Optimal. Leaf size=210 \[ \frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{2 d}-\frac{a b^2 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}-\frac{b^3 \sec ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.219708, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3090, 3768, 3770, 2606, 30, 2611, 14} \[ \frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a b^2 \tan (c+d x) \sec ^5(c+d x)}{2 d}-\frac{a b^2 \tan (c+d x) \sec ^3(c+d x)}{8 d}-\frac{3 a b^2 \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}-\frac{b^3 \sec ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 3090
Rule 3768
Rule 3770
Rule 2606
Rule 30
Rule 2611
Rule 14
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=\int \left (a^3 \sec ^5(c+d x)+3 a^2 b \sec ^5(c+d x) \tan (c+d x)+3 a b^2 \sec ^5(c+d x) \tan ^2(c+d x)+b^3 \sec ^5(c+d x) \tan ^3(c+d x)\right ) \, dx\\ &=a^3 \int \sec ^5(c+d x) \, dx+\left (3 a^2 b\right ) \int \sec ^5(c+d x) \tan (c+d x) \, dx+\left (3 a b^2\right ) \int \sec ^5(c+d x) \tan ^2(c+d x) \, dx+b^3 \int \sec ^5(c+d x) \tan ^3(c+d x) \, dx\\ &=\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}+\frac{1}{4} \left (3 a^3\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{2} \left (a b^2\right ) \int \sec ^5(c+d x) \, dx+\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int x^4 \, dx,x,\sec (c+d x)\right )}{d}+\frac{b^3 \operatorname{Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{3 a^2 b \sec ^5(c+d x)}{5 d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{a b^2 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}+\frac{1}{8} \left (3 a^3\right ) \int \sec (c+d x) \, dx-\frac{1}{8} \left (3 a b^2\right ) \int \sec ^3(c+d x) \, dx+\frac{b^3 \operatorname{Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^2 b \sec ^5(c+d x)}{5 d}-\frac{b^3 \sec ^5(c+d x)}{5 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{3 a b^2 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{a b^2 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}-\frac{1}{16} \left (3 a b^2\right ) \int \sec (c+d x) \, dx\\ &=\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 a^2 b \sec ^5(c+d x)}{5 d}-\frac{b^3 \sec ^5(c+d x)}{5 d}+\frac{b^3 \sec ^7(c+d x)}{7 d}+\frac{3 a^3 \sec (c+d x) \tan (c+d x)}{8 d}-\frac{3 a b^2 \sec (c+d x) \tan (c+d x)}{16 d}+\frac{a^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{a b^2 \sec ^3(c+d x) \tan (c+d x)}{8 d}+\frac{a b^2 \sec ^5(c+d x) \tan (c+d x)}{2 d}\\ \end{align*}
Mathematica [B] time = 2.02354, size = 637, normalized size = 3.03 \[ \frac{\sec ^7(c+d x) \left (3584 \left (3 a^2 b-b^3\right ) \cos (2 (c+d x))-3675 a \left (2 a^2-b^2\right ) \cos (c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+10752 a^2 b+4340 a^3 \sin (2 (c+d x))+2800 a^3 \sin (4 (c+d x))+420 a^3 \sin (6 (c+d x))-4410 a^3 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-1470 a^3 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-210 a^3 \cos (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+4410 a^3 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1470 a^3 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+210 a^3 \cos (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+6790 a b^2 \sin (2 (c+d x))-1400 a b^2 \sin (4 (c+d x))-210 a b^2 \sin (6 (c+d x))+2205 a b^2 \cos (3 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+735 a b^2 \cos (5 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 a b^2 \cos (7 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-2205 a b^2 \cos (3 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-735 a b^2 \cos (5 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-105 a b^2 \cos (7 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1536 b^3\right )}{35840 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.133, size = 328, normalized size = 1.6 \begin{align*}{\frac{{a}^{3} \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{3\,{a}^{2}b}{5\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,a{b}^{2}\sin \left ( dx+c \right ) }{16\,d}}-{\frac{3\,a{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{7\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{3\,{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{35\,d\cos \left ( dx+c \right ) }}-{\frac{\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}{b}^{3}}{35\,d}}-{\frac{2\,{b}^{3}\cos \left ( dx+c \right ) }{35\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.22831, size = 281, normalized size = 1.34 \begin{align*} \frac{35 \, a b^{2}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 8 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 70 \, a^{3}{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + \frac{672 \, a^{2} b}{\cos \left (d x + c\right )^{5}} - \frac{32 \,{\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} b^{3}}{\cos \left (d x + c\right )^{7}}}{1120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.554504, size = 410, normalized size = 1.95 \begin{align*} \frac{105 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 160 \, b^{3} + 224 \,{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} + 70 \,{\left (3 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{5} + 8 \, a b^{2} \cos \left (d x + c\right ) + 2 \,{\left (2 \, a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1120 \, d \cos \left (d x + c\right )^{7}} \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 = 1.20132, size = 628, normalized size = 2.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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