Integrand size = 41, antiderivative size = 230 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 (5 a A+3 b B+3 a C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 (7 A b+7 a B+5 b C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 (5 a A+3 b B+3 a C) \sqrt {\sec (c+d x)} \sin (c+d x)}{5 d}+\frac {2 (7 A b+7 a B+5 b C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 (b B+a C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {2 b C \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d} \] Output:
-2/5*(5*A*a+3*B*b+3*C*a)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^( 1/2))*sec(d*x+c)^(1/2)/d+2/21*(7*A*b+7*B*a+5*C*b)*cos(d*x+c)^(1/2)*Inverse JacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+2/5*(5*A*a+3*B*b+3*C*a) *sec(d*x+c)^(1/2)*sin(d*x+c)/d+2/21*(7*A*b+7*B*a+5*C*b)*sec(d*x+c)^(3/2)*s in(d*x+c)/d+2/5*(B*b+C*a)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*b*C*sec(d*x+c) ^(7/2)*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.95 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.37 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 e^{-i d x} \cos ^3(c+d x) \csc (c) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (7 \sqrt {2} (5 a A+3 b B+3 a C) e^{2 i d x} \left (-1+e^{2 i c}\right ) \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )-\frac {e^{-i (c-d x)} \left (-1+e^{2 i c}\right ) \left (35 A \left (1+e^{2 i (c+d x)}\right )^2 \left (b \left (-1+e^{2 i (c+d x)}\right )+3 a e^{i (c+d x)} \left (1+e^{2 i (c+d x)}\right )\right )+7 a \left (1+e^{2 i (c+d x)}\right ) \left (5 B \left (-1+e^{4 i (c+d x)}\right )+3 C e^{i (c+d x)} \left (1+8 e^{2 i (c+d x)}+3 e^{4 i (c+d x)}\right )\right )+b \left (21 B e^{i (c+d x)} \left (1+9 e^{2 i (c+d x)}+11 e^{4 i (c+d x)}+3 e^{6 i (c+d x)}\right )+5 C \left (-5-17 e^{2 i (c+d x)}+17 e^{4 i (c+d x)}+5 e^{6 i (c+d x)}\right )\right )\right ) \sqrt {\sec (c+d x)}}{\left (1+e^{2 i (c+d x)}\right )^3}+10 (7 A b+7 a B+5 b C) e^{i d x} \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} \sin (c)\right )}{105 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x)))} \] Input:
Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C* Sec[c + d*x]^2),x]
Output:
(2*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[ c + d*x]^2)*(7*Sqrt[2]*(5*a*A + 3*b*B + 3*a*C)*E^((2*I)*d*x)*(-1 + E^((2*I )*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(35*A*(1 + E^((2*I)*(c + d*x)))^2*(b*(-1 + E^((2*I)*(c + d*x) )) + 3*a*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))) + 7*a*(1 + E^((2*I)*(c + d*x)))*(5*B*(-1 + E^((4*I)*(c + d*x))) + 3*C*E^(I*(c + d*x))*(1 + 8*E^( (2*I)*(c + d*x)) + 3*E^((4*I)*(c + d*x)))) + b*(21*B*E^(I*(c + d*x))*(1 + 9*E^((2*I)*(c + d*x)) + 11*E^((4*I)*(c + d*x)) + 3*E^((6*I)*(c + d*x))) + 5*C*(-5 - 17*E^((2*I)*(c + d*x)) + 17*E^((4*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)))))*Sqrt[Sec[c + d*x]])/(E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^ 3) + 10*(7*A*b + 7*a*B + 5*b*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(105*d*E^(I*d*x)*(b + a*Cos[c + d *x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))
Time = 1.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.92, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3120, 4534, 3042, 4255, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {2}{7} \int \frac {1}{2} \sec ^{\frac {3}{2}}(c+d x) \left (7 (b B+a C) \sec ^2(c+d x)+(7 A b+5 C b+7 a B) \sec (c+d x)+7 a A\right )dx+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \int \sec ^{\frac {3}{2}}(c+d x) \left (7 (b B+a C) \sec ^2(c+d x)+(7 A b+5 C b+7 a B) \sec (c+d x)+7 a A\right )dx+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+5 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+7 a A\right )dx+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left ((7 a B+7 A b+5 b C) \int \sec ^{\frac {5}{2}}(c+d x)dx+\int \sec ^{\frac {3}{2}}(c+d x) \left (7 (b B+a C) \sec ^2(c+d x)+7 a A\right )dx\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left ((7 a B+7 A b+5 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx+\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 a A\right )dx\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 a A\right )dx+(7 a B+7 A b+5 b C) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 a A\right )dx+(7 a B+7 A b+5 b C) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 a A\right )dx+(7 a B+7 A b+5 b C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 a A\right )dx+(7 a B+7 A b+5 b C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{7} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 a A\right )dx+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} (5 a A+3 a C+3 b B) \int \sec ^{\frac {3}{2}}(c+d x)dx+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} (5 a A+3 a C+3 b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} (5 a A+3 a C+3 b B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} (5 a A+3 a C+3 b B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} (5 a A+3 a C+3 b B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {7}{5} (5 a A+3 a C+3 b B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+(7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{7} \left ((7 a B+7 A b+5 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+\frac {7}{5} (5 a A+3 a C+3 b B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {14 (a C+b B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\) |
