Integrand size = 33, antiderivative size = 169 \[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}+\frac {16 a^2 (21 A+13 C) \sqrt {a+a \sec (c+d x)} \tan (c+d x)}{315 d}+\frac {2 a (21 A+13 C) (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{105 d}-\frac {4 C (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{63 d}+\frac {2 C (a+a \sec (c+d x))^{7/2} \tan (c+d x)}{9 a d} \]
2/105*a*(21*A+13*C)*(a+a*sec(d*x+c))^(3/2)*tan(d*x+c)/d-4/63*C*(a+a*sec(d* x+c))^(5/2)*tan(d*x+c)/d+2/9*C*(a+a*sec(d*x+c))^(7/2)*tan(d*x+c)/a/d+64/31 5*a^3*(21*A+13*C)*tan(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+16/315*a^2*(21*A+13* C)*(a+a*sec(d*x+c))^(1/2)*tan(d*x+c)/d
Time = 2.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.53 \[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 a^3 \left (903 A+584 C+(294 A+292 C) \sec (c+d x)+3 (21 A+73 C) \sec ^2(c+d x)+130 C \sec ^3(c+d x)+35 C \sec ^4(c+d x)\right ) \tan (c+d x)}{315 d \sqrt {a (1+\sec (c+d x))}} \]
(2*a^3*(903*A + 584*C + (294*A + 292*C)*Sec[c + d*x] + 3*(21*A + 73*C)*Sec [c + d*x]^2 + 130*C*Sec[c + d*x]^3 + 35*C*Sec[c + d*x]^4)*Tan[c + d*x])/(3 15*d*Sqrt[a*(1 + Sec[c + d*x])])
Time = 0.88 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4571, 27, 3042, 4489, 3042, 4280, 3042, 4280, 3042, 4279}
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.
| \(\displaystyle \int \sec (c+d x) (a \sec (c+d x)+a)^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4571 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sec (c+d x) (\sec (c+d x) a+a)^{5/2} (a (9 A+7 C)-2 a C \sec (c+d x))dx}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sec (c+d x) (\sec (c+d x) a+a)^{5/2} (a (9 A+7 C)-2 a C \sec (c+d x))dx}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (9 A+7 C)-2 a C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 4489 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \int \sec (c+d x) (\sec (c+d x) a+a)^{5/2}dx-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2}dx-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 4280 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \left (\frac {8}{5} a \int \sec (c+d x) (\sec (c+d x) a+a)^{3/2}dx+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \left (\frac {8}{5} a \int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2}dx+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 4280 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \sec (c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {2 a \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \left (\frac {8}{5} a \left (\frac {4}{3} a \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {2 a \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
| \(\Big \downarrow \) 4279 |
| \(\displaystyle \frac {\frac {3}{7} a (21 A+13 C) \left (\frac {8}{5} a \left (\frac {8 a^2 \tan (c+d x)}{3 d \sqrt {a \sec (c+d x)+a}}+\frac {2 a \tan (c+d x) \sqrt {a \sec (c+d x)+a}}{3 d}\right )+\frac {2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}\right )-\frac {4 a C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}}{9 a}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}\) |
(2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d) + ((-4*a*C*(a + a*Se c[c + d*x])^(5/2)*Tan[c + d*x])/(7*d) + (3*a*(21*A + 13*C)*((2*a*(a + a*Se c[c + d*x])^(3/2)*Tan[c + d*x])/(5*d) + (8*a*((8*a^2*Tan[c + d*x])/(3*d*Sq rt[a + a*Sec[c + d*x]]) + (2*a*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d )))/5))/7)/(9*a)
3.2.75.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S ymbol] :> Simp[-2*b*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]])), x] /; Free Q[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_ Symbol] :> Simp[(-b)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Simp[a*((2*m - 1)/m) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && Intege rQ[2*m]
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(cs c[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*Cot[e + f*x]*(( a + b*Csc[e + f*x])^m/(f*(m + 1))), x] + Simp[(a*B*m + A*b*(m + 1))/(b*(m + 1)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x], x] /; FreeQ[{a, b, A, B , e, f, m}, 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)]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[ (e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x] *((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) In t[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[b*A*(m + 2) + b*C*(m + 1) - a*C* Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C, m}, x] && !LtQ[m, -1]
Time = 44.86 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(\frac {2 a^{2} \left (903 A \cos \left (d x +c \right )^{4}+584 C \cos \left (d x +c \right )^{4}+294 A \cos \left (d x +c \right )^{3}+292 C \cos \left (d x +c \right )^{3}+63 A \cos \left (d x +c \right )^{2}+219 C \cos \left (d x +c \right )^{2}+130 C \cos \left (d x +c \right )+35 C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}\) | \(124\) |
