Optimal. Leaf size=47 \[ \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan (c+d x)}{a^2 d}-\frac {\cot (c+d x)}{a^2 d} \]
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Rubi [A] time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3175, 2620, 270} \[ \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {2 \tan (c+d x)}{a^2 d}-\frac {\cot (c+d x)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2620
Rule 3175
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \csc ^2(c+d x) \sec ^4(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot (c+d x)}{a^2 d}+\frac {2 \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 = 50, normalized size = 1.06 \[ \frac {\frac {5 \tan (c+d x)}{3 d}-\frac {\cot (c+d x)}{d}+\frac {\tan (c+d x) \sec ^2(c+d x)}{3 d}}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 46, normalized size = 0.98 \[ -\frac {8 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 1}{3 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.02 \[ -\frac {\frac {3}{a^{2} \tan \left (d x + c\right )} - \frac {a^{4} \tan \left (d x + c\right )^{3} + 6 \, a^{4} \tan \left (d x + c\right )}{a^{6}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 37, normalized size = 0.79 \[ \frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 \tan \left (d x +c \right )-\frac {1}{\tan \left (d x +c \right )}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 40, normalized size = 0.85 \[ \frac {\frac {\tan \left (d x + c\right )^{3} + 6 \, \tan \left (d x + c\right )}{a^{2}} - \frac {3}{a^{2} \tan \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.58, size = 36, normalized size = 0.77 \[ \frac {{\mathrm {tan}\left (c+d\,x\right )}^4+6\,{\mathrm {tan}\left (c+d\,x\right )}^2-3}{3\,a^2\,d\,\mathrm {tan}\left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc ^{2}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} - 2 \sin ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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