Optimal. Leaf size=73 \[ -\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 73, 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,
76} \begin {gather*} \frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^6(c+d x)}{3 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 76
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) \sin ^7(c+d x)}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^3 x^2 (-a+x)}{a^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int (-a-x)^3 x^2 (-a+x) \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\text {Subst}\left (\int \left (a^4 x^2+2 a^3 x^3-2 a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=-\frac {\cos ^3(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 1.19, size = 53, normalized size = 0.73 \begin {gather*} \frac {4 (17 \cos (c+d x)+10 \cos (2 (c+d x))+3 (4+\cos (3 (c+d x)))) \sin ^8\left (\frac {1}{2} (c+d x)\right )}{21 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 50, normalized size = 0.68
method | result | size |
derivativedivides | \(-\frac {\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{2 \sec \left (d x +c \right )^{4}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}+\frac {1}{3 \sec \left (d x +c \right )^{6}}}{d \,a^{2}}\) | \(50\) |
default | \(-\frac {\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{2 \sec \left (d x +c \right )^{4}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}+\frac {1}{3 \sec \left (d x +c \right )^{6}}}{d \,a^{2}}\) | \(50\) |
risch | \(-\frac {11 \cos \left (d x +c \right )}{64 a^{2} d}+\frac {\cos \left (7 d x +7 c \right )}{448 d \,a^{2}}-\frac {\cos \left (6 d x +6 c \right )}{96 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{64 d \,a^{2}}-\frac {7 \cos \left (3 d x +3 c \right )}{192 d \,a^{2}}+\frac {3 \cos \left (2 d x +2 c \right )}{32 d \,a^{2}}\) | \(101\) |
norman | \(\frac {-\frac {16 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {8 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {8}{21 a d}-\frac {8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {40 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {16 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} a}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 49, normalized size = 0.67 \begin {gather*} \frac {6 \, \cos \left (d x + c\right )^{7} - 14 \, \cos \left (d x + c\right )^{6} + 21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3}}{42 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.78, size = 49, normalized size = 0.67 \begin {gather*} \frac {6 \, \cos \left (d x + c\right )^{7} - 14 \, \cos \left (d x + c\right )^{6} + 21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3}}{42 \, 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. 141 vs.
\(2 (65) = 130\).
time = 0.46, size = 141, normalized size = 1.93 \begin {gather*} -\frac {8 \, {\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 {14 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {42 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1\right )}}{21 \, a^{2} 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 = 0.92, size = 58, normalized size = 0.79 \begin {gather*} -\frac {\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}+\frac {{\cos \left (c+d\,x\right )}^6}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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