Optimal. Leaf size=119 \[ \frac{a \cos ^7(c+d x)}{7 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^6(c+d x)}{6 d}-\frac{3 b \cos ^4(c+d x)}{4 d}+\frac{3 b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.109538, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2837, 12, 766} \[ \frac{a \cos ^7(c+d x)}{7 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^6(c+d x)}{6 d}-\frac{3 b \cos ^4(c+d x)}{4 d}+\frac{3 b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2837
Rule 12
Rule 766
Rubi steps
\begin{align*} \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx &=-\int (-b-a \cos (c+d x)) \sin ^6(c+d x) \tan (c+d x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{a (-b+x) \left (a^2-x^2\right )^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-b+x) \left (a^2-x^2\right )^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^6-\frac{a^6 b}{x}+3 a^4 b x-3 a^4 x^2-3 a^2 b x^3+3 a^2 x^4+b 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 b \cos ^2(c+d x)}{2 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{3 b \cos ^4(c+d x)}{4 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{b \cos ^6(c+d x)}{6 d}+\frac{a \cos ^7(c+d x)}{7 d}-\frac{b \log (\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.13935, size = 115, normalized size = 0.97 \[ -\frac{35 a \cos (c+d x)}{64 d}+\frac{7 a \cos (3 (c+d x))}{64 d}-\frac{7 a \cos (5 (c+d x))}{320 d}+\frac{a \cos (7 (c+d x))}{448 d}-\frac{b \left (-\frac{1}{3} \cos ^6(c+d x)+\frac{3}{2} \cos ^4(c+d x)-3 \cos ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 129, normalized size = 1.1 \begin{align*} -{\frac{16\,a\cos \left ( dx+c \right ) }{35\,d}}-{\frac{a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{7\,d}}-{\frac{6\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d}}-{\frac{8\,a\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{35\,d}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96798, size = 123, normalized size = 1.03 \begin{align*} \frac{60 \, a \cos \left (d x + c\right )^{7} + 70 \, b \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, b \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, b \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, b \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83308, size = 261, normalized size = 2.19 \begin{align*} \frac{60 \, a \cos \left (d x + c\right )^{7} + 70 \, b \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, b \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, b \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, b \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \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 = 1.35996, size = 428, normalized size = 3.6 \begin{align*} \frac{420 \, b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{384 \, a + 1089 \, b - \frac{2688 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{8463 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{8064 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{28749 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{13440 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{56035 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{56035 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{28749 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{8463 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{1089 \, b{\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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