Optimal. Leaf size=267 \[ -\frac{3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}+\frac{b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}+\frac{\left (-9 a^2 b^2+3 a^4+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}-\frac{3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}-\frac{\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}+\frac{b^2 \left (a^2-b^2\right )^3}{a^9 d (a \cos (c+d x)+b)}+\frac{2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac{b \cos ^6(c+d x)}{3 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^2 d} \]
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Rubi [A] time = 0.372333, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2837, 12, 948} \[ -\frac{3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}+\frac{b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}+\frac{\left (-9 a^2 b^2+3 a^4+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}-\frac{3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}-\frac{\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}+\frac{b^2 \left (a^2-b^2\right )^3}{a^9 d (a \cos (c+d x)+b)}+\frac{2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac{b \cos ^6(c+d x)}{3 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2837
Rule 12
Rule 948
Rubi steps
\begin{align*} \int \frac{\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin ^7(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a^2-x^2\right )^3}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a^2-x^2\right )^3}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2-\frac{b^2 \left (-a^2+b^2\right )^3}{(b-x)^2}+\frac{2 b \left (-a^2+b^2\right )^2 \left (-a^2+4 b^2\right )}{b-x}-6 b \left (-a^2+b^2\right )^2 x-\left (3 a^4-9 a^2 b^2+5 b^4\right ) x^2-2 b \left (-3 a^2+2 b^2\right ) x^3+3 \left (a^2-b^2\right ) x^4-2 b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=-\frac{\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac{3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac{\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}+\frac{b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac{3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}-\frac{b \cos ^6(c+d x)}{3 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{b^2 \left (a^2-b^2\right )^3}{a^9 d (b+a \cos (c+d x))}+\frac{2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^9 d}\\ \end{align*}
Mathematica [A] time = 3.65723, size = 417, normalized size = 1.56 \[ \frac{-1848 a^6 b^2 \cos (4 (c+d x))+112 a^6 b^2 \cos (6 (c+d x))+8400 a^5 b^3 \cos (3 (c+d x))-336 a^5 b^3 \cos (5 (c+d x))+1120 a^4 b^4 \cos (4 (c+d x))-4480 a^3 b^5 \cos (3 (c+d x))-140 \left (-228 a^6 b^2+400 a^4 b^4-192 a^2 b^6+21 a^8\right ) \cos (2 (c+d x))+26880 a^6 b^2 \log (a \cos (c+d x)+b)-161280 a^4 b^4 \log (a \cos (c+d x)+b)+241920 a^2 b^6 \log (a \cos (c+d x)+b)+1680 a b \cos (c+d x) \left (16 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)+67 a^4 b^2-116 a^2 b^4-8 a^6+56 b^6\right )+61320 a^6 b^2-132720 a^4 b^4+87360 a^2 b^6-3780 a^7 b \cos (3 (c+d x))+476 a^7 b \cos (5 (c+d x))-40 a^7 b \cos (7 (c+d x))+588 a^8 \cos (4 (c+d x))-132 a^8 \cos (6 (c+d x))+15 a^8 \cos (8 (c+d x))-3675 a^8-107520 b^8 \log (a \cos (c+d x)+b)-13440 b^8}{13440 a^9 d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 456, normalized size = 1.7 \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7\,{a}^{2}d}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\,{a}^{3}d}}-{\frac{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{5\,{a}^{2}d}}+{\frac{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}{b}^{2}}{5\,d{a}^{4}}}+{\frac{3\,b \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{2\,{a}^{3}d}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}{b}^{3}}{d{a}^{5}}}+{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{{a}^{2}d}}-3\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}{b}^{2}}{d{a}^{4}}}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}{b}^{4}}{3\,d{a}^{6}}}-3\,{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{{a}^{3}d}}+6\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{d{a}^{5}}}-3\,{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}{b}^{5}}{d{a}^{7}}}-{\frac{\cos \left ( dx+c \right ) }{{a}^{2}d}}+9\,{\frac{{b}^{2}\cos \left ( dx+c \right ) }{d{a}^{4}}}-15\,{\frac{{b}^{4}\cos \left ( dx+c \right ) }{d{a}^{6}}}+7\,{\frac{{b}^{6}\cos \left ( dx+c \right ) }{d{a}^{8}}}+2\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{{a}^{3}d}}-12\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{5}}}+18\,{\frac{{b}^{5}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{7}}}-8\,{\frac{{b}^{7}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{9}}}+{\frac{{b}^{2}}{{a}^{3}d \left ( b+a\cos \left ( dx+c \right ) \right ) }}-3\,{\frac{{b}^{4}}{d{a}^{5} \left ( b+a\cos \left ( dx+c \right ) \right ) }}+3\,{\frac{{b}^{6}}{d{a}^{7} \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{{b}^{8}}{d{a}^{9} \left ( b+a\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03138, size = 366, normalized size = 1.37 \begin{align*} \frac{\frac{210 \,{\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )}}{a^{10} \cos \left (d x + c\right ) + a^{9} b} + \frac{30 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 126 \,{\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \,{\left (3 \, a^{5} b - 2 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 70 \,{\left (3 \, a^{6} - 9 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 630 \,{\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 210 \,{\left (a^{6} - 9 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{8}} + \frac{420 \,{\left (a^{6} b - 6 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 4 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{9}}}{210 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41865, size = 803, normalized size = 3.01 \begin{align*} \frac{120 \, a^{8} \cos \left (d x + c\right )^{8} - 160 \, a^{7} b \cos \left (d x + c\right )^{7} + 1715 \, a^{6} b^{2} - 4725 \, a^{4} b^{4} + 3780 \, a^{2} b^{6} - 840 \, b^{8} - 56 \,{\left (9 \, a^{8} - 4 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{6} + 84 \,{\left (9 \, a^{7} b - 4 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{5} + 140 \,{\left (6 \, a^{8} - 9 \, a^{6} b^{2} + 4 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 280 \,{\left (6 \, a^{7} b - 9 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 840 \,{\left (a^{8} - 6 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 35 \,{\left (a^{7} b + 153 \, a^{5} b^{3} - 324 \, a^{3} b^{5} + 168 \, a b^{7}\right )} \cos \left (d x + c\right ) + 1680 \,{\left (a^{6} b^{2} - 6 \, a^{4} b^{4} + 9 \, a^{2} b^{6} - 4 \, b^{8} +{\left (a^{7} b - 6 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{840 \,{\left (a^{10} d \cos \left (d x + c\right ) + a^{9} b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39638, size = 2512, normalized size = 9.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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