Optimal. Leaf size=86 \[ -\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {2 a \csc ^3(c+d x)}{3 d}+\frac {a \csc ^2(c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 86, 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 = {2836, 12, 88} \[ -\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^4(c+d x)}{4 d}+\frac {2 a \csc ^3(c+d x)}{3 d}+\frac {a \csc ^2(c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^6 (a-x)^2 (a+x)^3}{x^6} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x^6} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {a^5}{x^6}+\frac {a^4}{x^5}-\frac {2 a^3}{x^4}-\frac {2 a^2}{x^3}+\frac {a}{x^2}+\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc (c+d x)}{d}+\frac {a \csc ^2(c+d x)}{d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 92, normalized size = 1.07 \[ -\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}+\frac {a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 124, normalized size = 1.44 \[ -\frac {60 \, a \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{2} - 60 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 15 \, {\left (4 \, a \cos \left (d x + c\right )^{2} - 3 \, a\right )} \sin \left (d x + c\right ) + 32 \, a}{60 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 84, normalized size = 0.98 \[ \frac {60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {137 \, a \sin \left (d x + c\right )^{5} + 60 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 40 \, a \sin \left (d x + c\right )^{2} + 15 \, a \sin \left (d x + c\right ) + 12 \, a}{\sin \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 160, normalized size = 1.86 \[ -\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}+\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{15 d \sin \left (d x +c \right )^{3}}-\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )}-\frac {8 a \sin \left (d x +c \right )}{15 d}-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d}-\frac {4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 72, normalized size = 0.84 \[ \frac {60 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {60 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} - 40 \, a \sin \left (d x + c\right )^{2} + 15 \, a \sin \left (d x + c\right ) + 12 \, a}{\sin \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.90, size = 193, normalized size = 2.24 \[ \frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a}{5}\right )}{32\,d} \]
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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