ &=-\frac {161 x}{334084}-\frac {1}{34 d (5+3 \tan (c+d x))^3}-\frac {15}{1156 d (5+3 \tan (c+d x))^2}-\frac {99}{19652 d (5+3 \tan (c+d x))}+\frac {60 \int \frac {3-5 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx}{83521}\\ &=-\frac {161 x}{334084}+\frac {60 \log (5 \cos (c+d x)+3 \sin (c+d x))}{83521 d}-\frac {1}{34 d (5+3 \tan (c+d x))^3}-\frac {15}{1156 d (5+3 \tan (c+d x))^2}-\frac {99}{19652 d (5+3 \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.66, size = 95, normalized size = 1.08 \[ -\frac {\frac {3366}{3 \tan (c+d x)+5}+\frac {8670}{(3 \tan (c+d x)+5)^2}+\frac {19652}{(3 \tan (c+d x)+5)^3}+(240-161 i) \log (-\tan (c+d x)+i)+(240+161 i) \log (\tan (c+d x)+i)-480 \log (3 \tan (c+d x)+5)}{668168 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 157, normalized size = 1.78 \[ -\frac {27 \, {\left (161 \, d x - 305\right )} \tan \left (d x + c\right )^{3} + 27 \, {\left (805 \, d x - 964\right )} \tan \left (d x + c\right )^{2} + 20125 \, d x - 120 \, {\left (27 \, \tan \left (d x + c\right )^{3} + 135 \, \tan \left (d x + c\right )^{2} + 225 \, \tan \left (d x + c\right ) + 125\right )} \log \left (\frac {9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25}{\tan \left (d x + c\right )^{2} + 1}\right ) + 45 \, {\left (805 \, d x - 114\right )} \tan \left (d x + c\right ) + 35451}{334084 \, {\left (27 \, d \tan \left (d x + c\right )^{3} + 135 \, d \tan \left (d x + c\right )^{2} + 225 \, d \tan \left (d x + c\right ) + 125 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.73, size = 83, normalized size = 0.94 \[ -\frac {161 \, d x + 161 \, c + \frac {11880 \, \tan \left (d x + c\right )^{3} + 74547 \, \tan \left (d x + c\right )^{2} + 162495 \, \tan \left (d x + c\right ) + 128576}{{\left (3 \, \tan \left (d x + c\right ) + 5\right )}^{3}} + 120 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 \, \log \left ({\left | 3 \, \tan \left (d x + c\right ) + 5 \right |}\right )}{334084 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 97, normalized size = 1.10 \[ -\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521 d}-\frac {161 \arctan \left (\tan \left (d x +c \right )\right )}{334084 d}-\frac {1}{34 d \left (5+3 \tan \left (d x +c \right )\right )^{3}}-\frac {15}{1156 d \left (5+3 \tan \left (d x +c \right )\right )^{2}}-\frac {99}{19652 d \left (5+3 \tan \left (d x +c \right )\right )}+\frac {60 \ln \left (5+3 \tan \left (d x +c \right )\right )}{83521 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 93, normalized size = 1.06 \[ -\frac {161 \, d x + 161 \, c + \frac {17 \, {\left (891 \, \tan \left (d x + c\right )^{2} + 3735 \, \tan \left (d x + c\right ) + 4328\right )}}{27 \, \tan \left (d x + c\right )^{3} + 135 \, \tan \left (d x + c\right )^{2} + 225 \, \tan \left (d x + c\right ) + 125} + 120 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 240 \, \log \left (3 \, \tan \left (d x + c\right ) + 5\right )}{334084 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.07, size = 104, normalized size = 1.18 \[ \frac {60\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {5}{3}\right )}{83521\,d}-\frac {\frac {33\,{\mathrm {tan}\left (c+d\,x\right )}^2}{19652}+\frac {415\,\mathrm {tan}\left (c+d\,x\right )}{58956}+\frac {1082}{132651}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3+5\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {25\,\mathrm {tan}\left (c+d\,x\right )}{3}+\frac {125}{27}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {30}{83521}+\frac {161}{668168}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-\frac {30}{83521}-\frac {161}{668168}{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.14, size = 790, normalized size = 8.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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