Optimal. Leaf size=20 \[ \frac {\tan (c+d x)}{a d}-\frac {x}{a} \]
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Rubi [A] time = 0.06, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3171, 3175, 3767, 8} \[ \frac {\tan (c+d x)}{a d}-\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3171
Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=-\frac {x}{a}+\int \frac {1}{a-a \sin ^2(c+d x)} \, dx\\ &=-\frac {x}{a}+\frac {\int \sec ^2(c+d x) \, dx}{a}\\ &=-\frac {x}{a}-\frac {\operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a d}\\ &=-\frac {x}{a}+\frac {\tan (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.35 \[ \frac {\frac {\tan (c+d x)}{d}-\frac {\tan ^{-1}(\tan (c+d x))}{d}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 34, normalized size = 1.70 \[ -\frac {d x \cos \left (d x + c\right ) - \sin \left (d x + c\right )}{a d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 1.30 \[ -\frac {\frac {d x + c}{a} - \frac {\tan \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 30, normalized size = 1.50 \[ \frac {\tan \left (d x +c \right )}{a d}-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 26, normalized size = 1.30 \[ -\frac {\frac {d x + c}{a} - \frac {\tan \left (d x + c\right )}{a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.74, size = 20, normalized size = 1.00 \[ \frac {\mathrm {tan}\left (c+d\,x\right )}{a\,d}-\frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.57, size = 100, normalized size = 5.00 \[ \begin {cases} - \frac {d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} + \frac {d x}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} - \frac {2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - a d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{2}{\relax (c )}}{- a \sin ^{2}{\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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