Optimal. Leaf size=87 \[ -\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \cos ^4(c+d x)}{4 d}+\frac {2 a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^2(c+d x)}{d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2836, 12, 88} \[ -\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \cos ^4(c+d x)}{4 d}+\frac {2 a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^2(c+d x)}{d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^4(c+d x) \tan (c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a (-a-x)^2 (-a+x)^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^2 (-a+x)^3}{x} \, dx,x,-a \cos (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 \cos (c+d x)\right )}{a^4 d}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {a \cos ^2(c+d x)}{d}+\frac {2 a \cos ^3(c+d x)}{3 d}-\frac {a \cos ^4(c+d x)}{4 d}-\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.95 \[ -\frac {5 a \cos (c+d x)}{8 d}+\frac {5 a \cos (3 (c+d x))}{48 d}-\frac {a \cos (5 (c+d x))}{80 d}-\frac {a \left (\frac {1}{4} \cos ^4(c+d x)-\cos ^2(c+d x)+\log (\cos (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 71, normalized size = 0.82 \[ -\frac {12 \, a \cos \left (d x + c\right )^{5} + 15 \, a \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, a \cos \left (d x + c\right )^{2} + 60 \, a \cos \left (d x + c\right ) + 60 \, a \log \left (-\cos \left (d x + c\right )\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.58, size = 201, normalized size = 2.31 \[ \frac {60 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 60 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {201 \, a - \frac {1125 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2610 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1970 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {805 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {137 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 95, normalized size = 1.09 \[ -\frac {8 a \cos \left (d x +c \right )}{15 d}-\frac {a \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{5 d}-\frac {4 a \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{15 d}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 69, normalized size = 0.79 \[ -\frac {12 \, a \cos \left (d x + c\right )^{5} + 15 \, a \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, a \cos \left (d x + c\right )^{2} + 60 \, a \cos \left (d x + c\right ) + 60 \, a \log \left (\cos \left (d x + c\right )\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 67, normalized size = 0.77 \[ -\frac {a\,\cos \left (c+d\,x\right )-a\,{\cos \left (c+d\,x\right )}^2-\frac {2\,a\,{\cos \left (c+d\,x\right )}^3}{3}+\frac {a\,{\cos \left (c+d\,x\right )}^4}{4}+\frac {a\,{\cos \left (c+d\,x\right )}^5}{5}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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 \[ a \left (\int \sin ^{5}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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