Optimal. Leaf size=37 \[ \frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^3(c+d x)}{3 a d} \]
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Rubi [A]
time = 0.09, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3957, 2918,
2686, 30, 2687} \begin {gather*} \frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^3(c+d x)}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2686
Rule 2687
Rule 2918
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc (c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^3(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \, dx,x,\csc (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 66, normalized size = 1.78 \begin {gather*} \frac {\csc (c) \csc (2 (c+d x)) (-6 \sin (c)+4 \sin (d x)+2 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (c+2 d x))}{6 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 36, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{4 d a}\) | \(36\) |
default | \(\frac {-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{4 d a}\) | \(36\) |
norman | \(\frac {-\frac {1}{4 a d}-\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{12 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}\) | \(41\) |
risch | \(-\frac {2 i \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{3 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 49, normalized size = 1.32 \begin {gather*} -\frac {\frac {3 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a \sin \left (d x + c\right )} + \frac {\sin \left (d x + c\right )^{3}}{a {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.50, size = 41, normalized size = 1.11 \begin {gather*} -\frac {\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1}{3 \, {\left (a d \cos \left (d x + c\right ) + a 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 {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 37, normalized size = 1.00 \begin {gather*} -\frac {\frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a} + \frac {3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 32, normalized size = 0.86 \begin {gather*} -\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3}{12\,a\,d\,\mathrm {tan}\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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