Optimal. Leaf size=53 \[ \frac {\sec ^5(a+b x)}{5 b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}-\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 53, 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 = {2622, 302, 207} \[ \frac {\sec ^5(a+b x)}{5 b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}-\frac {\tanh ^{-1}(\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 207
Rule 302
Rule 2622
Rubi steps
\begin {align*} \int \csc (a+b x) \sec ^6(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+x^2+x^4+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac {\tanh ^{-1}(\cos (a+b x))}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 72, normalized size = 1.36 \[ \frac {\sec ^5(a+b x)}{5 b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 77, normalized size = 1.45 \[ -\frac {15 \, \cos \left (b x + a\right )^{5} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 15 \, \cos \left (b x + a\right )^{5} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 30 \, \cos \left (b x + a\right )^{4} - 10 \, \cos \left (b x + a\right )^{2} - 6}{30 \, b \cos \left (b x + a\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 145, normalized size = 2.74 \[ \frac {\frac {4 \, {\left (\frac {70 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {140 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {90 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + 23\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{5}} + 15 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{30 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 60, normalized size = 1.13 \[ \frac {1}{5 b \cos \left (b x +a \right )^{5}}+\frac {1}{3 b \cos \left (b x +a \right )^{3}}+\frac {1}{b \cos \left (b x +a \right )}+\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 60, normalized size = 1.13 \[ \frac {\frac {2 \, {\left (15 \, \cos \left (b x + a\right )^{4} + 5 \, \cos \left (b x + a\right )^{2} + 3\right )}}{\cos \left (b x + a\right )^{5}} - 15 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 15 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{30 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 45, normalized size = 0.85 \[ \frac {{\cos \left (a+b\,x\right )}^4+\frac {{\cos \left (a+b\,x\right )}^2}{3}+\frac {1}{5}}{b\,{\cos \left (a+b\,x\right )}^5}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{b} \]
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 \[ \int \frac {\sec ^{6}{\left (a + b x \right )}}{\sin {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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