| parts | \(\frac {2 A \,a^{2} \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \left (43 \sin \left (d x +c \right )+14 \tan \left (d x +c \right )+3 \sec \left (d x +c \right ) \tan \left (d x +c \right )\right )}{15 d \left (\cos \left (d x +c \right )+1\right )}+\frac {2 C \,a^{2} \left (584 \cos \left (d x +c \right )^{4}+292 \cos \left (d x +c \right )^{3}+219 \cos \left (d x +c \right )^{2}+130 \cos \left (d x +c \right )+35\right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}\, \tan \left (d x +c \right ) \sec \left (d x +c \right )^{3}}{315 d \left (\cos \left (d x +c \right )+1\right )}\) | \(149\) |
2/315*a^2/d*(903*A*cos(d*x+c)^4+584*C*cos(d*x+c)^4+294*A*cos(d*x+c)^3+292* C*cos(d*x+c)^3+63*A*cos(d*x+c)^2+219*C*cos(d*x+c)^2+130*C*cos(d*x+c)+35*C) *(a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)*tan(d*x+c)*sec(d*x+c)^3
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.77 \[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left ({\left (903 \, A + 584 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 2 \, {\left (147 \, A + 146 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 73 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 130 \, C a^{2} \cos \left (d x + c\right ) + 35 \, C a^{2}\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}} \]
2/315*((903*A + 584*C)*a^2*cos(d*x + c)^4 + 2*(147*A + 146*C)*a^2*cos(d*x + c)^3 + 3*(21*A + 73*C)*a^2*cos(d*x + c)^2 + 130*C*a^2*cos(d*x + c) + 35* C*a^2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c)/(d*cos(d*x + c )^5 + d*cos(d*x + c)^4)
\[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
\[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
2/315*(315*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*(5*(A*a^2*d*cos(2*d*x + 2*c)^4 + A*a^2*d*sin(2*d*x + 2*c)^4 + 4* A*a^2*d*cos(2*d*x + 2*c)^3 + 6*A*a^2*d*cos(2*d*x + 2*c)^2 + 4*A*a^2*d*cos( 2*d*x + 2*c) + A*a^2*d + 2*(A*a^2*d*cos(2*d*x + 2*c)^2 + 2*A*a^2*d*cos(2*d *x + 2*c) + A*a^2*d)*sin(2*d*x + 2*c)^2)*integrate((((cos(8*d*x + 8*c)*cos (2*d*x + 2*c) + 3*cos(6*d*x + 6*c)*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*c os(2*d*x + 2*c) + cos(2*d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 3*sin(6*d*x + 6*c)*sin(2*d*x + 2*c) + 3*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + sin(2*d*x + 2*c)^2)*cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)) ) + (cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 3*cos(2*d*x + 2*c)*sin(6*d*x + 6* c) + 3*cos(2*d*x + 2*c)*sin(4*d*x + 4*c) - cos(8*d*x + 8*c)*sin(2*d*x + 2* c) - 3*cos(6*d*x + 6*c)*sin(2*d*x + 2*c) - 3*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*sin(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))*cos(5/2*arctan 2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) - ((cos(2*d*x + 2*c)*sin(8*d*x + 8*c) + 3*cos(2*d*x + 2*c)*sin(6*d*x + 6*c) + 3*cos(2*d*x + 2*c)*sin(4*d* x + 4*c) - cos(8*d*x + 8*c)*sin(2*d*x + 2*c) - 3*cos(6*d*x + 6*c)*sin(2*d* x + 2*c) - 3*cos(4*d*x + 4*c)*sin(2*d*x + 2*c))*cos(9/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - (cos(8*d*x + 8*c)*cos(2*d*x + 2*c) + 3*cos(6* d*x + 6*c)*cos(2*d*x + 2*c) + 3*cos(4*d*x + 4*c)*cos(2*d*x + 2*c) + cos(2* d*x + 2*c)^2 + sin(8*d*x + 8*c)*sin(2*d*x + 2*c) + 3*sin(6*d*x + 6*c)*s...
\[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right ) \,d x } \]
Time = 23.62 (sec) , antiderivative size = 766, normalized size of antiderivative = 4.53 \[ \int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*((A*a^2*2i) /d - (a^2*exp(c*1i + d*x*1i)*(903*A + 584*C)*2i)/(315*d)))/(exp(c*1i + d*x *1i) + 1) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2) *(exp(c*1i + d*x*1i)*((A*a^2*2i)/d - (a^2*(A + 2*C)*4i)/d + (a^2*(21*A - 3 2*C)*2i)/(105*d)) - (A*a^2*2i)/(5*d) + (a^2*(5*A + 2*C)*4i)/(5*d) - (a^2*( 5*A + 32*C)*2i)/(5*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1) ^2) - ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp( c*1i + d*x*1i)*((A*a^2*4i)/(3*d) - (a^2*(3*A + C)*8i)/(9*d) + (a^2*(13*A + 20*C)*4i)/(9*d) - (a^2*(A + C)*40i)/(9*d)) - (A*a^2*4i)/(3*d) + (a^2*(3*A + C)*8i)/(9*d) - (a^2*(13*A + 20*C)*4i)/(9*d) + (a^2*(A + C)*40i)/(9*d))) /((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^4) + ((a + a/(exp(- c* 1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1i + d*x*1i)*((A*a^2* 10i)/(3*d) - (a^2*(189*A + 292*C)*2i)/(315*d)) - (A*a^2*2i)/(3*d) + (a^2*( 9*A + 4*C)*2i)/(3*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)) + ((a + a/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(exp(c*1 i + d*x*1i)*((A*a^2*10i)/(7*d) - (a^2*(3*A + 4*C)*10i)/(7*d) - (a^2*(9*A - 16*C)*2i)/(63*d) + (a^2*(11*A + 20*C)*2i)/(7*d)) - (A*a^2*2i)/(7*d) + (a^ 2*(11*A + 4*C)*2i)/(7*d) - (a^2*(5*A + 12*C)*6i)/(7*d) + (a^2*(5*A - 16*C) *2i)/(7*d)))/((exp(c*1i + d*x*1i) + 1)*(exp(c*2i + d*x*2i) + 1)^3)