Integrand size = 27, antiderivative size = 61 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=\frac {(2 a c+b d) \text {arctanh}(\sin (e+f x))}{2 f}+\frac {(b c+a d) \tan (e+f x)}{f}+\frac {b d \sec (e+f x) \tan (e+f x)}{2 f} \]
1/2*(2*a*c+b*d)*arctanh(sin(f*x+e))/f+(a*d+b*c)*tan(f*x+e)/f+1/2*b*d*sec(f *x+e)*tan(f*x+e)/f
Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=\frac {a c \text {arctanh}(\sin (e+f x))}{f}+\frac {b d \text {arctanh}(\sin (e+f x))}{2 f}+\frac {b c \tan (e+f x)}{f}+\frac {a d \tan (e+f x)}{f}+\frac {b d \sec (e+f x) \tan (e+f x)}{2 f} \]
(a*c*ArcTanh[Sin[e + f*x]])/f + (b*d*ArcTanh[Sin[e + f*x]])/(2*f) + (b*c*T an[e + f*x])/f + (a*d*Tan[e + f*x])/f + (b*d*Sec[e + f*x]*Tan[e + f*x])/(2 *f)
Time = 0.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3042, 4485, 3042, 4274, 3042, 4254, 24, 4257}
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.
\(\displaystyle \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (a+b \csc \left (e+f x+\frac {\pi }{2}\right )\right ) \left (c+d \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 4485 |
\(\displaystyle \frac {1}{2} \int \sec (e+f x) (2 a c+b d+2 (b c+a d) \sec (e+f x))dx+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int \csc \left (e+f x+\frac {\pi }{2}\right ) \left (2 a c+b d+2 (b c+a d) \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
\(\Big \downarrow \) 4274 |
\(\displaystyle \frac {1}{2} \left (2 (a d+b c) \int \sec ^2(e+f x)dx+(2 a c+b d) \int \sec (e+f x)dx\right )+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left ((2 a c+b d) \int \csc \left (e+f x+\frac {\pi }{2}\right )dx+2 (a d+b c) \int \csc \left (e+f x+\frac {\pi }{2}\right )^2dx\right )+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {1}{2} \left ((2 a c+b d) \int \csc \left (e+f x+\frac {\pi }{2}\right )dx-\frac {2 (a d+b c) \int 1d(-\tan (e+f x))}{f}\right )+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{2} \left ((2 a c+b d) \int \csc \left (e+f x+\frac {\pi }{2}\right )dx+\frac {2 (a d+b c) \tan (e+f x)}{f}\right )+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {1}{2} \left (\frac {(2 a c+b d) \text {arctanh}(\sin (e+f x))}{f}+\frac {2 (a d+b c) \tan (e+f x)}{f}\right )+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f}\) |
(b*d*Sec[e + f*x]*Tan[e + f*x])/(2*f) + (((2*a*c + b*d)*ArcTanh[Sin[e + f* x]])/f + (2*(b*c + a*d)*Tan[e + f*x])/f)/2
3.3.47.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-b)*B*Cot[ e + f*x]*((d*Csc[e + f*x])^n/(f*(n + 1))), x] + Simp[1/(n + 1) Int[(d*Csc [e + f*x])^n*Simp[A*a*(n + 1) + B*b*n + (A*b + B*a)*(n + 1)*Csc[e + f*x], x ], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && !LeQ[ n, -1]
Time = 2.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23
method | result | size |
derivativedivides | \(\frac {a c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+a d \tan \left (f x +e \right )+b c \tan \left (f x +e \right )+b d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(75\) |
default | \(\frac {a c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+a d \tan \left (f x +e \right )+b c \tan \left (f x +e \right )+b d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(75\) |
parts | \(\frac {\left (a d +b c \right ) \tan \left (f x +e \right )}{f}+\frac {a c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {b d \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )}{f}\) | \(76\) |
parallelrisch | \(\frac {-\left (a c +\frac {b d}{2}\right ) \left (1+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )+\left (a c +\frac {b d}{2}\right ) \left (1+\cos \left (2 f x +2 e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )+\left (a d +b c \right ) \sin \left (2 f x +2 e \right )+\sin \left (f x +e \right ) b d}{f \left (1+\cos \left (2 f x +2 e \right )\right )}\) | \(110\) |
norman | \(\frac {\frac {\left (2 a d +2 b c +b d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (2 a d +2 b c -b d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{2}}-\frac {\left (2 a c +b d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 f}+\frac {\left (2 a c +b d \right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 f}\) | \(123\) |
risch | \(-\frac {i \left (b d \,{\mathrm e}^{3 i \left (f x +e \right )}-2 a d \,{\mathrm e}^{2 i \left (f x +e \right )}-2 b c \,{\mathrm e}^{2 i \left (f x +e \right )}-b d \,{\mathrm e}^{i \left (f x +e \right )}-2 a d -2 b c \right )}{f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}-\frac {a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) b d}{2 f}+\frac {a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) b d}{2 f}\) | \(160\) |
1/f*(a*c*ln(sec(f*x+e)+tan(f*x+e))+a*d*tan(f*x+e)+b*c*tan(f*x+e)+b*d*(1/2* sec(f*x+e)*tan(f*x+e)+1/2*ln(sec(f*x+e)+tan(f*x+e))))
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.57 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=\frac {{\left (2 \, a c + b d\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, a c + b d\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (b d + 2 \, {\left (b c + a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f \cos \left (f x + e\right )^{2}} \]
1/4*((2*a*c + b*d)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (2*a*c + b*d)*co s(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*(b*d + 2*(b*c + a*d)*cos(f*x + e)) *sin(f*x + e))/(f*cos(f*x + e)^2)
\[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=\int \left (a + b \sec {\left (e + f x \right )}\right ) \left (c + d \sec {\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}\, dx \]
Time = 0.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.44 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=-\frac {b d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 4 \, a c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 4 \, b c \tan \left (f x + e\right ) - 4 \, a d \tan \left (f x + e\right )}{4 \, f} \]
-1/4*(b*d*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + l og(sin(f*x + e) - 1)) - 4*a*c*log(sec(f*x + e) + tan(f*x + e)) - 4*b*c*tan (f*x + e) - 4*a*d*tan(f*x + e))/f
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (57) = 114\).
Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.51 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=\frac {{\left (2 \, a c + b d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - {\left (2 \, a c + b d\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (2 \, b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - b d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, b c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2}}}{2 \, f} \]
1/2*((2*a*c + b*d)*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - (2*a*c + b*d)*log( abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*(2*b*c*tan(1/2*f*x + 1/2*e)^3 + 2*a*d*t an(1/2*f*x + 1/2*e)^3 - b*d*tan(1/2*f*x + 1/2*e)^3 - 2*b*c*tan(1/2*f*x + 1 /2*e) - 2*a*d*tan(1/2*f*x + 1/2*e) - b*d*tan(1/2*f*x + 1/2*e))/(tan(1/2*f* x + 1/2*e)^2 - 1)^2)/f
Time = 14.56 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.70 \[ \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (2\,a\,c+b\,d\right )}{f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,a\,d+2\,b\,c+b\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a\,d+2\,b\,c-b\,d\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]