Optimal. Leaf size=46 \[ \frac {\tan (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {2 \cot (c+d x)}{a d} \]
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Rubi [A] time = 0.08, antiderivative size = 46, 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} \[ \frac {\tan (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}-\frac {2 \cot (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2620
Rule 3175
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{a-a \sin ^2(c+d x)} \, dx &=\frac {\int \csc ^4(c+d x) \sec ^2(c+d x) \, dx}{a}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac {2 \cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\tan (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 1.07 \[ \frac {\frac {\tan (c+d x)}{d}-\frac {5 \cot (c+d x)}{3 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 d}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 56, normalized size = 1.22 \[ -\frac {8 \, \cos \left (d x + c\right )^{4} - 12 \, \cos \left (d x + c\right )^{2} + 3}{3 \, {\left (a d \cos \left (d x + c\right )^{3} - a d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 42, normalized size = 0.91 \[ \frac {\frac {3 \, \tan \left (d x + c\right )}{a} - \frac {6 \, \tan \left (d x + c\right )^{2} + 1}{a \tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 35, normalized size = 0.76 \[ \frac {\tan \left (d x +c \right )-\frac {2}{\tan \left (d x +c \right )}-\frac {1}{3 \tan \left (d x +c \right )^{3}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 42, normalized size = 0.91 \[ \frac {\frac {3 \, \tan \left (d x + c\right )}{a} - \frac {6 \, \tan \left (d x + c\right )^{2} + 1}{a \tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.73, size = 38, normalized size = 0.83 \[ -\frac {-{\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {1}{3}}{a\,d\,{\mathrm {tan}\left (c+d\,x\right )}^3} \]
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 \[ - \frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sin ^{2}{\left (c + d x \right )} - 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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