Optimal. Leaf size=86 \[ -\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}-\frac {b \csc ^4(c+d x)}{4 d}+\frac {b \csc ^2(c+d x)}{d}+\frac {b \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.08, 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 = {2837, 12, 766} \[ -\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}-\frac {b \csc ^4(c+d x)}{4 d}+\frac {b \csc ^2(c+d x)}{d}+\frac {b \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 766
Rule 2837
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc (c+d x) (a+b \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^6 (a+x) \left (b^2-x^2\right )^2}{x^6} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x) \left (b^2-x^2\right )^2}{x^6} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (\frac {a b^4}{x^6}+\frac {b^4}{x^5}-\frac {2 a b^2}{x^4}-\frac {2 b^2}{x^3}+\frac {a}{x^2}+\frac {1}{x}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {a \csc (c+d x)}{d}+\frac {b \csc ^2(c+d x)}{d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {b \csc ^4(c+d x)}{4 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {b \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 92, normalized size = 1.07 \[ -\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^3(c+d x)}{3 d}-\frac {a \csc (c+d x)}{d}+\frac {b \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 124, normalized size = 1.44 \[ -\frac {60 \, a \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{2} - 60 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 15 \, {\left (4 \, b \cos \left (d x + c\right )^{2} - 3 \, b\right )} \sin \left (d x + c\right ) + 32 \, a}{60 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 84, normalized size = 0.98 \[ \frac {60 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {137 \, b \sin \left (d x + c\right )^{5} + 60 \, a \sin \left (d x + c\right )^{4} - 60 \, b \sin \left (d x + c\right )^{3} - 40 \, a \sin \left (d x + c\right )^{2} + 15 \, b \sin \left (d x + c\right ) + 12 \, a}{\sin \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 160, normalized size = 1.86 \[ -\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}+\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{15 d \sin \left (d x +c \right )^{3}}-\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )}-\frac {8 a \sin \left (d x +c \right )}{15 d}-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d}-\frac {4 a \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{15 d}-\frac {b \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {b \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 72, normalized size = 0.84 \[ \frac {60 \, b \log \left (\sin \left (d x + c\right )\right ) - \frac {60 \, a \sin \left (d x + c\right )^{4} - 60 \, b \sin \left (d x + c\right )^{3} - 40 \, a \sin \left (d x + c\right )^{2} + 15 \, b \sin \left (d x + c\right ) + 12 \, a}{\sin \left (d x + c\right )^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.80, size = 193, normalized size = 2.24 \[ \frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {b\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a}{5}\right )}{32\,d} \]
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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