Optimal. Leaf size=37 \[ -\frac {a \cot (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3872, 2838, 2621, 321, 207, 3767, 8} \[ -\frac {a \cot (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 207
Rule 321
Rule 2621
Rule 2838
Rule 3767
Rule 3872
Rubi steps
\begin {align*} \int \csc ^2(c+d x) (a+a \sec (c+d x)) \, dx &=-\int (-a-a \cos (c+d x)) \csc ^2(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^2(c+d x) \, dx+a \int \csc ^2(c+d x) \sec (c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}-\frac {a \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot (c+d x)}{d}-\frac {a \csc (c+d x)}{d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=\frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 41, normalized size = 1.11 \[ -\frac {a \csc (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\sin ^2(c+d x)\right )}{d}-\frac {a \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 63, normalized size = 1.70 \[ \frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - a \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 2 \, a}{2 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 50, normalized size = 1.35 \[ \frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 47, normalized size = 1.27 \[ -\frac {a \cot \left (d x +c \right )}{d}-\frac {a}{d \sin \left (d x +c \right )}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 50, normalized size = 1.35 \[ -\frac {a {\left (\frac {2}{\sin \left (d x + c\right )} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac {2 \, a}{\tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 29, normalized size = 0.78 \[ \frac {a\,\left (2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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 \[ a \left (\int \csc ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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