Integrand size = 33, antiderivative size = 244 \[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\frac {4 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^3 (13 A+11 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {4 a^3 (24 A+23 B) \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (13 A+11 B) \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 a B (a+a \sec (c+d x))^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (9 A+13 B) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)} \] Output:
4/15*a^3*(21*A+17*B)*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+4/21*a^3*(13*A+11*B)*cos(d*x+c)^(1/2)*InverseJacobiAM (1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+4/105*a^3*(24*A+23*B)*sin(d*x+c )/d/sec(d*x+c)^(3/2)+4/21*a^3*(13*A+11*B)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/ 9*a*B*(a+a*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/63*(9*A+13*B)*(a^ 3+a^3*sec(d*x+c))*sin(d*x+c)/d/sec(d*x+c)^(5/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.80 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.80 \[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\frac {a^3 \sqrt {\sec (c+d x)} \left (240 (13 A+11 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-112 i (21 A+17 B) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (7056 i A+5712 i B+30 (107 A+97 B) \sin (c+d x)+14 (54 A+73 B) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+270 B \sin (3 (c+d x))+35 B \sin (4 (c+d x)))\right )}{1260 d} \] Input:
Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]] ,x]
Output:
(a^3*Sqrt[Sec[c + d*x]]*(240*(13*A + 11*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(21*A + 17*B)*E^(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))] + Cos[c + d*x]*((7056*I)*A + (5712*I)*B + 30*(107*A + 97*B)*Sin[c + d*x] + 14*(54*A + 73*B)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 270*B*Sin[3*(c + d*x)] + 35*B*Sin[4*(c + d*x)])))/(1260*d)
Time = 1.63 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.02, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3439, 3042, 4505, 27, 3042, 4505, 27, 3042, 4484, 27, 3042, 4274, 3042, 4256, 3042, 4258, 3042, 3119, 3120}
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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3439 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^3 (A \sec (c+d x)+B)}{\sec ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A \csc \left (c+d x+\frac {\pi }{2}\right )+B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {2}{9} \int \frac {(\sec (c+d x) a+a)^2 (a (9 A+13 B)+3 a (3 A+B) \sec (c+d x))}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(\sec (c+d x) a+a)^2 (a (9 A+13 B)+3 a (3 A+B) \sec (c+d x))}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (9 A+13 B)+3 a (3 A+B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {3 (\sec (c+d x) a+a) \left ((24 A+23 B) a^2+5 (3 A+2 B) \sec (c+d x) a^2\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \int \frac {(\sec (c+d x) a+a) \left ((24 A+23 B) a^2+5 (3 A+2 B) \sec (c+d x) a^2\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((24 A+23 B) a^2+5 (3 A+2 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2}{5} \int -\frac {15 (13 A+11 B) a^3+7 (21 A+17 B) \sec (c+d x) a^3}{2 \sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {15 (13 A+11 B) a^3+7 (21 A+17 B) \sec (c+d x) a^3}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {15 (13 A+11 B) a^3+7 (21 A+17 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^3 (13 A+11 B) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+7 a^3 (21 A+17 B) \int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^3 (13 A+11 B) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+7 a^3 (21 A+17 B) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (7 a^3 (21 A+17 B) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 a^3 (13 A+11 B) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (7 a^3 (21 A+17 B) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 a^3 (13 A+11 B) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (7 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+15 a^3 (13 A+11 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^3 (13 A+11 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+7 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^3 (13 A+11 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {14 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 (9 A+13 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {6}{7} \left (\frac {2 a^3 (24 A+23 B) \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\frac {14 a^3 (21 A+17 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+15 a^3 (13 A+11 B) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )\right )\right )+\frac {2 a B \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
Input:
Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x]))/Sqrt[Sec[c + d*x]],x]
Output:
(2*a*B*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2 *(9*A + 13*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/ 2)) + (6*((2*a^3*(24*A + 23*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (( 14*a^3*(21*A + 17*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec [c + d*x]])/d + 15*a^3*(13*A + 11*B)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])))/5))/7)/9
