Optimal. Leaf size=18 \[ \frac {\tan ^3(c+d x)}{3 a^2 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 2687, 30}
\begin {gather*} \frac {\tan ^3(c+d x)}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2687
Rule 3254
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \sec ^2(c+d x) \tan ^2(c+d x) \, dx}{a^2}\\ &=\frac {\text {Subst}\left (\int x^2 \, dx,x,\tan (c+d x)\right )}{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 = 18, normalized size = 1.00 \begin {gather*} \frac {\tan ^3(c+d x)}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\tan ^{3}\left (d x +c \right )}{3 a^{2} d}\) | \(17\) |
default | \(\frac {\tan ^{3}\left (d x +c \right )}{3 a^{2} d}\) | \(17\) |
risch | \(-\frac {2 i \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}+1\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(36\) |
norman | \(\frac {-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {16 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {8 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{a \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 16, normalized size = 0.89 \begin {gather*} \frac {\tan \left (d x + c\right )^{3}}{3 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 32, normalized size = 1.78 \begin {gather*} -\frac {{\left (\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (14) = 28\).
time = 3.27, size = 94, normalized size = 5.22 \begin {gather*} \begin {cases} - \frac {8 \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} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\left (c \right )}}{\left (- a \sin ^{2}{\left (c \right )} + a\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 16, normalized size = 0.89 \begin {gather*} \frac {\tan \left (d x + c\right )^{3}}{3 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.38, size = 16, normalized size = 0.89 \begin {gather*} \frac {{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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