Optimal. Leaf size=69 \[ -\frac {a^3}{d (a-a \cos (c+d x))}+\frac {a^2 \sec (c+d x)}{d}+\frac {2 a^2 \log (1-\cos (c+d x))}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3872, 2836, 12, 44} \[ -\frac {a^3}{d (a-a \cos (c+d x))}+\frac {a^2 \sec (c+d x)}{d}+\frac {2 a^2 \log (1-\cos (c+d x))}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int \csc ^3(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^3(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {a^2}{(-a-x)^2 x^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \frac {1}{(-a-x)^2 x^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^5 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2}{a^3 x}+\frac {1}{a^2 (a+x)^2}+\frac {2}{a^3 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^3}{d (a-a \cos (c+d x))}+\frac {2 a^2 \log (1-\cos (c+d x))}{d}-\frac {2 a^2 \log (\cos (c+d x))}{d}+\frac {a^2 \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 75, normalized size = 1.09 \[ -\frac {a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )-2 \sec (c+d x)-8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 \log (\cos (c+d x))\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 112, normalized size = 1.62 \[ \frac {2 \, a^{2} \cos \left (d x + c\right ) - a^{2} - 2 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) + 2 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.67, size = 135, normalized size = 1.96 \[ \frac {4 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 4 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {a^{2} + \frac {5 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + \frac {{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 50, normalized size = 0.72 \[ \frac {a^{2} \sec \left (d x +c \right )}{d}-\frac {a^{2}}{d \left (-1+\sec \left (d x +c \right )\right )}+\frac {2 a^{2} \ln \left (-1+\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 68, normalized size = 0.99 \[ \frac {2 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) - 2 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) + \frac {2 \, a^{2} \cos \left (d x + c\right ) - a^{2}}{\cos \left (d x + c\right )^{2} - \cos \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 61, normalized size = 0.88 \[ -\frac {2\,a^2\,\cos \left (c+d\,x\right )-a^2}{d\,\left (\cos \left (c+d\,x\right )-{\cos \left (c+d\,x\right )}^2\right )}-\frac {4\,a^2\,\mathrm {atanh}\left (2\,\cos \left (c+d\,x\right )-1\right )}{d} \]
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 \[ a^{2} \left (\int 2 \csc ^{3}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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