Optimal. Leaf size=73 \[ \frac {\sin ^6(c+d x)}{6 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.15, 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 = {3872, 2835, 2564, 30, 2565, 270} \[ \frac {\sin ^6(c+d x)}{6 a d}+\frac {\cos ^7(c+d x)}{7 a d}-\frac {2 \cos ^5(c+d x)}{5 a d}+\frac {\cos ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2564
Rule 2565
Rule 2835
Rule 3872
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 {\operatorname {Subst}\left (\int x^5 \, dx,x,\sin (c+d x)\right )}{a d}+\frac {\operatorname {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 {\operatorname {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 = 1.72, size = 52, normalized size = 0.71 \[ \frac {4 \sin ^8\left (\frac {1}{2} (c+d x)\right ) (197 \cos (c+d x)+85 \cos (2 (c+d x))+15 \cos (3 (c+d x))+123)}{105 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 69, normalized size = 0.95 \[ \frac {30 \, \cos \left (d x + c\right )^{7} - 35 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 105 \, \cos \left (d x + c\right )^{2}}{210 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 119, normalized size = 1.63 \[ \frac {16 \, {\left (\frac {7 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {21 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {140 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 1\right )}}{105 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 70, normalized size = 0.96 \[ -\frac {\frac {1}{6 \sec \left (d x +c \right )^{6}}+\frac {1}{2 \sec \left (d x +c \right )^{2}}-\frac {1}{7 \sec \left (d x +c \right )^{7}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}-\frac {1}{2 \sec \left (d x +c \right )^{4}}+\frac {2}{5 \sec \left (d x +c \right )^{5}}}{d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 69, normalized size = 0.95 \[ \frac {30 \, \cos \left (d x + c\right )^{7} - 35 \, \cos \left (d x + c\right )^{6} - 84 \, \cos \left (d x + c\right )^{5} + 105 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 105 \, \cos \left (d x + c\right )^{2}}{210 \, a d} \]
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 \[ -\frac {\frac {{\cos \left (c+d\,x\right )}^2}{2\,a}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}-\frac {{\cos \left (c+d\,x\right )}^4}{2\,a}+\frac {2\,{\cos \left (c+d\,x\right )}^5}{5\,a}+\frac {{\cos \left (c+d\,x\right )}^6}{6\,a}-\frac {{\cos \left (c+d\,x\right )}^7}{7\,a}}{d} \]
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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