Integrand size = 14, antiderivative size = 40 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=a^2 x+\frac {b (2 a+b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4213, 398, 209} \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=a^2 x+\frac {b (2 a+b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Rule 209
Rule 398
Rule 4213
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b+b x^2\right )^2}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \left (b (2 a+b)+b^2 x^2+\frac {a^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {b (2 a+b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = a^2 x+\frac {b (2 a+b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 a^2 f x+3 b (2 a+b) \tan (e+f x)+b^2 \tan ^3(e+f x)}{3 f} \]
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Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15
method | result | size |
parts | \(a^{2} x -\frac {b^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}+\frac {2 a b \tan \left (f x +e \right )}{f}\) | \(46\) |
derivativedivides | \(\frac {a^{2} \left (f x +e \right )+2 a b \tan \left (f x +e \right )-b^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(48\) |
default | \(\frac {a^{2} \left (f x +e \right )+2 a b \tan \left (f x +e \right )-b^{2} \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(48\) |
risch | \(a^{2} x +\frac {4 i b \left (3 a \,{\mathrm e}^{4 i \left (f x +e \right )}+6 a \,{\mathrm e}^{2 i \left (f x +e \right )}+3 b \,{\mathrm e}^{2 i \left (f x +e \right )}+3 a +b \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(68\) |
norman | \(\frac {a^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-a^{2} x +3 a^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3 a^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}-\frac {2 b \left (2 a +b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 b \left (2 a +b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{f}+\frac {4 b \left (6 a +b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}\) | \(138\) |
parallelrisch | \(\frac {3 x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6} a^{2} f +\left (-12 a b -6 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}-9 x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} a^{2} f +\left (24 a b +4 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}+9 x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} a^{2} f +\left (-12 a b -6 b^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-3 a^{2} f x}{3 f \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(158\) |
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Time = 0.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.45 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {3 \, a^{2} f x \cos \left (f x + e\right )^{3} + {\left (2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}{3 \, f \cos \left (f x + e\right )^{3}} \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}\, dx \]
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Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=a^{2} x + \frac {{\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} b^{2}}{3 \, f} + \frac {2 \, a b \tan \left (f x + e\right )}{f} \]
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Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.22 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {b^{2} \tan \left (f x + e\right )^{3} + 3 \, {\left (f x + e\right )} a^{2} + 6 \, a b \tan \left (f x + e\right ) + 3 \, b^{2} \tan \left (f x + e\right )}{3 \, f} \]
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Time = 18.82 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \left (a+b \sec ^2(e+f x)\right )^2 \, dx=\frac {\frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}-\mathrm {tan}\left (e+f\,x\right )\,\left (b^2-2\,b\,\left (a+b\right )\right )+a^2\,f\,x}{f} \]
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