Optimal. Leaf size=119 \[ \frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^6(c+d x)}{6 d}-\frac {3 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{4 d}+\frac {a \cos ^3(c+d x)}{d}+\frac {3 a \cos ^2(c+d x)}{2 d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2836, 12, 88} \[ \frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^6(c+d x)}{6 d}-\frac {3 a \cos ^5(c+d x)}{5 d}-\frac {3 a \cos ^4(c+d x)}{4 d}+\frac {a \cos ^3(c+d x)}{d}+\frac {3 a \cos ^2(c+d x)}{2 d}-\frac {a \cos (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^6(c+d x) \tan (c+d x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {a (-a-x)^3 (-a+x)^4}{x} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(-a-x)^3 (-a+x)^4}{x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^6-\frac {a^7}{x}+3 a^5 x-3 a^4 x^2-3 a^3 x^3+3 a^2 x^4+a x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {3 a \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 a \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 86, normalized size = 0.72 \[ \frac {a \left (1120 \cos ^6(c+d x)-5040 \cos ^4(c+d x)+10080 \cos ^2(c+d x)-3675 \cos (c+d x)+735 \cos (3 (c+d x))-147 \cos (5 (c+d x))+15 \cos (7 (c+d x))-6720 \log (\cos (c+d x))\right )}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 93, normalized size = 0.78 \[ \frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 247, normalized size = 2.08 \[ \frac {420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {1473 \, a - \frac {11151 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {36813 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {69475 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {28749 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 129, normalized size = 1.08 \[ -\frac {16 a \cos \left (d x +c \right )}{35 d}-\frac {a \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}-\frac {6 a \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}-\frac {8 a \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d}-\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{6 d}-\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 91, normalized size = 0.76 \[ \frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, a \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, a \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, a \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, a \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 89, normalized size = 0.75 \[ -\frac {a\,\cos \left (c+d\,x\right )-\frac {3\,a\,{\cos \left (c+d\,x\right )}^2}{2}-a\,{\cos \left (c+d\,x\right )}^3+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{4}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {a\,{\cos \left (c+d\,x\right )}^6}{6}-\frac {a\,{\cos \left (c+d\,x\right )}^7}{7}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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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