Integrand size = 23, antiderivative size = 42 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a-b+b \sec ^2(e+f x)}}\right )}{\sqrt {a} f} \] Output:
-arctanh(a^(1/2)*sec(f*x+e)/(a-b+b*sec(f*x+e)^2)^(1/2))/a^(1/2)/f
Leaf count is larger than twice the leaf count of optimal. \(221\) vs. \(2(42)=84\).
Time = 1.59 (sec) , antiderivative size = 221, normalized size of antiderivative = 5.26 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\frac {\cos (e+f x) \left (2 \text {arctanh}\left (\tan ^2\left (\frac {1}{2} (e+f x)\right )-\frac {\sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}}{\sqrt {a}}\right )+\log \left (a-2 b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sqrt {4 b \tan ^2\left (\frac {1}{2} (e+f x)\right )+a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^2}\right )\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{2 \sqrt {a} f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^4\left (\frac {1}{2} (e+f x)\right )}} \] Input:
Integrate[Csc[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]
Output:
(Cos[e + f*x]*(2*ArcTanh[Tan[(e + f*x)/2]^2 - Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2]/Sqrt[a]] + Log[a - 2*b - a*Tan[(e + f*x)/ 2]^2 + Sqrt[a]*Sqrt[4*b*Tan[(e + f*x)/2]^2 + a*(-1 + Tan[(e + f*x)/2]^2)^2 ]])*Sec[(e + f*x)/2]^2*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[e + f*x ]^2])/(2*Sqrt[a]*f*Sqrt[(a + b + (a - b)*Cos[2*(e + f*x)])*Sec[(e + f*x)/2 ]^4])
Time = 0.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4147, 25, 291, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\sin (e+f x) \sqrt {a+b \tan (e+f x)^2}}dx\) |
\(\Big \downarrow \) 4147 |
\(\displaystyle \frac {\int -\frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {1}{\left (1-\sec ^2(e+f x)\right ) \sqrt {b \sec ^2(e+f x)+a-b}}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle -\frac {\int \frac {1}{1-\frac {a \sec ^2(e+f x)}{b \sec ^2(e+f x)+a-b}}d\frac {\sec (e+f x)}{\sqrt {b \sec ^2(e+f x)+a-b}}}{f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {a} \sec (e+f x)}{\sqrt {a+b \sec ^2(e+f x)-b}}\right )}{\sqrt {a} f}\) |
Input:
Int[Csc[e + f*x]/Sqrt[a + b*Tan[e + f*x]^2],x]
Output:
-(ArcTanh[(Sqrt[a]*Sec[e + f*x])/Sqrt[a - b + b*Sec[e + f*x]^2]]/(Sqrt[a]* f))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Simp[1/(f*ff^ m) Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a - b + b*ff^2*x^2)^p/x^(m + 1 )), x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[( m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(301\) vs. \(2(36)=72\).
Time = 3.14 (sec) , antiderivative size = 302, normalized size of antiderivative = 7.19
method | result | size |
default | \(\frac {\sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \left (\ln \left (\frac {2 \sqrt {a}\, \sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right )+2 \sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {a}-2 a \cos \left (f x +e \right )+2 \cos \left (f x +e \right ) b +2 b}{\sqrt {a}\, \left (\cos \left (f x +e \right )+1\right )}\right )+\ln \left (\frac {-2 a \left (1-\cos \left (f x +e \right )\right )^{2}+4 \left (1-\cos \left (f x +e \right )\right )^{2} b +4 \sqrt {\frac {a \cos \left (f x +e \right )^{2}+b \sin \left (f x +e \right )^{2}}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {a}\, \sin \left (f x +e \right )^{2}+2 a \sin \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2}}\right )\right )}{f \sqrt {a}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )}\) | \(302\) |
Input:
int(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/f/a^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*(ln(2 /a^(1/2)*(a^(1/2)*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2) *cos(f*x+e)+((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/(cos(f*x+e)+1)^2)^(1/2)*a^(1/ 2)-a*cos(f*x+e)+cos(f*x+e)*b+b)/(cos(f*x+e)+1))+ln(2/(1-cos(f*x+e))^2*(-a* (1-cos(f*x+e))^2+2*(1-cos(f*x+e))^2*b+2*((a*cos(f*x+e)^2+b*sin(f*x+e)^2)/( cos(f*x+e)+1)^2)^(1/2)*a^(1/2)*sin(f*x+e)^2+a*sin(f*x+e)^2)))/(a+b*tan(f*x +e)^2)^(1/2)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)
Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 3.60 \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\left [\frac {\log \left (-\frac {2 \, {\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) + a + b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right )}{2 \, \sqrt {a} f}, -\frac {\sqrt {-a} \arctan \left (-\frac {\sqrt {-a} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}\right )}{a f}\right ] \] Input:
integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="fricas")
Output:
[1/2*log(-2*((a - b)*cos(f*x + e)^2 - 2*sqrt(a)*sqrt(((a - b)*cos(f*x + e) ^2 + b)/cos(f*x + e)^2)*cos(f*x + e) + a + b)/(cos(f*x + e)^2 - 1))/(sqrt( a)*f), -sqrt(-a)*arctan(-sqrt(-a)*sqrt(((a - b)*cos(f*x + e)^2 + b)/cos(f* x + e)^2)*cos(f*x + e)/((a - b)*cos(f*x + e)^2 + b))/(a*f)]
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {\csc {\left (e + f x \right )}}{\sqrt {a + b \tan ^{2}{\left (e + f x \right )}}}\, dx \] Input:
integrate(csc(f*x+e)/(a+b*tan(f*x+e)**2)**(1/2),x)
Output:
Integral(csc(e + f*x)/sqrt(a + b*tan(e + f*x)**2), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int { \frac {\csc \left (f x + e\right )}{\sqrt {b \tan \left (f x + e\right )^{2} + a}} \,d x } \] Input:
integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="maxima")
Output:
integrate(csc(f*x + e)/sqrt(b*tan(f*x + e)^2 + a), x)
Exception generated. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}} \,d x \] Input:
int(1/(sin(e + f*x)*(a + b*tan(e + f*x)^2)^(1/2)),x)
Output:
int(1/(sin(e + f*x)*(a + b*tan(e + f*x)^2)^(1/2)), x)
\[ \int \frac {\csc (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx=\int \frac {\sqrt {\tan \left (f x +e \right )^{2} b +a}\, \csc \left (f x +e \right )}{\tan \left (f x +e \right )^{2} b +a}d x \] Input:
int(csc(f*x+e)/(a+b*tan(f*x+e)^2)^(1/2),x)
Output:
int((sqrt(tan(e + f*x)**2*b + a)*csc(e + f*x))/(tan(e + f*x)**2*b + a),x)