Optimal. Leaf size=86 \[ \frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^4(c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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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} \[ \frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^4(c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^2 (a+x)^3}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^3}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^4+\frac {a^5}{x}-2 a^3 x-2 a^2 x^2+a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {a \log (\sin (c+d x))}{d}+\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 86, normalized size = 1.00 \[ \frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^4(c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}-\frac {a \sin ^2(c+d x)}{d}+\frac {a \sin (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 74, normalized size = 0.86 \[ \frac {15 \, a \cos \left (d x + c\right )^{4} + 30 \, a \cos \left (d x + c\right )^{2} + 60 \, a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 70, normalized size = 0.81 \[ \frac {12 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 40 \, a \sin \left (d x + c\right )^{3} - 60 \, a \sin \left (d x + c\right )^{2} + 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 94, normalized size = 1.09 \[ \frac {8 a \sin \left (d x +c \right )}{15 d}+\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d}+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{15 d}+\frac {a \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 69, normalized size = 0.80 \[ \frac {12 \, a \sin \left (d x + c\right )^{5} + 15 \, a \sin \left (d x + c\right )^{4} - 40 \, a \sin \left (d x + c\right )^{3} - 60 \, a \sin \left (d x + c\right )^{2} + 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 60 \, a \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.87, size = 126, normalized size = 1.47 \[ \frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^4}{4\,d}+\frac {8\,a\,\sin \left (c+d\,x\right )}{15\,d}+\frac {4\,a\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{15\,d}+\frac {a\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{5\,d} \]
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 \[ a \left (\int \cos ^{5}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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