Optimal. Leaf size=81 \[ \frac {a \sin (c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {a \csc ^2(c+d x)}{d}+\frac {2 a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2707, 88} \[ \frac {a \sin (c+d x)}{d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {a \csc ^2(c+d x)}{d}+\frac {2 a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^5}{x^5}+\frac {a^4}{x^4}-\frac {2 a^3}{x^3}-\frac {2 a^2}{x^2}+\frac {a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {2 a \csc (c+d x)}{d}+\frac {a \csc ^2(c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 87, normalized size = 1.07 \[ \frac {a \sin (c+d x)}{d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc (c+d x)}{d}+\frac {a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 110, normalized size = 1.36 \[ -\frac {12 \, a \cos \left (d x + c\right )^{2} - 12 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (3 \, a \cos \left (d x + c\right )^{4} - 12 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) - 9 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 82, normalized size = 1.01 \[ \frac {12 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 12 \, a \sin \left (d x + c\right ) - \frac {25 \, a \sin \left (d x + c\right )^{4} - 24 \, a \sin \left (d x + c\right )^{3} - 12 \, a \sin \left (d x + c\right )^{2} + 4 \, a \sin \left (d x + c\right ) + 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 136, normalized size = 1.68 \[ -\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}+\frac {8 a \sin \left (d x +c \right )}{3 d}+\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{d}+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{3 d}-\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.79, size = 69, normalized size = 0.85 \[ \frac {12 \, a \log \left (\sin \left (d x + c\right )\right ) + 12 \, a \sin \left (d x + c\right ) + \frac {24 \, a \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 4 \, a \sin \left (d x + c\right ) - 3 \, a}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.03, size = 207, normalized size = 2.56 \[ \frac {7\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {46\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {40\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}-\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {a}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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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