Optimal. Leaf size=140 \[ -\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^6(c+d x)}{6 d}-\frac{3 b \cot ^4(c+d x)}{4 d}-\frac{3 b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d} \]
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Rubi [A] time = 0.144351, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {3872, 2834, 2620, 266, 43, 3768, 3770} \[ -\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^6(c+d x)}{6 d}-\frac{3 b \cot ^4(c+d x)}{4 d}-\frac{3 b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2834
Rule 2620
Rule 266
Rule 43
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx &=-\int (-b-a \cos (c+d x)) \csc ^7(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^7(c+d x) \, dx+b \int \csc ^7(c+d x) \sec (c+d x) \, dx\\ &=-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{6} (5 a) \int \csc ^5(c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^7} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{8} (5 a) \int \csc ^3(c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int \frac{(1+x)^3}{x^4} \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{1}{16} (5 a) \int \csc (c+d x) \, dx+\frac{b \operatorname{Subst}\left (\int \left (\frac{1}{x^4}+\frac{3}{x^3}+\frac{3}{x^2}+\frac{1}{x}\right ) \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=-\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{3 b \cot ^2(c+d x)}{2 d}-\frac{3 b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^6(c+d x)}{6 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac{b \log (\tan (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.601131, size = 216, normalized size = 1.54 \[ -\frac{a \csc ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}-\frac{a \csc ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}-\frac{5 a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{a \sec ^6\left (\frac{1}{2} (c+d x)\right )}{384 d}+\frac{a \sec ^4\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{5 a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{64 d}+\frac{5 a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{5 a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{16 d}-\frac{b \left (2 \csc ^6(c+d x)+3 \csc ^4(c+d x)+6 \csc ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.097, size = 136, normalized size = 1. \begin{align*} -{\frac{a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{5}}{6\,d}}-{\frac{5\,a\cot \left ( dx+c \right ) \left ( \csc \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{5\,a\cot \left ( dx+c \right ) \csc \left ( dx+c \right ) }{16\,d}}+{\frac{5\,a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{16\,d}}-{\frac{b}{6\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{b}{4\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{b}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b\ln \left ( \tan \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 = 0.964925, size = 193, normalized size = 1.38 \begin{align*} -\frac{3 \,{\left (5 \, a - 16 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \,{\left (5 \, a + 16 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 96 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac{2 \,{\left (15 \, a \cos \left (d x + c\right )^{5} + 24 \, b \cos \left (d x + c\right )^{4} - 40 \, a \cos \left (d x + c\right )^{3} - 60 \, b \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) + 44 \, b\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.18366, size = 749, normalized size = 5.35 \begin{align*} \frac{30 \, a \cos \left (d x + c\right )^{5} + 48 \, b \cos \left (d x + c\right )^{4} - 80 \, a \cos \left (d x + c\right )^{3} - 120 \, b \cos \left (d x + c\right )^{2} + 66 \, a \cos \left (d x + c\right ) - 96 \,{\left (b \cos \left (d x + c\right )^{6} - 3 \, b \cos \left (d x + c\right )^{4} + 3 \, b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\cos \left (d x + c\right )\right ) - 3 \,{\left ({\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (5 \, a - 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a + 16 \, b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 3 \,{\left ({\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{6} - 3 \,{\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (5 \, a + 16 \, b\right )} \cos \left (d x + c\right )^{2} - 5 \, a - 16 \, b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 88 \, b}{96 \,{\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - 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 = 1.42245, size = 482, normalized size = 3.44 \begin{align*} \frac{12 \,{\left (5 \, a + 16 \, b\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 384 \, b \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac{{\left (a + b - \frac{9 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{12 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{45 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{87 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{110 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{352 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}} - \frac{45 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{87 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{9 \, a{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{12 \, b{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{384 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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