Optimal. Leaf size=199 \[ -\frac{2 a^2 \sin ^7(c+d x)}{7 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{139 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{245 a^2 x}{128} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.360779, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3872, 2872, 2637, 2635, 8, 2633, 3770, 3767} \[ -\frac{2 a^2 \sin ^7(c+d x)}{7 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{139 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{245 a^2 x}{128} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2872
Rule 2637
Rule 2635
Rule 8
Rule 2633
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx &=\int (-a-a \cos (c+d x))^2 \sin ^6(c+d x) \tan ^2(c+d x) \, dx\\ &=\frac{\int \left (-3 a^{10}-8 a^{10} \cos (c+d x)+2 a^{10} \cos ^2(c+d x)+12 a^{10} \cos ^3(c+d x)+2 a^{10} \cos ^4(c+d x)-8 a^{10} \cos ^5(c+d x)-3 a^{10} \cos ^6(c+d x)+2 a^{10} \cos ^7(c+d x)+a^{10} \cos ^8(c+d x)+2 a^{10} \sec (c+d x)+a^{10} \sec ^2(c+d x)\right ) \, dx}{a^8}\\ &=-3 a^2 x+a^2 \int \cos ^8(c+d x) \, dx+a^2 \int \sec ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^7(c+d x) \, dx+\left (2 a^2\right ) \int \sec (c+d x) \, dx-\left (3 a^2\right ) \int \cos ^6(c+d x) \, dx-\left (8 a^2\right ) \int \cos (c+d x) \, dx-\left (8 a^2\right ) \int \cos ^5(c+d x) \, dx+\left (12 a^2\right ) \int \cos ^3(c+d x) \, dx\\ &=-3 a^2 x+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{8 a^2 \sin (c+d x)}{d}+\frac{a^2 \cos (c+d x) \sin (c+d x)}{d}+\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{2 d}-\frac{a^2 \cos ^5(c+d x) \sin (c+d x)}{2 d}+\frac{a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} \left (7 a^2\right ) \int \cos ^6(c+d x) \, dx+a^2 \int 1 \, dx+\frac{1}{2} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac{1}{2} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx-\frac{a^2 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}+\frac{\left (8 a^2\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{\left (12 a^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-2 a^2 x+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{7 a^2 \cos (c+d x) \sin (c+d x)}{4 d}-\frac{a^2 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^7(c+d x)}{7 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{1}{48} \left (35 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{4} \left (3 a^2\right ) \int 1 \, dx-\frac{1}{8} \left (15 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{5 a^2 x}{4}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{13 a^2 \cos (c+d x) \sin (c+d x)}{16 d}+\frac{11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^7(c+d x)}{7 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{1}{64} \left (35 a^2\right ) \int \cos ^2(c+d x) \, dx-\frac{1}{16} \left (15 a^2\right ) \int 1 \, dx\\ &=-\frac{35 a^2 x}{16}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{139 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^7(c+d x)}{7 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{1}{128} \left (35 a^2\right ) \int 1 \, dx\\ &=-\frac{245 a^2 x}{128}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{139 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{11 a^2 \cos ^3(c+d x) \sin (c+d x)}{192 d}-\frac{17 a^2 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{a^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^7(c+d x)}{7 d}+\frac{a^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.883454, size = 144, normalized size = 0.72 \[ -\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left (\frac{1}{2} (c+d x)\right ) \left (30720 \sin ^7(c+d x)+43008 \sin ^5(c+d x)+71680 \sin ^3(c+d x)+215040 \sin (c+d x)-55440 \sin (2 (c+d x))+2520 \sin (4 (c+d x))+560 \sin (6 (c+d x))-105 \sin (8 (c+d x))+37800 \tan ^{-1}(\tan (c+d x))-107520 \tan (c+d x)-215040 \tanh ^{-1}(\sin (c+d x))+168000 c+168000 d x\right )}{430080 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.049, size = 210, normalized size = 1.1 \begin{align*}{\frac{7\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}\cos \left ( dx+c \right ) }{8\,d}}+{\frac{49\,{a}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{48\,d}}+{\frac{245\,{a}^{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{192\,d}}+{\frac{245\,{a}^{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{128\,d}}-{\frac{245\,{a}^{2}x}{128}}-{\frac{245\,{a}^{2}c}{128\,d}}-{\frac{2\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{2\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{2\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-2\,{\frac{{a}^{2}\sin \left ( dx+c \right ) }{d}}+2\,{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{d\cos \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.53875, size = 290, normalized size = 1.46 \begin{align*} -\frac{1024 \,{\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{2} - 35 \,{\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 840 \, d x + 840 \, c + 3 \, \sin \left (8 \, d x + 8 \, c\right ) + 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} + 2240 \,{\left (105 \, d x + 105 \, c - \frac{87 \, \tan \left (d x + c\right )^{5} + 136 \, \tan \left (d x + c\right )^{3} + 57 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1} - 48 \, \tan \left (d x + c\right )\right )} a^{2}}{107520 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.95531, size = 522, normalized size = 2.62 \begin{align*} -\frac{25725 \, a^{2} d x \cos \left (d x + c\right ) - 13440 \, a^{2} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + 13440 \, a^{2} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (1680 \, a^{2} \cos \left (d x + c\right )^{8} + 3840 \, a^{2} \cos \left (d x + c\right )^{7} - 4760 \, a^{2} \cos \left (d x + c\right )^{6} - 16896 \, a^{2} \cos \left (d x + c\right )^{5} + 770 \, a^{2} \cos \left (d x + c\right )^{4} + 31232 \, a^{2} \cos \left (d x + c\right )^{3} + 14595 \, a^{2} \cos \left (d x + c\right )^{2} - 45056 \, a^{2} \cos \left (d x + c\right ) + 13440 \, a^{2}\right )} \sin \left (d x + c\right )}{13440 \, d \cos \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.41883, size = 304, normalized size = 1.53 \begin{align*} -\frac{25725 \,{\left (d x + c\right )} a^{2} - 26880 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) + 26880 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + \frac{26880 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1} + \frac{2 \,{\left (39165 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} + 300265 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 989261 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1791073 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 1814943 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 670131 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 147735 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 14595 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{8}}}{13440 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]