Optimal. Leaf size=87 \[ -\frac {a \cos (c+d x)}{d}+\frac {a \cos ^2(c+d x)}{d}+\frac {2 a \cos ^3(c+d x)}{3 d}-\frac {a \cos ^4(c+d x)}{4 d}-\frac {a \cos ^5(c+d x)}{5 d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A]
time = 0.07, 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 = {3957, 2915, 12,
90} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rule 3957
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 {\text {Subst}\left (\int \frac {a (-a-x)^2 (-a+x)^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^2 (-a+x)^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=\frac {\text {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 = 0.06, size = 83, normalized size = 0.95 \begin {gather*} -\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 (-\cos ^2(c+d x)+\frac {1}{4} \cos ^4(c+d x)+\log (\cos (c+d x))\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 67, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\) | \(67\) |
default | \(\frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\) | \(67\) |
risch | \(i a x +\frac {3 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {3 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}+\frac {2 i a c}{d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {5 a \cos \left (d x +c \right )}{8 d}-\frac {a \cos \left (5 d x +5 c \right )}{80 d}-\frac {a \cos \left (4 d x +4 c \right )}{32 d}+\frac {5 a \cos \left (3 d x +3 c \right )}{48 d}\) | \(120\) |
norman | \(\frac {-\frac {16 a}{15 d}-\frac {22 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {62 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {10 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {a \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(148\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 69, normalized size = 0.79 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 71, normalized size = 0.82 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 201 vs.
\(2 (81) = 162\).
time = 0.43, size = 201, normalized size = 2.31 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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