Integrand size = 15, antiderivative size = 38 \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2702, 308, 213} \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{b}+\frac {\sec ^3(a+b x)}{3 b}+\frac {\sec (a+b x)}{b} \]
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Rule 213
Rule 308
Rule 2702
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (1+x^2+\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 {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{b} \\ & = -\frac {\text {arctanh}(\cos (a+b x))}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.50 \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{b}+\frac {\sec (a+b x)}{b}+\frac {\sec ^3(a+b x)}{3 b} \]
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Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(\frac {\frac {1}{3 \cos \left (b x +a \right )^{3}}+\frac {1}{\cos \left (b x +a \right )}+\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b}\) | \(40\) |
default | \(\frac {\frac {1}{3 \cos \left (b x +a \right )^{3}}+\frac {1}{\cos \left (b x +a \right )}+\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{b}\) | \(40\) |
norman | \(\frac {\frac {4 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {8}{3 b}-\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3}}+\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}\) | \(70\) |
risch | \(\frac {2 \,{\mathrm e}^{5 i \left (b x +a \right )}+\frac {20 \,{\mathrm e}^{3 i \left (b x +a \right )}}{3}+2 \,{\mathrm e}^{i \left (b x +a \right )}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b}\) | \(87\) |
parallelrisch | \(\frac {3 \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3} \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+12 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-8}{3 b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{3} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{3}}\) | \(98\) |
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Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.76 \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=-\frac {3 \, \cos \left (b x + a\right )^{3} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, \cos \left (b x + a\right )^{3} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (b x + a\right )^{2} - 2}{6 \, b \cos \left (b x + a\right )^{3}} \]
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\[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=\int \frac {\sec ^{4}{\left (a + b x \right )}}{\sin {\left (a + b x \right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=\frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{2} + 1\right )}}{\cos \left (b x + a\right )^{3}} - 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{6 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (36) = 72\).
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66 \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=\frac {\frac {8 \, {\left (\frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {3 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 2\right )}}{{\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} + 1\right )}^{3}} + 3 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{6 \, b} \]
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Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \csc (a+b x) \sec ^4(a+b x) \, dx=-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )-\frac {{\cos \left (a+b\,x\right )}^2+\frac {1}{3}}{{\cos \left (a+b\,x\right )}^3}}{b} \]
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