Integrand size = 24, antiderivative size = 77 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=a^3 c x+\frac {a^3 c \text {arctanh}(\sin (e+f x))}{f}-\frac {a^3 c \tan (e+f x)}{f}-\frac {a^3 c \sec (e+f x) \tan (e+f x)}{f}-\frac {a^3 c \tan ^3(e+f x)}{3 f} \]
a^3*c*x+a^3*c*arctanh(sin(f*x+e))/f-a^3*c*tan(f*x+e)/f-a^3*c*sec(f*x+e)*ta n(f*x+e)/f-1/3*a^3*c*tan(f*x+e)^3/f
Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.31 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {a^3 c \sec ^3(e+f x) \left (9 (e+f x) \cos (e+f x)+12 \text {arctanh}(\sin (e+f x)) \cos ^3(e+f x)+3 e \cos (3 (e+f x))+3 f x \cos (3 (e+f x))-6 \sin (e+f x)-6 \sin (2 (e+f x))-2 \sin (3 (e+f x))\right )}{12 f} \]
(a^3*c*Sec[e + f*x]^3*(9*(e + f*x)*Cos[e + f*x] + 12*ArcTanh[Sin[e + f*x]] *Cos[e + f*x]^3 + 3*e*Cos[3*(e + f*x)] + 3*f*x*Cos[3*(e + f*x)] - 6*Sin[e + f*x] - 6*Sin[2*(e + f*x)] - 2*Sin[3*(e + f*x)]))/(12*f)
Time = 0.39 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 4392, 3042, 4374, 2009}
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.
\(\displaystyle \int (a \sec (e+f x)+a)^3 (c-c \sec (e+f x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \csc \left (e+f x+\frac {\pi }{2}\right )+a\right )^3 \left (c-c \csc \left (e+f x+\frac {\pi }{2}\right )\right )dx\) |
\(\Big \downarrow \) 4392 |
\(\displaystyle -a c \int (\sec (e+f x) a+a)^2 \tan ^2(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a c \int \cot \left (e+f x+\frac {\pi }{2}\right )^2 \left (\csc \left (e+f x+\frac {\pi }{2}\right ) a+a\right )^2dx\) |
\(\Big \downarrow \) 4374 |
\(\displaystyle -a c \int \left (a^2 \tan ^2(e+f x)+a^2 \sec ^2(e+f x) \tan ^2(e+f x)+2 a^2 \sec (e+f x) \tan ^2(e+f x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a c \left (-\frac {a^2 \text {arctanh}(\sin (e+f x))}{f}+\frac {a^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 \tan (e+f x)}{f}+\frac {a^2 \tan (e+f x) \sec (e+f x)}{f}-a^2 x\right )\) |
-(a*c*(-(a^2*x) - (a^2*ArcTanh[Sin[e + f*x]])/f + (a^2*Tan[e + f*x])/f + ( a^2*Sec[e + f*x]*Tan[e + f*x])/f + (a^2*Tan[e + f*x]^3)/(3*f)))
3.1.15.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*(csc[(e_.) + (f_.)*(x_)]*( d_.) + (c_))^(n_.), x_Symbol] :> Simp[((-a)*c)^m Int[Cot[e + f*x]^(2*m)*( c + d*Csc[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && RationalQ[n] && !( IntegerQ[n] && GtQ[m - n, 0])
Time = 1.80 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00
method | result | size |
parts | \(a^{3} c x +\frac {a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}-\frac {a^{3} c \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}+\frac {a^{3} c \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )}{f}\) | \(77\) |
derivativedivides | \(\frac {a^{3} c \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-2 a^{3} c \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+a^{3} c \left (f x +e \right )}{f}\) | \(96\) |
default | \(\frac {a^{3} c \left (-\frac {2}{3}-\frac {\sec \left (f x +e \right )^{2}}{3}\right ) \tan \left (f x +e \right )-2 a^{3} c \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )+2 a^{3} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+a^{3} c \left (f x +e \right )}{f}\) | \(96\) |
risch | \(a^{3} c x +\frac {2 i a^{3} c \left (3 \,{\mathrm e}^{5 i \left (f x +e \right )}-6 \,{\mathrm e}^{2 i \left (f x +e \right )}-3 \,{\mathrm e}^{i \left (f x +e \right )}-2\right )}{3 f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{3}}+\frac {a^{3} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{f}-\frac {a^{3} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{f}\) | \(109\) |
