Optimal. Leaf size=73 \[ -\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2790,
2687, 30, 2686, 14} \begin {gather*} -\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {\cot ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2686
Rule 2687
Rule 2790
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\int \left (a^2 \cot ^4(c+d x) \csc ^2(c+d x)-2 a^2 \cot ^3(c+d x) \csc ^3(c+d x)+a^2 \cot ^2(c+d x) \csc ^4(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \cot ^4(c+d x) \csc ^2(c+d x) \, dx}{a^2}+\frac {\int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^2}-\frac {2 \int \cot ^3(c+d x) \csc ^3(c+d x) \, dx}{a^2}\\ &=\frac {\text {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot ^5(c+d x)}{5 a^2 d}+\frac {\text {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^2 d}+\frac {2 \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\csc (c+d x)\right )}{a^2 d}\\ &=-\frac {\cot ^3(c+d x)}{3 a^2 d}-\frac {2 \cot ^5(c+d x)}{5 a^2 d}-\frac {2 \csc ^3(c+d x)}{3 a^2 d}+\frac {2 \csc ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 105, normalized size = 1.44 \begin {gather*} \frac {\csc (c) \csc (c+d x) \sec ^2(c+d x) (-80 \sin (c)+80 \sin (d x)+55 \sin (c+d x)+44 \sin (2 (c+d x))+11 \sin (3 (c+d x))-60 \sin (2 c+d x)+16 \sin (c+2 d x)+4 \sin (2 c+3 d x))}{240 a^2 d (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 60, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{8 d \,a^{2}}\) | \(60\) |
default | \(\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{8 d \,a^{2}}\) | \(60\) |
norman | \(\frac {-\frac {1}{8 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 a d}+\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{40 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}\) | \(82\) |
risch | \(-\frac {2 i \left (15 \,{\mathrm e}^{4 i \left (d x +c \right )}+20 \,{\mathrm e}^{3 i \left (d x +c \right )}+20 \,{\mathrm e}^{2 i \left (d x +c \right )}+4 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{15 a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 90, normalized size = 1.23 \begin {gather*} -\frac {\frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{2}} + \frac {15 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{2} \sin \left (d x + c\right )}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.06, size = 71, normalized size = 0.97 \begin {gather*} -\frac {\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )^{2} + 8 \, \cos \left (d x + c\right ) + 4}{15 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\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 ^{2}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\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 = 0.48, size = 74, normalized size = 1.01 \begin {gather*} -\frac {\frac {15}{a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 5 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}}}{120 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.98, size = 71, normalized size = 0.97 \begin {gather*} -\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+14\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-3}{120\,a^2\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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