Optimal. Leaf size=32 \[ \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3175, 3767} \[ \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {\tan (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \sec ^4(c+d x) \, dx}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a^2 d}\\ &=\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 26, normalized size = 0.81 \[ \frac {\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 1.06 \[ \frac {{\left (2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right )}{3 \, a^{2} d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 25, normalized size = 0.78 \[ \frac {\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 25, normalized size = 0.78 \[ \frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 25, normalized size = 0.78 \[ \frac {\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )}{3 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.45, size = 24, normalized size = 0.75 \[ \frac {\mathrm {tan}\left (c+d\,x\right )\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2+3\right )}{3\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.17, size = 238, normalized size = 7.44 \[ \begin {cases} - \frac {6 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} + \frac {4 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} - \frac {6 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{3 a^{2} d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 9 a^{2} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 9 a^{2} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} - 3 a^{2} d} & \text {for}\: d \neq 0 \\\frac {x}{\left (- a \sin ^{2}{\relax (c )} + a\right )^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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