Integrand size = 21, antiderivative size = 42 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {(a+b) \cot (e+f x)}{f}-\frac {a \cot ^3(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \]
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Time = 0.05 (sec) , antiderivative size = 42, 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 = {3744, 459} \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {(a+b) \cot (e+f x)}{f}-\frac {a \cot ^3(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \]
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Rule 459
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b x^2\right )}{x^4} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (b+\frac {a}{x^4}+\frac {a+b}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(a+b) \cot (e+f x)}{f}-\frac {a \cot ^3(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.43 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {2 a \cot (e+f x)}{3 f}-\frac {b \cot (e+f x)}{f}-\frac {a \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f} \]
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Time = 0.50 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {2}{3}-\frac {\csc \left (f x +e \right )^{2}}{3}\right ) \cot \left (f x +e \right )+b \left (\frac {1}{\sin \left (f x +e \right ) \cos \left (f x +e \right )}-2 \cot \left (f x +e \right )\right )}{f}\) | \(54\) |
default | \(\frac {a \left (-\frac {2}{3}-\frac {\csc \left (f x +e \right )^{2}}{3}\right ) \cot \left (f x +e \right )+b \left (\frac {1}{\sin \left (f x +e \right ) \cos \left (f x +e \right )}-2 \cot \left (f x +e \right )\right )}{f}\) | \(54\) |
risch | \(\frac {4 i \left (3 a \,{\mathrm e}^{4 i \left (f x +e \right )}-3 b \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+6 b \,{\mathrm e}^{2 i \left (f x +e \right )}-a -3 b \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}\) | \(88\) |
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Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.57 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=-\frac {2 \, {\left (a + 3 \, b\right )} \cos \left (f x + e\right )^{4} - 3 \, {\left (a + 3 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
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\[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=\int \left (a + b \tan ^{2}{\left (e + f x \right )}\right ) \csc ^{4}{\left (e + f x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=\frac {3 \, b \tan \left (f x + e\right ) - \frac {3 \, {\left (a + b\right )} \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
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Time = 0.42 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=\frac {3 \, b \tan \left (f x + e\right ) - \frac {3 \, a \tan \left (f x + e\right )^{2} + 3 \, b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
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Time = 10.72 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.98 \[ \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx=\frac {b\,\mathrm {tan}\left (e+f\,x\right )}{f}-\frac {\left (a+b\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {a}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3} \]
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