Optimal. Leaf size=79 \[ \frac {\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac {5 x}{16 a^3} \]
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Rubi [A] time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3657, 12, 2635, 8} \[ \frac {\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}+\frac {5 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}+\frac {5 \sin (c+d x) \cos (c+d x)}{16 a^3 d}+\frac {5 x}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2635
Rule 3657
Rubi steps
\begin {align*} \int \frac {1}{\left (a+a \tan ^2(c+d x)\right )^3} \, dx &=\int \frac {\cos ^6(c+d x)}{a^3} \, dx\\ &=\frac {\int \cos ^6(c+d x) \, dx}{a^3}\\ &=\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {5 \int \cos ^4(c+d x) \, dx}{6 a^3}\\ &=\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {5 \int \cos ^2(c+d x) \, dx}{8 a^3}\\ &=\frac {5 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {5 \int 1 \, dx}{16 a^3}\\ &=\frac {5 x}{16 a^3}+\frac {5 \cos (c+d x) \sin (c+d x)}{16 a^3 d}+\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}+\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 0.58 \[ \frac {45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x}{192 a^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 120, normalized size = 1.52 \[ \frac {15 \, d x \tan \left (d x + c\right )^{6} + 45 \, d x \tan \left (d x + c\right )^{4} + 15 \, \tan \left (d x + c\right )^{5} + 45 \, d x \tan \left (d x + c\right )^{2} + 40 \, \tan \left (d x + c\right )^{3} + 15 \, d x + 33 \, \tan \left (d x + c\right )}{48 \, {\left (a^{3} d \tan \left (d x + c\right )^{6} + 3 \, a^{3} d \tan \left (d x + c\right )^{4} + 3 \, a^{3} d \tan \left (d x + c\right )^{2} + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.63, size = 61, normalized size = 0.77 \[ \frac {\frac {15 \, {\left (d x + c\right )}}{a^{3}} + \frac {15 \, \tan \left (d x + c\right )^{5} + 40 \, \tan \left (d x + c\right )^{3} + 33 \, \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3} a^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 95, normalized size = 1.20 \[ \frac {\tan \left (d x +c \right )}{6 d \,a^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {5 \tan \left (d x +c \right )}{24 d \,a^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {5 \tan \left (d x +c \right )}{16 d \,a^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right )}{16 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 90, normalized size = 1.14 \[ \frac {\frac {15 \, \tan \left (d x + c\right )^{5} + 40 \, \tan \left (d x + c\right )^{3} + 33 \, \tan \left (d x + c\right )}{a^{3} \tan \left (d x + c\right )^{6} + 3 \, a^{3} \tan \left (d x + c\right )^{4} + 3 \, a^{3} \tan \left (d x + c\right )^{2} + a^{3}} + \frac {15 \, {\left (d x + c\right )}}{a^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.93, size = 51, normalized size = 0.65 \[ \frac {5\,x}{16\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6\,\left (\frac {5\,{\mathrm {tan}\left (c+d\,x\right )}^5}{16}+\frac {5\,{\mathrm {tan}\left (c+d\,x\right )}^3}{6}+\frac {11\,\mathrm {tan}\left (c+d\,x\right )}{16}\right )}{a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.40, size = 454, normalized size = 5.75 \[ \begin {cases} \frac {15 d x \tan ^{6}{\left (c + d x \right )}}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} + \frac {45 d x \tan ^{4}{\left (c + d x \right )}}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} + \frac {45 d x \tan ^{2}{\left (c + d x \right )}}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} + \frac {15 d x}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} + \frac {15 \tan ^{5}{\left (c + d x \right )}}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} + \frac {40 \tan ^{3}{\left (c + d x \right )}}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} + \frac {33 \tan {\left (c + d x \right )}}{48 a^{3} d \tan ^{6}{\left (c + d x \right )} + 144 a^{3} d \tan ^{4}{\left (c + d x \right )} + 144 a^{3} d \tan ^{2}{\left (c + d x \right )} + 48 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (a \tan ^{2}{\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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