Optimal. Leaf size=73 \[ \frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.16, 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 = {3872, 2836, 12, 75} \[ \frac {\cos ^7(c+d x)}{7 a^2 d}-\frac {\cos ^6(c+d x)}{3 a^2 d}+\frac {\cos ^4(c+d x)}{2 a^2 d}-\frac {\cos ^3(c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 2836
Rule 3872
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 {\operatorname {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 {\operatorname {Subst}\left (\int (-a-x)^3 x^2 (-a+x) \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\operatorname {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 = 1.97, size = 53, normalized size = 0.73 \[ \frac {4 \sin ^8\left (\frac {1}{2} (c+d x)\right ) (17 \cos (c+d x)+10 \cos (2 (c+d x))+3 (\cos (3 (c+d x))+4))}{21 a^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 49, normalized size = 0.67 \[ \frac {6 \, \cos \left (d x + c\right )^{7} - 14 \, \cos \left (d x + c\right )^{6} + 21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3}}{42 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 141, normalized size = 1.93 \[ -\frac {8 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {21 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {42 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1\right )}}{21 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 50, normalized size = 0.68 \[ -\frac {\frac {1}{3 \sec \left (d x +c \right )^{6}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}+\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{2 \sec \left (d x +c \right )^{4}}}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 49, normalized size = 0.67 \[ \frac {6 \, \cos \left (d x + c\right )^{7} - 14 \, \cos \left (d x + c\right )^{6} + 21 \, \cos \left (d x + c\right )^{4} - 14 \, \cos \left (d x + c\right )^{3}}{42 \, a^{2} d} \]
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 \[ -\frac {\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}+\frac {{\cos \left (c+d\,x\right )}^6}{3\,a^2}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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