Optimal. Leaf size=18 \[ \frac {\sec ^3(c+d x)}{3 a^2 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3254, 2686, 30}
\begin {gather*} \frac {\sec ^3(c+d x)}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 3254
Rubi steps
\begin {align*} \int \frac {\sin (c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \sec ^3(c+d x) \tan (c+d x) \, dx}{a^2}\\ &=\frac {\text {Subst}\left (\int x^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {\sec ^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 {\sec ^3(c+d x)}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 17, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {1}{3 d \,a^{2} \cos \left (d x +c \right )^{3}}\) | \(17\) |
default | \(\frac {1}{3 d \,a^{2} \cos \left (d x +c \right )^{3}}\) | \(17\) |
risch | \(\frac {8 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(31\) |
norman | \(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2}{3 a d}-\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(101\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{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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Fricas [A]
time = 0.37, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{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. 156 vs.
\(2 (14) = 28\).
time = 2.15, size = 156, normalized size = 8.67 \begin {gather*} \begin {cases} - \frac {6 \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 {2}{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 {\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 = 0.52, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{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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Mupad [B]
time = 13.59, size = 16, normalized size = 0.89 \begin {gather*} \frac {1}{3\,a^2\,d\,{\cos \left (c+d\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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