Integrand size = 21, antiderivative size = 120 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec ^3(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}+\frac {3 \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac {8 \tan (c+d x)}{35 d \left (a^2+a^2 \sec (c+d x)\right )^2}+\frac {\tan (c+d x)}{5 d \left (a^4+a^4 \sec (c+d x)\right )} \]
1/7*sec(d*x+c)^3*tan(d*x+c)/d/(a+a*sec(d*x+c))^4+3/35*tan(d*x+c)/a/d/(a+a* sec(d*x+c))^3-8/35*tan(d*x+c)/d/(a^2+a^2*sec(d*x+c))^2+1/5*tan(d*x+c)/d/(a ^4+a^4*sec(d*x+c))
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.58 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (35 \sin \left (\frac {1}{2} (c+d x)\right )+21 \sin \left (\frac {3}{2} (c+d x)\right )+7 \sin \left (\frac {5}{2} (c+d x)\right )+\sin \left (\frac {7}{2} (c+d x)\right )\right )}{1120 a^4 d} \]
(Sec[(c + d*x)/2]^7*(35*Sin[(c + d*x)/2] + 21*Sin[(3*(c + d*x))/2] + 7*Sin [(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]))/(1120*a^4*d)
Time = 0.61 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 4297, 3042, 4286, 25, 3042, 4488, 3042, 4281}
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.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a \sec (c+d x)+a)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^4}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
| \(\Big \downarrow \) 4297 |
| \(\displaystyle \frac {3 \int \frac {\sec ^3(c+d x)}{(\sec (c+d x) a+a)^3}dx}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4286 |
| \(\displaystyle \frac {3 \left (\frac {\int -\frac {\sec (c+d x) (3 a-5 a \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{5 a^2}+\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}\right )}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac {\int \frac {\sec (c+d x) (3 a-5 a \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{5 a^2}\right )}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (3 a-5 a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{5 a^2}\right )}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4488 |
| \(\displaystyle \frac {3 \left (\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac {\frac {8 a \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}-\frac {7}{3} \int \frac {\sec (c+d x)}{\sec (c+d x) a+a}dx}{5 a^2}\right )}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac {\frac {8 a \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}-\frac {7}{3} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{5 a^2}\right )}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
| \(\Big \downarrow \) 4281 |
| \(\displaystyle \frac {3 \left (\frac {\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac {\frac {8 a \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}-\frac {7 \tan (c+d x)}{3 d (a \sec (c+d x)+a)}}{5 a^2}\right )}{7 a}+\frac {\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}\) |
(Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (3*(Tan[c + d *x]/(5*d*(a + a*Sec[c + d*x])^3) - ((8*a*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (7*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])))/(5*a^2)))/(7*a)
3.1.73.3.1 Defintions of rubi rules used
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[-Cot[e + f*x]/(f*(b + a*Csc[e + f*x])), x] /; FreeQ[{a, b, e, f} , x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^3*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[b*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(a*f*(2*m + 1))), x] - Simp[1/(a^2*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1) *(a*m - b*(2*m + 1)*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[ a^2 - b^2, 0] && LtQ[m, -2^(-1)]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_), x_Symbol] :> Simp[b*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs c[e + f*x])^(n - 1)/(a*f*(2*m + 1))), x] + Simp[d*((m + 1)/(b*(2*m + 1))) Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 1), x], x] /; FreeQ [{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && EqQ[m + n, 0] && LtQ[m, -2^(-1)] && IntegerQ[2*m]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(A*b - a*B)*Cot[e + f*x]*((a + b*Csc[e + f*x])^m/(a*f*(2*m + 1))), x] + Simp[(a*B*m + A*b*(m + 1))/(a*b*(2*m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[a*B*m + A*b*(m + 1), 0] && LtQ[m, -2^(-1)]
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.47
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(56\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\) | \(56\) |
| risch | \(\frac {4 i \left (35 \,{\mathrm e}^{3 i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{35 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(58\) |
| parallelrisch | \(\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+21 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{280 a^{4} d}\) | \(60\) |
| norman | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{70 a d}-\frac {13 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{280 a d}+\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{140 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{56 a d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} a^{3}}\) | \(152\) |
1/8/d/a^4*(1/7*tan(1/2*d*x+1/2*c)^7+3/5*tan(1/2*d*x+1/2*c)^5+tan(1/2*d*x+1 /2*c)^3+tan(1/2*d*x+1/2*c))
Time = 0.25 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.82 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {{\left (2 \, \cos \left (d x + c\right )^{3} + 8 \, \cos \left (d x + c\right )^{2} + 13 \, \cos \left (d x + c\right ) + 12\right )} \sin \left (d x + c\right )}{35 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
1/35*(2*cos(d*x + c)^3 + 8*cos(d*x + c)^2 + 13*cos(d*x + c) + 12)*sin(d*x + c)/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c) ^2 + 4*a^4*d*cos(d*x + c) + a^4*d)
\[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\int \frac {\sec ^{4}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Integral(sec(c + d*x)**4/(sec(c + d*x)**4 + 4*sec(c + d*x)**3 + 6*sec(c + d*x)**2 + 4*sec(c + d*x) + 1), x)/a**4
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{280 \, a^{4} d} \]
1/280*(35*sin(d*x + c)/(cos(d*x + c) + 1) + 35*sin(d*x + c)^3/(cos(d*x + c ) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos( d*x + c) + 1)^7)/(a^4*d)
Time = 0.32 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.49 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{280 \, a^{4} d} \]
1/280*(5*tan(1/2*d*x + 1/2*c)^7 + 21*tan(1/2*d*x + 1/2*c)^5 + 35*tan(1/2*d *x + 1/2*c)^3 + 35*tan(1/2*d*x + 1/2*c))/(a^4*d)
Time = 13.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.48 \[ \int \frac {\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx=\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+35\right )}{280\,a^4\,d} \]