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 = {3175, 3767, 8} \[ \frac {\tan (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3175
Rule 3767
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 {\operatorname {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 \[ \frac {\tan (c+d x)}{a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 21, normalized size = 1.62 \[ \frac {\sin \left (d x + c\right )}{a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 13, normalized size = 1.00 \[ \frac {\tan \left (d x + c\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 14, normalized size = 1.08 \[ \frac {\tan \left (d x +c \right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 13, normalized size = 1.00 \[ \frac {\tan \left (d x + c\right )}{a d} \]
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 \[ \frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.89, size = 41, normalized size = 3.15 \[ \begin {cases} - \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} & \text {for}\: d \neq 0 \\\frac {x}{- a \sin ^{2}{\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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