Integrand size = 25, antiderivative size = 30 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3744, 270} \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \]
[In]
[Out]
Rule 270
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {\cot (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{a f} \\ \end{align*}
Time = 0.69 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\cot (e+f x) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{\sqrt {2} a f} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(-\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{f a \tan \left (f x +e \right )}\) | \(31\) |
default | \(-\frac {\sqrt {a +b \tan \left (f x +e \right )^{2}}}{f a \tan \left (f x +e \right )}\) | \(31\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a f \sin \left (f x + e\right )} \]
[In]
[Out]
\[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {\csc ^{2}{\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{a f \tan \left (f x + e\right )} \]
[In]
[Out]
\[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )^{2}}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \]
[In]
[Out]
Time = 11.83 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\mathrm {cot}\left (e+f\,x\right )\,\sqrt {a+\frac {b\,{\sin \left (e+f\,x\right )}^2}{{\cos \left (e+f\,x\right )}^2}}}{a\,f} \]
[In]
[Out]