Integrand size = 23, antiderivative size = 82 \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {(a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 b^{3/2} (a+b)^{3/2} f}-\frac {a \tan (e+f x)}{2 b (a+b) f \left (a+b+b \tan ^2(e+f x)\right )} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4231, 393, 211} \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {(a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 b^{3/2} f (a+b)^{3/2}}-\frac {a \tan (e+f x)}{2 b f (a+b) \left (a+b \tan ^2(e+f x)+b\right )} \]
[In]
[Out]
Rule 211
Rule 393
Rule 4231
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1+x^2}{\left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a \tan (e+f x)}{2 b (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(a+2 b) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{2 b (a+b) f} \\ & = \frac {(a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{2 b^{3/2} (a+b)^{3/2} f}-\frac {a \tan (e+f x)}{2 b (a+b) f \left (a+b+b \tan ^2(e+f x)\right )} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02 \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {(a+2 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {a \sqrt {b} \sin (2 (e+f x))}{(a+b) (a+2 b+a \cos (2 (e+f x)))}}{2 b^{3/2} f} \]
[In]
[Out]
Time = 0.62 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {a \tan \left (f x +e \right )}{2 \left (a +b \right ) b \left (a +b +b \tan \left (f x +e \right )^{2}\right )}+\frac {\left (a +2 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) b \sqrt {\left (a +b \right ) b}}}{f}\) | \(76\) |
default | \(\frac {-\frac {a \tan \left (f x +e \right )}{2 \left (a +b \right ) b \left (a +b +b \tan \left (f x +e \right )^{2}\right )}+\frac {\left (a +2 b \right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{2 \left (a +b \right ) b \sqrt {\left (a +b \right ) b}}}{f}\) | \(76\) |
risch | \(-\frac {i \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}{b f \left (a +b \right ) \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )}-\frac {a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i b a +2 i b^{2}+a \sqrt {-a b -b^{2}}+2 b \sqrt {-a b -b^{2}}}{a \sqrt {-a b -b^{2}}}\right )}{4 \sqrt {-a b -b^{2}}\, \left (a +b \right ) f b}-\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i b a +2 i b^{2}+a \sqrt {-a b -b^{2}}+2 b \sqrt {-a b -b^{2}}}{a \sqrt {-a b -b^{2}}}\right )}{2 \sqrt {-a b -b^{2}}\, \left (a +b \right ) f}+\frac {a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i b a +2 i b^{2}-a \sqrt {-a b -b^{2}}-2 b \sqrt {-a b -b^{2}}}{a \sqrt {-a b -b^{2}}}\right )}{4 \sqrt {-a b -b^{2}}\, \left (a +b \right ) f b}+\frac {\ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i b a +2 i b^{2}-a \sqrt {-a b -b^{2}}-2 b \sqrt {-a b -b^{2}}}{a \sqrt {-a b -b^{2}}}\right )}{2 \sqrt {-a b -b^{2}}\, \left (a +b \right ) f}\) | \(452\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (70) = 140\).
Time = 0.28 (sec) , antiderivative size = 406, normalized size of antiderivative = 4.95 \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\left [-\frac {4 \, {\left (a^{2} b + a b^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{2} + a b + 2 \, b^{2}\right )} \sqrt {-a b - b^{2}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b - b^{2}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right )}{8 \, {\left ({\left (a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} f\right )}}, -\frac {2 \, {\left (a^{2} b + a b^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{2} + a b + 2 \, b^{2}\right )} \sqrt {a b + b^{2}} \arctan \left (\frac {{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{2 \, \sqrt {a b + b^{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right )}{4 \, {\left ({\left (a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} b^{3} + 2 \, a b^{4} + b^{5}\right )} f\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\int \frac {\sec ^{4}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2}}\, dx \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.07 \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=-\frac {\frac {a \tan \left (f x + e\right )}{a^{2} b + 2 \, a b^{2} + b^{3} + {\left (a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{2}} - \frac {{\left (a + 2 \, b\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} {\left (a b + b^{2}\right )}}}{2 \, f} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.09 \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )} {\left (a + 2 \, b\right )}}{{\left (a b + b^{2}\right )}^{\frac {3}{2}}} - \frac {a \tan \left (f x + e\right )}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )} {\left (a b + b^{2}\right )}}}{2 \, f} \]
[In]
[Out]
Time = 19.61 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\sec ^4(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\mathrm {tan}\left (e+f\,x\right )}{\sqrt {a+b}}\right )\,\left (a+2\,b\right )}{2\,b^{3/2}\,f\,{\left (a+b\right )}^{3/2}}-\frac {a\,\mathrm {tan}\left (e+f\,x\right )}{2\,b\,f\,\left (a+b\right )\,\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a+b\right )} \]
[In]
[Out]