Optimal. Leaf size=65 \[ -\frac {3 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {3 \tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 65, 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 = {3254, 2700,
276} \begin {gather*} \frac {\tan ^3(c+d x)}{3 a^2 d}+\frac {3 \tan (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {3 \cot (c+d x)}{a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2700
Rule 3254
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{\left (a-a \sin ^2(c+d x)\right )^2} \, dx &=\frac {\int \csc ^4(c+d x) \sec ^4(c+d x) \, dx}{a^2}\\ &=\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^4} \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac {\text {Subst}\left (\int \left (3+\frac {1}{x^4}+\frac {3}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=-\frac {3 \cot (c+d x)}{a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {3 \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 = 0.02, size = 46, normalized size = 0.71 \begin {gather*} \frac {16 \left (-\frac {\cot (2 (c+d x))}{3 d}-\frac {\cot (2 (c+d x)) \csc ^2(2 (c+d x))}{6 d}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 47, normalized size = 0.72
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 \tan \left (d x +c \right )-\frac {1}{3 \tan \left (d x +c \right )^{3}}-\frac {3}{\tan \left (d x +c \right )}}{d \,a^{2}}\) | \(47\) |
default | \(\frac {\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 \tan \left (d x +c \right )-\frac {1}{3 \tan \left (d x +c \right )^{3}}-\frac {3}{\tan \left (d x +c \right )}}{d \,a^{2}}\) | \(47\) |
risch | \(\frac {32 i \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}-1\right )}{3 d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(49\) |
norman | \(\frac {\frac {1}{24 a d}+\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}-\frac {91 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {35 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d}-\frac {91 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}+\frac {5 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a d}+\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 52, normalized size = 0.80 \begin {gather*} \frac {\frac {\tan \left (d x + c\right )^{3} + 9 \, \tan \left (d x + c\right )}{a^{2}} - \frac {9 \, \tan \left (d x + c\right )^{2} + 1}{a^{2} \tan \left (d x + c\right )^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 72, normalized size = 1.11 \begin {gather*} -\frac {16 \, \cos \left (d x + c\right )^{6} - 24 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{2} + 1}{3 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - a^{2} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} - 2 \sin ^{2}{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 34, normalized size = 0.52 \begin {gather*} -\frac {8 \, {\left (3 \, \tan \left (2 \, d x + 2 \, c\right )^{2} + 1\right )}}{3 \, a^{2} d \tan \left (2 \, d x + 2 \, c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.68, size = 48, normalized size = 0.74 \begin {gather*} -\frac {-{\mathrm {tan}\left (c+d\,x\right )}^6-9\,{\mathrm {tan}\left (c+d\,x\right )}^4+9\,{\mathrm {tan}\left (c+d\,x\right )}^2+1}{3\,a^2\,d\,{\mathrm {tan}\left (c+d\,x\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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