Integrand size = 21, antiderivative size = 46 \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=-\frac {(a+2 b) \cot (e+f x)}{f}-\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \]
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Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4217, 459} \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=-\frac {(a+b) \cot ^3(e+f x)}{3 f}-\frac {(a+2 b) \cot (e+f x)}{f}+\frac {b \tan (e+f x)}{f} \]
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Rule 459
Rule 4217
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b+b x^2\right )}{x^4} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (b+\frac {a+b}{x^4}+\frac {a+2 b}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(a+2 b) \cot (e+f x)}{f}-\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.83 \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=-\frac {2 a \cot (e+f x)}{3 f}-\frac {5 b \cot (e+f x)}{3 f}-\frac {a \cot (e+f x) \csc ^2(e+f x)}{3 f}-\frac {b \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \]
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Time = 0.49 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {2}{3}-\frac {\csc \left (f x +e \right )^{2}}{3}\right ) \cot \left (f x +e \right )+b \left (-\frac {1}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {4}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {8 \cot \left (f x +e \right )}{3}\right )}{f}\) | \(73\) |
default | \(\frac {a \left (-\frac {2}{3}-\frac {\csc \left (f x +e \right )^{2}}{3}\right ) \cot \left (f x +e \right )+b \left (-\frac {1}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {4}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {8 \cot \left (f x +e \right )}{3}\right )}{f}\) | \(73\) |
risch | \(\frac {4 i \left (3 a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+8 b \,{\mathrm e}^{2 i \left (f x +e \right )}-a -4 b \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(76\) |
parallelrisch | \(\frac {\cot \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (\left (a +b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}+\left (8 a +20 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}+\left (-18 a -90 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}+\left (8 a +20 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+a +b \right )}{24 f \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-24 f}\) | \(108\) |
norman | \(\frac {\frac {a +b}{24 f}+\frac {\left (a +b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{8}}{24 f}-\frac {3 \left (a +5 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{4 f}+\frac {\left (2 a +5 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}{6 f}+\frac {\left (2 a +5 b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}}{6 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )}\) | \(123\) |
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Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.43 \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=-\frac {2 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{4} - 3 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \csc ^{4}{\left (e + f x \right )}\, dx \]
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Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.93 \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {3 \, b \tan \left (f x + e\right ) - \frac {3 \, {\left (a + 2 \, b\right )} \tan \left (f x + e\right )^{2} + a + b}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
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Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.09 \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {3 \, b \tan \left (f x + e\right ) - \frac {3 \, a \tan \left (f x + e\right )^{2} + 6 \, b \tan \left (f x + e\right )^{2} + a + b}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
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Time = 19.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx=\frac {b\,\mathrm {tan}\left (e+f\,x\right )}{f}-\frac {\left (a+2\,b\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {a}{3}+\frac {b}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3} \]
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