Optimal. Leaf size=73 \[ \frac {\cos ^3(c+d x)}{3 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^6(c+d x)}{6 a d} \]
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Rubi [A]
time = 0.12, antiderivative size = 73, 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, 276} \begin {gather*} \frac {\sin ^6(c+d x)}{6 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \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 30
Rule 276
Rule 2644
Rule 2645
Rule 2914
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^7(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=\frac {\int \cos (c+d x) \sin ^5(c+d x) \, dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^5(c+d x) \, dx}{a}\\ &=\frac {\text {Subst}\left (\int x^5 \, dx,x,\sin (c+d x)\right )}{a d}+\frac {\text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\sin ^6(c+d x)}{6 a d}+\frac {\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{a d}\\ &=\frac {\cos ^3(c+d x)}{3 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^7(c+d x)}{7 a d}+\frac {\sin ^6(c+d x)}{6 a d}\\ \end {align*}
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Mathematica [A]
time = 1.00, size = 52, normalized size = 0.71 \begin {gather*} \frac {4 (123+197 \cos (c+d x)+85 \cos (2 (c+d x))+15 \cos (3 (c+d x))) \sin ^8\left (\frac {1}{2} (c+d x)\right )}{105 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 70, normalized size = 0.96
method | result | size |
derivativedivides | \(-\frac {\frac {1}{2 \sec \left (d x +c \right )^{2}}+\frac {2}{5 \sec \left (d x +c \right )^{5}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}+\frac {1}{6 \sec \left (d x +c \right )^{6}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}-\frac {1}{2 \sec \left (d x +c \right )^{4}}}{d a}\) | \(70\) |
default | \(-\frac {\frac {1}{2 \sec \left (d x +c \right )^{2}}+\frac {2}{5 \sec \left (d x +c \right )^{5}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}+\frac {1}{6 \sec \left (d x +c \right )^{6}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}-\frac {1}{2 \sec \left (d x +c \right )^{4}}}{d a}\) | \(70\) |
norman | \(\frac {\frac {16}{105 a d}+\frac {16 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {16 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}+\frac {16 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {64 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(102\) |
risch | \(\frac {5 \cos \left (d x +c \right )}{64 a d}+\frac {\cos \left (7 d x +7 c \right )}{448 a d}-\frac {\cos \left (6 d x +6 c \right )}{192 a d}-\frac {3 \cos \left (5 d x +5 c \right )}{320 a d}+\frac {\cos \left (4 d x +4 c \right )}{32 a d}+\frac {\cos \left (3 d x +3 c \right )}{192 a d}-\frac {5 \cos \left (2 d x +2 c \right )}{64 a d}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 69, normalized size = 0.95 \begin {gather*} \frac {30 \, \cos \left (d x + c\right )^{7} - 35 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 105 \, \cos \left (d x + c\right )^{2}}{210 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.83, size = 69, normalized size = 0.95 \begin {gather*} \frac {30 \, \cos \left (d x + c\right )^{7} - 35 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 105 \, \cos \left (d x + c\right )^{2}}{210 \, 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 = 0.46, size = 119, normalized size = 1.63 \begin {gather*} \frac {16 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {21 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {140 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )}}{105 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 84, normalized size = 1.15 \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}{2\,a}+\frac {2\,{\cos \left (c+d\,x\right )}^5}{5\,a}+\frac {{\cos \left (c+d\,x\right )}^6}{6\,a}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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