Optimal. Leaf size=38 \[ \frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}+\frac {x}{a^2} \]
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Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3175, 3473, 8} \[ \frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}+\frac {x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3175
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \tan ^4(c+d x) \, dx}{a^2}\\ &=\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\int \tan ^2(c+d x) \, dx}{a^2}\\ &=-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\int 1 \, dx}{a^2}\\ &=\frac {x}{a^2}-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 1.11 \[ \frac {\frac {\tan ^{-1}(\tan (c+d x))}{d}+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 49, normalized size = 1.29 \[ \frac {3 \, d x \cos \left (d x + c\right )^{3} - {\left (4 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right )}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 44, normalized size = 1.16 \[ \frac {\frac {3 \, {\left (d x + c\right )}}{a^{2}} + \frac {a^{4} \tan \left (d x + c\right )^{3} - 3 \, a^{4} \tan \left (d x + c\right )}{a^{6}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 46, normalized size = 1.21 \[ \frac {\tan ^{3}\left (d x +c \right )}{3 a^{2} d}-\frac {\tan \left (d x +c \right )}{a^{2} d}+\frac {\arctan \left (\tan \left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 37, normalized size = 0.97 \[ \frac {\frac {\tan \left (d x + c\right )^{3} - 3 \, \tan \left (d x + c\right )}{a^{2}} + \frac {3 \, {\left (d x + c\right )}}{a^{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.48, size = 31, normalized size = 0.82 \[ \frac {x}{a^2}-\frac {\mathrm {tan}\left (c+d\,x\right )-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3}}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 29.68, size = 551, normalized size = 14.50 \[ \begin {cases} \frac {3 d x \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {9 d x \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {9 d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {3 d x}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {20 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{4}{\relax (c )}}{\left (- a \sin ^{2}{\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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