Optimal. Leaf size=55 \[ -\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^5(c+d x)}{5 a^2 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12,
45} \begin {gather*} -\frac {\cos ^5(c+d x)}{5 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^5(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^2 x^2}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int (-a-x)^2 x^2 \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \left (a^2 x^2+2 a x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=-\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^5(c+d x)}{5 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 42, normalized size = 0.76 \begin {gather*} \frac {4 (4+3 \cos (c+d x)+3 \cos (2 (c+d x))) \sin ^6\left (\frac {1}{2} (c+d x)\right )}{15 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 39, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {\frac {1}{2 \sec \left (d x +c \right )^{4}}-\frac {1}{5 \sec \left (d x +c \right )^{5}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}}{d \,a^{2}}\) | \(39\) |
default | \(\frac {\frac {1}{2 \sec \left (d x +c \right )^{4}}-\frac {1}{5 \sec \left (d x +c \right )^{5}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}}{d \,a^{2}}\) | \(39\) |
risch | \(-\frac {3 \cos \left (d x +c \right )}{8 a^{2} d}-\frac {\cos \left (5 d x +5 c \right )}{80 d \,a^{2}}+\frac {\cos \left (4 d x +4 c \right )}{16 d \,a^{2}}-\frac {7 \cos \left (3 d x +3 c \right )}{48 d \,a^{2}}+\frac {\cos \left (2 d x +2 c \right )}{4 d \,a^{2}}\) | \(84\) |
norman | \(\frac {-\frac {8 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {16}{15 a d}-\frac {8 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {32 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} a}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 39, normalized size = 0.71 \begin {gather*} -\frac {6 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3}}{30 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.85, size = 39, normalized size = 0.71 \begin {gather*} -\frac {6 \, \cos \left (d x + c\right )^{5} - 15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3}}{30 \, a^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (49) = 98\).
time = 0.48, size = 119, normalized size = 2.16 \begin {gather*} -\frac {8 \, {\left (\frac {10 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {20 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 2\right )}}{15 \, a^{2} 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 = 0.06, size = 36, normalized size = 0.65 \begin {gather*} -\frac {{\cos \left (c+d\,x\right )}^3\,\left (6\,{\cos \left (c+d\,x\right )}^2-15\,\cos \left (c+d\,x\right )+10\right )}{30\,a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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