Input:
Int[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(2*b*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d) + ((14*(b*B + a*C)*Sec[c + d *x]^(5/2)*Sin[c + d*x])/(5*d) + (7*(5*a*A + 3*b*B + 3*a*C)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d))/5 + (7*A*b + 7*a*B + 5*b*C)*((2*Sqrt[Cos[c + d*x] ]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sec[c + d*x]^(3 /2)*Sin[c + d*x])/(3*d)))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] && !LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(823\) vs. \(2(205)=410\).
Time = 14.62 (sec) , antiderivative size = 824, normalized size of antiderivative = 3.58
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(824\) |
| parts | \(\text {Expression too large to display}\) | \(1012\) |
Input:
int(sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*(A*b+B*a)*(- 1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2 )/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/ 2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/ 2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2/5*(B*b+C*a)/sin(1/2*d*x+1/2*c) ^2/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2- 1)*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+12*(2*si n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d *x+1/2*c)-3*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d* x+1/2*c)^2)^(1/2)+2*C*b*(-1/56*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4 +sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^4-5/42*cos(1/2*d*x +1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+ 1/2*c)^2-1/2)^2+5/21*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(c os(1/2*d*x+1/2*c),2^(1/2)))+2*a*A/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2* c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.25 \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, \sqrt {2} {\left (7 i \, B a + i \, {\left (7 \, A + 5 \, C\right )} b\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-7 i \, B a - i \, {\left (7 \, A + 5 \, C\right )} b\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (5 \, A + 3 \, C\right )} a + 3 i \, B b\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 3 \, C\right )} a - 3 i \, B b\right )} \cos \left (d x + c\right )^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left ({\left (5 \, A + 3 \, C\right )} a + 3 \, B b\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, B a + {\left (7 \, A + 5 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 15 \, C b + 21 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}} \] Input:
integrate(sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="fricas")
Output:
-1/105*(5*sqrt(2)*(7*I*B*a + I*(7*A + 5*C)*b)*cos(d*x + c)^3*weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-7*I*B*a - I*(7* A + 5*C)*b)*cos(d*x + c)^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin (d*x + c)) + 21*sqrt(2)*(I*(5*A + 3*C)*a + 3*I*B*b)*cos(d*x + c)^3*weierst rassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-I*(5*A + 3*C)*a - 3*I*B*b)*cos(d*x + c)^3*weierstrassZeta( -4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(21* ((5*A + 3*C)*a + 3*B*b)*cos(d*x + c)^3 + 5*(7*B*a + (7*A + 5*C)*b)*cos(d*x + c)^2 + 15*C*b + 21*(C*a + B*b)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^3)
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**(3/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)* *2),x)
Output:
Timed out
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="maxima")
Output:
Timed out
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \] Input:
integrate(sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)*sec (d*x + c)^(3/2), x)
Timed out. \[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int((a + b/cos(c + d*x))*(1/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/co s(c + d*x)^2),x)
Output:
int((a + b/cos(c + d*x))*(1/cos(c + d*x))^(3/2)*(A + B/cos(c + d*x) + C/co s(c + d*x)^2), x)
\[ \int \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}d x \right ) b^{2}+2 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) a b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) a^{2} \] Input:
int(sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x))*sec(c + d*x)**4,x)*b*c + int(sqrt(sec(c + d*x))*sec (c + d*x)**3,x)*a*c + int(sqrt(sec(c + d*x))*sec(c + d*x)**3,x)*b**2 + 2*i nt(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*a*b + int(sqrt(sec(c + d*x))*sec( c + d*x),x)*a**2