Optimal. Leaf size=55 \[ \frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d} \]
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Rubi [A]
time = 0.11, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2914,
2644, 30, 2645, 14} \begin {gather*} \frac {\sin ^4(c+d x)}{4 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 2914
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^5(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac {\int \cos (c+d x) \sin ^3(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^3(c+d x) \, dx}{a}\\ &=\frac {\text {Subst}\left (\int x^3 \, dx,x,\sin (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\sin ^4(c+d x)}{4 a d}+\frac {\text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^3(c+d x)}{3 a d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 42, normalized size = 0.76 \begin {gather*} \frac {2 (13+21 \cos (c+d x)+6 \cos (2 (c+d x))) \sin ^6\left (\frac {1}{2} (c+d x)\right )}{15 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 49, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {1}{5 \sec \left (d x +c \right )^{5}}+\frac {1}{4 \sec \left (d x +c \right )^{4}}+\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{2 \sec \left (d x +c \right )^{2}}}{d a}\) | \(49\) |
default | \(\frac {-\frac {1}{5 \sec \left (d x +c \right )^{5}}+\frac {1}{4 \sec \left (d x +c \right )^{4}}+\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{2 \sec \left (d x +c \right )^{2}}}{d a}\) | \(49\) |
norman | \(\frac {\frac {4}{15 a d}+\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(83\) |
risch | \(\frac {\cos \left (d x +c \right )}{8 a d}-\frac {\cos \left (5 d x +5 c \right )}{80 a d}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {\cos \left (3 d x +3 c \right )}{48 a d}-\frac {\cos \left (2 d x +2 c \right )}{8 a d}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 49, normalized size = 0.89 \begin {gather*} -\frac {12 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right )^{2}}{60 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.44, size = 49, normalized size = 0.89 \begin {gather*} -\frac {12 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right )^{2}}{60 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 97, normalized size = 1.76 \begin {gather*} \frac {4 \, {\left (\frac {5 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {10 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {30 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}}{15 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 58, normalized size = 1.05 \begin {gather*} -\frac {\frac {{\cos \left (c+d\,x\right )}^2}{2\,a}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}-\frac {{\cos \left (c+d\,x\right )}^4}{4\,a}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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