parallelrisch | \(-\frac {2 \left (\frac {3 \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2}+\frac {3 \left (-\cos \left (3 f x +3 e \right )-3 \cos \left (f x +e \right )\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2}-\frac {9 f x \cos \left (f x +e \right )}{2}-\frac {3 f x \cos \left (3 f x +3 e \right )}{2}+\sin \left (3 f x +3 e \right )+3 \sin \left (f x +e \right )+3 \sin \left (2 f x +2 e \right )\right ) a^{3} c}{3 f \left (\cos \left (3 f x +3 e \right )+3 \cos \left (f x +e \right )\right )}\) | \(148\) |
norman | \(\frac {a^{3} c x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{6}-a^{3} c x +\frac {4 a^{3} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {4 a^{3} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 f}+3 a^{3} c x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-3 a^{3} c x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}-1\right )^{3}}+\frac {a^{3} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{f}-\frac {a^{3} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{f}\) | \(158\) |
a^3*c*x+a^3*c/f*ln(sec(f*x+e)+tan(f*x+e))-a^3*c*sec(f*x+e)*tan(f*x+e)/f+a^ 3*c/f*(-2/3-1/3*sec(f*x+e)^2)*tan(f*x+e)
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.53 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {6 \, a^{3} c f x \cos \left (f x + e\right )^{3} + 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, a^{3} c \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (2 \, a^{3} c \cos \left (f x + e\right )^{2} + 3 \, a^{3} c \cos \left (f x + e\right ) + a^{3} c\right )} \sin \left (f x + e\right )}{6 \, f \cos \left (f x + e\right )^{3}} \]
1/6*(6*a^3*c*f*x*cos(f*x + e)^3 + 3*a^3*c*cos(f*x + e)^3*log(sin(f*x + e) + 1) - 3*a^3*c*cos(f*x + e)^3*log(-sin(f*x + e) + 1) - 2*(2*a^3*c*cos(f*x + e)^2 + 3*a^3*c*cos(f*x + e) + a^3*c)*sin(f*x + e))/(f*cos(f*x + e)^3)
\[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=- a^{3} c \left (\int \left (-1\right )\, dx + \int \left (- 2 \sec {\left (e + f x \right )}\right )\, dx + \int 2 \sec ^{3}{\left (e + f x \right )}\, dx + \int \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
-a**3*c*(Integral(-1, x) + Integral(-2*sec(e + f*x), x) + Integral(2*sec(e + f*x)**3, x) + Integral(sec(e + f*x)**4, x))
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=-\frac {2 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c - 6 \, {\left (f x + e\right )} a^{3} c - 3 \, a^{3} c {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{3} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{6 \, f} \]
-1/6*(2*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c - 6*(f*x + e)*a^3*c - 3*a^ 3*c*(2*sin(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin (f*x + e) - 1)) - 12*a^3*c*log(sec(f*x + e) + tan(f*x + e)))/f
Time = 0.34 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {3 \, {\left (f x + e\right )} a^{3} c + 3 \, a^{3} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 3 \, a^{3} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {4 \, {\left (a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{3} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}}{3 \, f} \]
1/3*(3*(f*x + e)*a^3*c + 3*a^3*c*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 3*a^ 3*c*log(abs(tan(1/2*f*x + 1/2*e) - 1)) - 4*(a^3*c*tan(1/2*f*x + 1/2*e)^3 - 3*a^3*c*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^3)/f
Time = 15.48 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.35 \[ \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x)) \, dx=\frac {4\,a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-\frac {4\,a^3\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}+a^3\,c\,x+\frac {2\,a^3\,c\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{f} \]