Optimal. Leaf size=60 \[ -\frac {\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac {1}{4 d (a+a \cos (c+d x))^2}-\frac {3}{4 d \left (a^2+a^2 \cos (c+d x)\right )} \]
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Rubi [A]
time = 0.09, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3957, 2915, 12,
90, 212} \begin {gather*} -\frac {3}{4 d \left (a^2 \cos (c+d x)+a^2\right )}-\frac {\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac {1}{4 d (a \cos (c+d x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 212
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc (c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos (c+d x) \cot (c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {a \text {Subst}\left (\int \frac {x^2}{a^2 (-a-x) (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x^2}{(-a-x) (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a}{2 (a-x)^3}-\frac {3}{4 (a-x)^2}+\frac {1}{4 \left (a^2-x^2\right )}\right ) \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac {1}{4 d (a+a \cos (c+d x))^2}-\frac {3}{4 d \left (a^2+a^2 \cos (c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,-a \cos (c+d x)\right )}{4 a d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac {1}{4 d (a+a \cos (c+d x))^2}-\frac {3}{4 d \left (a^2+a^2 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 83, normalized size = 1.38 \begin {gather*} -\frac {\left (-1+6 \cos ^2\left (\frac {1}{2} (c+d x)\right )+4 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )\right ) \sec ^2(c+d x)}{4 a^2 d (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 55, normalized size = 0.92
method | result | size |
derivativedivides | \(\frac {\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{8}+\frac {1}{4 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {3}{4 \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{8}}{d \,a^{2}}\) | \(55\) |
default | \(\frac {\frac {\ln \left (-1+\cos \left (d x +c \right )\right )}{8}+\frac {1}{4 \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {3}{4 \left (1+\cos \left (d x +c \right )\right )}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{8}}{d \,a^{2}}\) | \(55\) |
norman | \(\frac {-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}}{a}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{2} d}\) | \(63\) |
risch | \(-\frac {3 \,{\mathrm e}^{3 i \left (d x +c \right )}+4 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}}{2 a^{2} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{4 a^{2} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{4 a^{2} d}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 74, normalized size = 1.23 \begin {gather*} -\frac {\frac {2 \, {\left (3 \, \cos \left (d x + c\right ) + 2\right )}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac {\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} - \frac {\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.87, size = 106, normalized size = 1.77 \begin {gather*} -\frac {{\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 6 \, \cos \left (d x + c\right ) + 4}{8 \, {\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 )}} \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 {\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 = 0.46, size = 87, normalized size = 1.45 \begin {gather*} \frac {\frac {2 \, \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2}} + \frac {\frac {4 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{a^{4}}}{16 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 60, normalized size = 1.00 \begin {gather*} -\frac {\frac {3\,\cos \left (c+d\,x\right )}{4}+\frac {1}{2}}{d\,\left (a^2\,{\cos \left (c+d\,x\right )}^2+2\,a^2\,\cos \left (c+d\,x\right )+a^2\right )}-\frac {\mathrm {atanh}\left (\cos \left (c+d\,x\right )\right )}{4\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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