Optimal. Leaf size=13 \[ \frac {\tan (c+d x)}{a d} \]
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Rubi [A]
time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3254, 3852, 8}
\begin {gather*} \frac {\tan (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=\frac {\tan (c+d x)}{a d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {\tan (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 14, normalized size = 1.08
method | result | size |
derivativedivides | \(\frac {\tan \left (d x +c \right )}{a d}\) | \(14\) |
default | \(\frac {\tan \left (d x +c \right )}{a d}\) | \(14\) |
risch | \(\frac {2 i}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(23\) |
norman | \(-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 13, normalized size = 1.00 \begin {gather*} \frac {\tan \left (d x + c\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 21, normalized size = 1.62 \begin {gather*} \frac {\sin \left (d x + c\right )}{a d \cos \left (d x + c\right )} \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. 41 vs.
\(2 (8) = 16\).
time = 0.47, size = 41, normalized size = 3.15 \begin {gather*} \begin {cases} - \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} & \text {for}\: d \neq 0 \\\frac {x}{- a \sin ^{2}{\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 13, normalized size = 1.00 \begin {gather*} \frac {\tan \left (d x + c\right )}{a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.59, size = 13, normalized size = 1.00 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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