Optimal. Leaf size=43 \[ \frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2833, 12, 43} \[ \frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {(a+x)^2}{x^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (1+\frac {a^2}{x^2}+\frac {2 a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d}+\frac {a^2 \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.88 \[ a^2 \left (\frac {\sin (c+d x)}{d}-\frac {\csc (c+d x)}{d}+\frac {2 \log (\sin (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 46, normalized size = 1.07 \[ -\frac {a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 41, normalized size = 0.95 \[ \frac {2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + a^{2} \sin \left (d x + c\right ) - \frac {a^{2}}{\sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 46, normalized size = 1.07 \[ \frac {a^{2} \sin \left (d x +c \right )}{d}-\frac {a^{2}}{d \sin \left (d x +c \right )}+\frac {2 a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 40, normalized size = 0.93 \[ \frac {2 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + a^{2} \sin \left (d x + c\right ) - \frac {a^{2}}{\sin \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.89, size = 111, normalized size = 2.58 \[ \frac {2\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {2\,a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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 \[ a^{2} \left (\int \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int 2 \sin {\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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