Optimal. Leaf size=40 \[ -\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{b}+\frac {\log (\tan (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2620, 266, 43} \[ -\frac {\cot ^4(a+b x)}{4 b}-\frac {\cot ^2(a+b x)}{b}+\frac {\log (\tan (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2620
Rubi steps
\begin {align*} \int \csc ^5(a+b x) \sec (a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^5} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(1+x)^2}{x^3} \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {2}{x^2}+\frac {1}{x}\right ) \, dx,x,\tan ^2(a+b x)\right )}{2 b}\\ &=-\frac {\cot ^2(a+b x)}{b}-\frac {\cot ^4(a+b x)}{4 b}+\frac {\log (\tan (a+b x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 44, normalized size = 1.10 \[ -\frac {\csc ^4(a+b x)+2 \csc ^2(a+b x)-4 \log (\sin (a+b x))+4 \log (\cos (a+b x))}{4 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 105, normalized size = 2.62 \[ \frac {2 \, \cos \left (b x + a\right )^{2} - 2 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\cos \left (b x + a\right )^{2}\right ) + 2 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{4} \, \cos \left (b x + a\right )^{2} + \frac {1}{4}\right ) - 3}{4 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 165, normalized size = 4.12 \[ \frac {\frac {{\left (\frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {48 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} + \frac {12 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 32 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right ) - 64 \, \log \left ({\left | -\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1 \right |}\right )}{64 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 39, normalized size = 0.98 \[ -\frac {1}{4 \sin \left (b x +a \right )^{4} b}-\frac {1}{2 \sin \left (b x +a \right )^{2} b}+\frac {\ln \left (\tan \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 51, normalized size = 1.28 \[ -\frac {\frac {2 \, \sin \left (b x + a\right )^{2} + 1}{\sin \left (b x + a\right )^{4}} + 2 \, \log \left (\sin \left (b x + a\right )^{2} - 1\right ) - 2 \, \log \left (\sin \left (b x + a\right )^{2}\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.44, size = 79, normalized size = 1.98 \[ \frac {\ln \left (\frac {\cos \left (2\,a+2\,b\,x\right )}{2}-\frac {1}{2}\right )}{2\,b}-\frac {\ln \left (\cos \left (a+b\,x\right )\right )}{b}-\frac {\frac {\cos \left (2\,a+2\,b\,x\right )}{4}-\frac {1}{2}}{b\,\left (\cos \left (2\,a+2\,b\,x\right )-{\left (\frac {\cos \left (2\,a+2\,b\,x\right )}{2}+\frac {1}{2}\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\left (a + b x \right )}}{\sin ^{5}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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