Integrand size = 17, antiderivative size = 37 \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=-\frac {2 \cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \]
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Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2700, 276} \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=\frac {\tan (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}-\frac {2 \cot (a+b x)}{b} \]
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Rule 276
Rule 2700
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (a+b x)\right )}{b} \\ & = -\frac {2 \cot (a+b x)}{b}-\frac {\cot ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=-\frac {5 \cot (a+b x)}{3 b}-\frac {\cot (a+b x) \csc ^2(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \]
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Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {16 i \left (2 \,{\mathrm e}^{2 i \left (b x +a \right )}-1\right )}{3 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}\) | \(46\) |
derivativedivides | \(\frac {-\frac {1}{3 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}}+\frac {4}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {8 \cot \left (b x +a \right )}{3}}{b}\) | \(50\) |
default | \(\frac {-\frac {1}{3 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3}}+\frac {4}{3 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {8 \cot \left (b x +a \right )}{3}}{b}\) | \(50\) |
parallelrisch | \(\frac {\tan ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )+20 \left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\cot ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-90 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+20 \cot \left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 b}\) | \(80\) |
norman | \(\frac {\frac {1}{24 b}+\frac {5 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{6 b}-\frac {15 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{4 b}+\frac {5 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{6 b}+\frac {\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )}{24 b}}{\left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}}\) | \(98\) |
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Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.46 \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=-\frac {8 \, \cos \left (b x + a\right )^{4} - 12 \, \cos \left (b x + a\right )^{2} + 3}{3 \, {\left (b \cos \left (b x + a\right )^{3} - b \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )} \]
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\[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=\int \frac {\sec ^{2}{\left (a + b x \right )}}{\sin ^{4}{\left (a + b x \right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=-\frac {\frac {6 \, \tan \left (b x + a\right )^{2} + 1}{\tan \left (b x + a\right )^{3}} - 3 \, \tan \left (b x + a\right )}{3 \, b} \]
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Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.95 \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=-\frac {\frac {6 \, \tan \left (b x + a\right )^{2} + 1}{\tan \left (b x + a\right )^{3}} - 3 \, \tan \left (b x + a\right )}{3 \, b} \]
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Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int \csc ^4(a+b x) \sec ^2(a+b x) \, dx=\frac {\mathrm {tan}\left (a+b\,x\right )}{b}-\frac {2\,{\mathrm {tan}\left (a+b\,x\right )}^2+\frac {1}{3}}{b\,{\mathrm {tan}\left (a+b\,x\right )}^3} \]
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