Optimal. Leaf size=68 \[ -\frac {(a-a \sin (c+d x))^4}{a^5 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^6}{6 a^7 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2746, 45}
\begin {gather*} -\frac {(a-a \sin (c+d x))^6}{6 a^7 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^4}{a^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int (a-x)^3 (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \left (4 a^2 (a-x)^3-4 a (a-x)^4+(a-x)^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=-\frac {(a-a \sin (c+d x))^4}{a^5 d}+\frac {4 (a-a \sin (c+d x))^5}{5 a^6 d}-\frac {(a-a \sin (c+d x))^6}{6 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 66, normalized size = 0.97 \begin {gather*} -\frac {\sin (c+d x) \left (-30+15 \sin (c+d x)+20 \sin ^2(c+d x)-15 \sin ^3(c+d x)-6 \sin ^4(c+d x)+5 \sin ^5(c+d x)\right )}{30 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 65, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a}\) | \(65\) |
default | \(\frac {-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{2}-\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}+\sin \left (d x +c \right )}{d a}\) | \(65\) |
risch | \(\frac {5 \sin \left (d x +c \right )}{8 a d}+\frac {\cos \left (6 d x +6 c \right )}{192 a d}+\frac {\sin \left (5 d x +5 c \right )}{80 a d}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {5 \sin \left (3 d x +3 c \right )}{48 a d}+\frac {5 \cos \left (2 d x +2 c \right )}{64 a d}\) | \(101\) |
norman | \(\frac {\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {14 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {14 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {14 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {14 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {42 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}+\frac {42 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}+\frac {196 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {196 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {212 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {212 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(257\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 67, normalized size = 0.99 \begin {gather*} -\frac {5 \, \sin \left (d x + c\right )^{6} - 6 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right )}{30 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 49, normalized size = 0.72 \begin {gather*} \frac {5 \, \cos \left (d x + c\right )^{6} + 2 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right )}{30 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1096 vs.
\(2 (54) = 108\).
time = 24.65, size = 1096, normalized size = 16.12 \begin {gather*} \begin {cases} \frac {30 \tan ^{11}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {30 \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {70 \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {156 \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {100 \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {156 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {70 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} - \frac {30 \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} + \frac {30 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a d \tan ^{12}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{10}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{8}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 300 a d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 225 a d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 90 a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{7}{\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 67, normalized size = 0.99 \begin {gather*} -\frac {5 \, \sin \left (d x + c\right )^{6} - 6 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} + 20 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 30 \, \sin \left (d x + c\right )}{30 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.14, size = 80, normalized size = 1.18 \begin {gather*} \frac {\frac {\sin \left (c+d\,x\right )}{a}-\frac {{\sin \left (c+d\,x\right )}^2}{2\,a}-\frac {2\,{\sin \left (c+d\,x\right )}^3}{3\,a}+\frac {{\sin \left (c+d\,x\right )}^4}{2\,a}+\frac {{\sin \left (c+d\,x\right )}^5}{5\,a}-\frac {{\sin \left (c+d\,x\right )}^6}{6\,a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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