Optimal. Leaf size=279 \[ -\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{7 b^3 \csc (c+d x)}{2 d}+\frac{7 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.316217, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3872, 2912, 3767, 2621, 302, 207, 2620, 270, 288} \[ -\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{7 b^3 \csc (c+d x)}{2 d}+\frac{7 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2912
Rule 3767
Rule 2621
Rule 302
Rule 207
Rule 2620
Rule 270
Rule 288
Rubi steps
\begin{align*} \int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx &=-\int (-b-a \cos (c+d x))^3 \csc ^6(c+d x) \sec ^3(c+d x) \, dx\\ &=\int \left (a^3 \csc ^6(c+d x)+3 a^2 b \csc ^6(c+d x) \sec (c+d x)+3 a b^2 \csc ^6(c+d x) \sec ^2(c+d x)+b^3 \csc ^6(c+d x) \sec ^3(c+d x)\right ) \, dx\\ &=a^3 \int \csc ^6(c+d x) \, dx+\left (3 a^2 b\right ) \int \csc ^6(c+d x) \sec (c+d x) \, dx+\left (3 a b^2\right ) \int \csc ^6(c+d x) \sec ^2(c+d x) \, dx+b^3 \int \csc ^6(c+d x) \sec ^3(c+d x) \, dx\\ &=-\frac{a^3 \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^6} \, dx,x,\tan (c+d x)\right )}{d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{x^8}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac{a^3 \cot (c+d x)}{d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot ^5(c+d x)}{5 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \left (1+x^2+x^4+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (1+\frac{1}{x^6}+\frac{3}{x^4}+\frac{3}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}-\frac{\left (7 b^3\right ) \operatorname{Subst}\left (\int \frac{x^6}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=-\frac{a^3 \cot (c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc ^5(c+d x)}{5 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{\left (3 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}-\frac{\left (7 b^3\right ) \operatorname{Subst}\left (\int \left (1+x^2+x^4+\frac{1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^3 \cot (c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{7 b^3 \csc (c+d x)}{2 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{\left (7 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{2 d}\\ &=\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{7 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 b \csc (c+d x)}{d}-\frac{7 b^3 \csc (c+d x)}{2 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}+\frac{3 a b^2 \tan (c+d x)}{d}\\ \end{align*}
Mathematica [B] time = 1.41209, size = 812, normalized size = 2.91 \[ -\frac{\csc ^9\left (\frac{1}{2} (c+d x)\right ) \sec ^5\left (\frac{1}{2} (c+d x)\right ) \left (16 \cos (3 (c+d x)) a^3-48 \cos (5 (c+d x)) a^3+16 \cos (7 (c+d x)) a^3+1176 b a^2-600 b \cos (4 (c+d x)) a^2+180 b \cos (6 (c+d x)) a^2+450 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (c+d x) a^2-450 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (c+d x) a^2+90 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (3 (c+d x)) a^2-90 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (3 (c+d x)) a^2-270 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (5 (c+d x)) a^2+270 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (5 (c+d x)) a^2+90 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (7 (c+d x)) a^2-90 b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (7 (c+d x)) a^2+80 \left (5 a^2+18 b^2\right ) \cos (c+d x) a+288 b^2 \cos (3 (c+d x)) a-864 b^2 \cos (5 (c+d x)) a+288 b^2 \cos (7 (c+d x)) a+412 b^3+66 \left (7 b^3+6 a^2 b\right ) \cos (2 (c+d x))-700 b^3 \cos (4 (c+d x))+210 b^3 \cos (6 (c+d x))+525 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (c+d x)-525 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (c+d x)+105 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-105 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-315 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+315 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+105 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))-105 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{1}{2} (c+d x)\right )\right ) \sin (7 (c+d x))\right )}{61440 d \left (\cot ^2\left (\frac{1}{2} (c+d x)\right )-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 334, normalized size = 1.2 \begin{align*} -{\frac{8\,{a}^{3}\cot \left ( dx+c \right ) }{15\,d}}-{\frac{{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{4}}{5\,d}}-{\frac{4\,{a}^{3}\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{15\,d}}-{\frac{3\,{a}^{2}b}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{2}b}{d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-3\,{\frac{{a}^{2}b}{d\sin \left ( dx+c \right ) }}+3\,{\frac{{a}^{2}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}-{\frac{3\,a{b}^{2}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}\cos \left ( dx+c \right ) }}-{\frac{6\,a{b}^{2}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) }}+{\frac{24\,a{b}^{2}}{5\,d\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) }}-{\frac{48\,a{b}^{2}\cot \left ( dx+c \right ) }{5\,d}}-{\frac{{b}^{3}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{7\,{b}^{3}}{15\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{7\,{b}^{3}}{6\,d\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{7\,{b}^{3}}{2\,d\sin \left ( dx+c \right ) }}+{\frac{7\,{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997119, size = 311, normalized size = 1.11 \begin{align*} -\frac{b^{3}{\left (\frac{2 \,{\left (105 \, \sin \left (d x + c\right )^{6} - 70 \, \sin \left (d x + c\right )^{4} - 14 \, \sin \left (d x + c\right )^{2} - 6\right )}}{\sin \left (d x + c\right )^{7} - \sin \left (d x + c\right )^{5}} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{2} b{\left (\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} + 3\right )}}{\sin \left (d x + c\right )^{5}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a b^{2}{\left (\frac{15 \, \tan \left (d x + c\right )^{4} + 5 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{5}} - 5 \, \tan \left (d x + c\right )\right )} + \frac{4 \,{\left (15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} + 3\right )} a^{3}}{\tan \left (d x + c\right )^{5}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8618, size = 857, normalized size = 3.07 \begin{align*} -\frac{32 \,{\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 30 \,{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 80 \,{\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} - 70 \,{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} - 180 \, a b^{2} \cos \left (d x + c\right ) + 60 \,{\left (a^{3} + 18 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 30 \, b^{3} + 46 \,{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2} - 15 \,{\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} +{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 15 \,{\left ({\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{6} - 2 \,{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{4} +{\left (6 \, a^{2} b + 7 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{60 \,{\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\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 [A] time = 1.33031, size = 672, normalized size = 2.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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