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[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* (x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim p[g^(m + n) Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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_)]*(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[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*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]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Time = 18.20 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.69
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-560 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (360 A +2200 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1296 A -3412 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (1806 A +2702 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-624 A -738 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+195 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-441 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+165 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-357 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{315 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(413\) |
| parts | \(\text {Expression too large to display}\) | \(967\) |
Input:
int((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x,method=_RETURNV ERBOSE)
Output:
-4/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-560*B *cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(360*A+2200*B)*sin(1/2*d*x+1/2*c )^8*cos(1/2*d*x+1/2*c)+(-1296*A-3412*B)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1 /2*c)+(1806*A+2702*B)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-624*A-738* B)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+195*A*(sin(1/2*d*x+1/2*c)^2)^(1 /2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)) -441*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellip ticE(cos(1/2*d*x+1/2*c),2^(1/2))+165*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin (1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-357*B*(si n(1/2*d*x+1/2*c)^2)^(1/2)*(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)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/ 2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.95 \[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (13 \, A + 11 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (13 \, A + 11 \, B\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (21 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (21 \, A + 17 \, B\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (35 \, B a^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, A + 34 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 30 \, {\left (13 \, A + 11 \, B\right )} a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{315 \, d} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x, algorith m="fricas")
Output:
-2/315*(15*I*sqrt(2)*(13*A + 11*B)*a^3*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(13*A + 11*B)*a^3*weierstrassPInvers e(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*I*sqrt(2)*(21*A + 17*B)*a^3*w eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*I*sqrt(2)*(21*A + 17*B)*a^3*weierstrassZeta(-4, 0, weierstrass PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (35*B*a^3*cos(d*x + c)^4 + 45*(A + 3*B)*a^3*cos(d*x + c)^3 + 7*(27*A + 34*B)*a^3*cos(d*x + c)^2 + 30*(13*A + 11*B)*a^3*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=a^{3} \left (\int \frac {A}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 A \cos {\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 A \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {A \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {B \cos {\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 B \cos ^{2}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {3 B \cos ^{3}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx + \int \frac {B \cos ^{4}{\left (c + d x \right )}}{\sqrt {\sec {\left (c + d x \right )}}}\, dx\right ) \] Input:
integrate((a+a*cos(d*x+c))**3*(A+B*cos(d*x+c))/sec(d*x+c)**(1/2),x)
Output:
a**3*(Integral(A/sqrt(sec(c + d*x)), x) + Integral(3*A*cos(c + d*x)/sqrt(s ec(c + d*x)), x) + Integral(3*A*cos(c + d*x)**2/sqrt(sec(c + d*x)), x) + I ntegral(A*cos(c + d*x)**3/sqrt(sec(c + d*x)), x) + Integral(B*cos(c + d*x) /sqrt(sec(c + d*x)), x) + Integral(3*B*cos(c + d*x)**2/sqrt(sec(c + d*x)), x) + Integral(3*B*cos(c + d*x)**3/sqrt(sec(c + d*x)), x) + Integral(B*cos (c + d*x)**4/sqrt(sec(c + d*x)), x))
\[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x, algorith m="maxima")
Output:
integrate((B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)), x)
\[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x, algorith m="giac")
Output:
integrate((B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3/sqrt(sec(d*x + c)), x)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3)/(1/cos(c + d*x))^(1/2),x )
Output:
int(((A + B*cos(c + d*x))*(a + a*cos(c + d*x))^3)/(1/cos(c + d*x))^(1/2), x)
\[ \int \frac {(a+a \cos (c+d x))^3 (A+B \cos (c+d x))}{\sqrt {\sec (c+d x)}} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) b +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) b \right ) \] Input:
int((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c))/sec(d*x+c)^(1/2),x)
Output:
a**3*(int(sqrt(sec(c + d*x))/sec(c + d*x),x)*a + 3*int((sqrt(sec(c + d*x)) *cos(c + d*x))/sec(c + d*x),x)*a + int((sqrt(sec(c + d*x))*cos(c + d*x))/s ec(c + d*x),x)*b + int((sqrt(sec(c + d*x))*cos(c + d*x)**4)/sec(c + d*x),x )*b + int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x),x)*a + 3*int(( sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x),x)*b + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x),x)*a + 3*int((sqrt(sec(c + d*x))*cos (c + d*x)**2)/sec(c + d*x),x)*b)