Optimal. Leaf size=118 \[ -\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x)}{2 d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {3 a \csc (c+d x)}{d}+\frac {3 a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2836, 12, 88} \[ -\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x)}{2 d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {3 a \csc (c+d x)}{d}+\frac {3 a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^5 (a-x)^3 (a+x)^4}{x^5} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^5} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 a^2+\frac {a^7}{x^5}+\frac {a^6}{x^4}-\frac {3 a^5}{x^3}-\frac {3 a^4}{x^2}+\frac {3 a^3}{x}-a x-x^2\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {3 a \csc (c+d x)}{d}+\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {3 a \log (\sin (c+d x))}{d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.54, size = 105, normalized size = 0.89 \[ -\frac {a \sin ^3(c+d x)}{3 d}+\frac {3 a \sin (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {3 a \csc (c+d x)}{d}+\frac {a \left (-2 \sin ^2(c+d x)-\csc ^4(c+d x)+6 \csc ^2(c+d x)+12 \log (\sin (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 142, normalized size = 1.20 \[ \frac {6 \, a \cos \left (d x + c\right )^{6} - 15 \, a \cos \left (d x + c\right )^{4} - 6 \, a \cos \left (d x + c\right )^{2} + 36 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} - 24 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right ) + 12 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 103, normalized size = 0.87 \[ -\frac {4 \, a \sin \left (d x + c\right )^{3} + 6 \, a \sin \left (d x + c\right )^{2} - 36 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 36 \, a \sin \left (d x + c\right ) + \frac {75 \, a \sin \left (d x + c\right )^{4} - 36 \, a \sin \left (d x + c\right )^{3} - 18 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 217, normalized size = 1.84 \[ -\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {5 a \left (\cos ^{8}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}+\frac {16 a \sin \left (d x +c \right )}{3 d}+\frac {5 \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{3 d}+\frac {2 a \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{d}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{3 d}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}+\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{2 d}+\frac {3 a \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {3 a \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 92, normalized size = 0.78 \[ -\frac {4 \, a \sin \left (d x + c\right )^{3} + 6 \, a \sin \left (d x + c\right )^{2} - 36 \, a \log \left (\sin \left (d x + c\right )\right ) - 36 \, a \sin \left (d x + c\right ) - \frac {36 \, a \sin \left (d x + c\right )^{3} + 18 \, a \sin \left (d x + c\right )^{2} - 4 \, a \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.01, size = 290, normalized size = 2.46 \[ \frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {118\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-27\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {644\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {69\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{4}+160\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {57\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {17\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {a}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}+\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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