Optimal. Leaf size=57 \[ \frac{\tan ^6(a+b x)}{6 b}+\frac{3 \tan ^4(a+b x)}{4 b}+\frac{3 \tan ^2(a+b x)}{2 b}+\frac{\log (\tan (a+b x))}{b} \]
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Rubi [A] time = 0.0312009, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2620, 266, 43} \[ \frac{\tan ^6(a+b x)}{6 b}+\frac{3 \tan ^4(a+b x)}{4 b}+\frac{3 \tan ^2(a+b x)}{2 b}+\frac{\log (\tan (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 2620
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \csc (a+b x) \sec ^7(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(1+x)^3}{x} \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (3+\frac{1}{x}+3 x+x^2\right ) \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=\frac{\log (\tan (a+b x))}{b}+\frac{3 \tan ^2(a+b x)}{2 b}+\frac{3 \tan ^4(a+b x)}{4 b}+\frac{\tan ^6(a+b x)}{6 b}\\ \end{align*}
Mathematica [A] time = 0.145671, size = 56, normalized size = 0.98 \[ -\frac{-2 \sec ^6(a+b x)-3 \sec ^4(a+b x)-6 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{12 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{6\,b \left ( \cos \left ( bx+a \right ) \right ) ^{6}}}+{\frac{1}{4\,b \left ( \cos \left ( bx+a \right ) \right ) ^{4}}}+{\frac{1}{2\,b \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00839, size = 115, normalized size = 2.02 \begin{align*} -\frac{\frac{6 \, \sin \left (b x + a\right )^{4} - 15 \, \sin \left (b x + a\right )^{2} + 11}{\sin \left (b x + a\right )^{6} - 3 \, \sin \left (b x + a\right )^{4} + 3 \, \sin \left (b x + a\right )^{2} - 1} + 6 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 6 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71733, size = 212, normalized size = 3.72 \begin{align*} -\frac{6 \, \cos \left (b x + a\right )^{6} \log \left (\cos \left (b x + a\right )^{2}\right ) - 6 \, \cos \left (b x + a\right )^{6} \log \left (-\frac{1}{4} \, \cos \left (b x + a\right )^{2} + \frac{1}{4}\right ) - 6 \, \cos \left (b x + a\right )^{4} - 3 \, \cos \left (b x + a\right )^{2} - 2}{12 \, b \cos \left (b x + a\right )^{6}} \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 = 1.22457, size = 289, normalized size = 5.07 \begin{align*} \frac{\frac{\frac{522 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac{1485 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{1580 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{1485 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac{522 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac{147 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + 147}{{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{6}} + 30 \, \log \left (\frac{{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 60 \, \log \left ({\left | -\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{60 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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