Integrand size = 33, antiderivative size = 239 \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{3 d}-\frac {2 b \left (2 a^2 A-3 A b^2-9 a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 d}-\frac {2 b^2 (a A-b B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3 d}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}} \] Output:
2*(3*A*a^2*b-A*b^3+B*a^3-3*B*a*b^2)*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+2/3*(A*a^3+9*A*a*b^2+9*B*a^2*b+B*b^3)* cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d -2/3*b*(2*A*a^2-3*A*b^2-9*B*a*b)*sec(d*x+c)^(1/2)*sin(d*x+c)/d-2/3*b^2*(A* a-B*b)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/3*a*A*(a+b*sec(d*x+c))^2*sin(d*x+c) /d/sec(d*x+c)^(1/2)
Time = 7.17 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (6 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \left (a^3 A+9 a A b^2+9 a^2 b B+b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {\left (a^3 A+2 b^3 B+6 b^2 (A b+3 a B) \cos (c+d x)+a^3 A \cos (2 (c+d x))\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}\right )}{3 d} \] Input:
Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2) ,x]
Output:
(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(6*(3*a^2*A*b - A*b^3 + a^3*B - 3*a *b^2*B)*EllipticE[(c + d*x)/2, 2] + 2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3 *B)*EllipticF[(c + d*x)/2, 2] + ((a^3*A + 2*b^3*B + 6*b^2*(A*b + 3*a*B)*Co s[c + d*x] + a^3*A*Cos[2*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3 *d)
Time = 1.56 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4513, 27, 3042, 4564, 27, 3042, 4535, 3042, 4258, 3042, 3120, 4534, 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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}-\frac {2}{3} \int -\frac {(a+b \sec (c+d x)) \left (-3 b (a A-b B) \sec ^2(c+d x)+\left (A a^2+6 b B a+3 A b^2\right ) \sec (c+d x)+a (7 A b+3 a B)\right )}{2 \sqrt {\sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {(a+b \sec (c+d x)) \left (-3 b (a A-b B) \sec ^2(c+d x)+\left (A a^2+6 b B a+3 A b^2\right ) \sec (c+d x)+a (7 A b+3 a B)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-3 b (a A-b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (A a^2+6 b B a+3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (7 A b+3 a B)\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left ((7 A b+3 a B) a^2-b \left (2 A a^2-9 b B a-3 A b^2\right ) \sec ^2(c+d x)+\left (A a^3+9 b B a^2+9 A b^2 a+b^3 B\right ) \sec (c+d x)\right )}{2 \sqrt {\sec (c+d x)}}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {(7 A b+3 a B) a^2-b \left (2 A a^2-9 b B a-3 A b^2\right ) \sec ^2(c+d x)+\left (A a^3+9 b B a^2+9 A b^2 a+b^3 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {(7 A b+3 a B) a^2-b \left (2 A a^2-9 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (A a^3+9 b B a^2+9 A b^2 a+b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {a^2 (7 A b+3 a B)-b \left (2 A a^2-9 b B a-3 A b^2\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}}dx+\left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \int \sqrt {\sec (c+d x)}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {a^2 (7 A b+3 a B)-b \left (2 A a^2-9 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {a^2 (7 A b+3 a B)-b \left (2 A a^2-9 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {a^2 (7 A b+3 a B)-b \left (2 A a^2-9 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{3} \left (\int \frac {a^2 (7 A b+3 a B)-b \left (2 A a^2-9 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\frac {2 b \left (2 a^2 A-9 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b \left (2 a^2 A-9 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx-\frac {2 b \left (2 a^2 A-9 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} \left (3 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b \left (2 a^2 A-9 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{3} \left (-\frac {2 b \left (2 a^2 A-9 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{d}+\frac {2 \left (a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {6 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\) |
Input:
Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]
Output:
(2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((6 *(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^ 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (2*b*(2*a^2*A - 3*A*b^2 - 9*a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d - (2 *b^2*(a*A - b*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/d)/3
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*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_))^(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[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & & LeQ[n, -1]
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. \(885\) vs. \(2(220)=440\).
Time = 9.12 (sec) , antiderivative size = 886, normalized size of antiderivative = 3.71
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(886\) |
| default | \(\text {Expression too large to display}\) | \(1212\) |
Input:
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(3/2),x,method=_RETURNV ERBOSE)
Output:
-2*(A*b^3+3*B*a*b^2)*(-2*(-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)*sin(1/2*d*x+1/2*c)^2+(sin(1/2*d*x+1/2*c)^2)^(1/2)*( 2*sin(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)*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-2*(3*A*a*b^2+3*B*a^2*b)*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1 /2*c)^2)^(1/2)*(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))/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d +2*(3*A*a^2*b+B*a^3)*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*Elliptic E(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-2/3*B*b^3*( -2*(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))*sin(1/2*d*x+1/2*c)^2-2*cos(1/2*d*x+1/2*c)*sin (1/2*d*x+1/2*c)^2+(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)))*((2*cos(1/2*d*x+1/2*c)^2-1)*s in(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*cos(1/2*d*x+1/2*c)^2-1)^(3/2)/sin(1/2*d*x+1/2*c)/d-2/3*a^3*A*((2* cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*cos(1/2*d*x+1/2*...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {2} {\left (-i \, A a^{3} - 9 i \, B a^{2} b - 9 i \, A a b^{2} - i \, B b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + \sqrt {2} {\left (i \, A a^{3} + 9 i \, B a^{2} b + 9 i \, A a b^{2} + i \, B b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 3 \, \sqrt {2} {\left (-i \, B a^{3} - 3 i \, A a^{2} b + 3 i \, B a b^{2} + i \, A b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {2} {\left (i \, B a^{3} + 3 i \, A a^{2} b - 3 i \, B a b^{2} - i \, A b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + \frac {2 \, {\left (A a^{3} \cos \left (d x + c\right )^{2} + B b^{3} + 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3 \, d \cos \left (d x + c\right )} \] Input:
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(3/2),x, algorith m="fricas")
Output:
1/3*(sqrt(2)*(-I*A*a^3 - 9*I*B*a^2*b - 9*I*A*a*b^2 - I*B*b^3)*cos(d*x + c) *weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + sqrt(2)*(I*A* a^3 + 9*I*B*a^2*b + 9*I*A*a*b^2 + I*B*b^3)*cos(d*x + c)*weierstrassPInvers e(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 3*sqrt(2)*(-I*B*a^3 - 3*I*A*a^2* b + 3*I*B*a*b^2 + I*A*b^3)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstrass PInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 3*sqrt(2)*(I*B*a^3 + 3*I *A*a^2*b - 3*I*B*a*b^2 - I*A*b^3)*cos(d*x + c)*weierstrassZeta(-4, 0, weie rstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) + 2*(A*a^3*cos(d*x + c)^2 + B*b^3 + 3*(3*B*a*b^2 + A*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos (d*x + c)))/(d*cos(d*x + c))
\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{3}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*sec(d*x+c))**3*(A+B*sec(d*x+c))/sec(d*x+c)**(3/2),x)
Output:
Integral((A + B*sec(c + d*x))*(a + b*sec(c + d*x))**3/sec(c + d*x)**(3/2), x)
\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(3/2),x, algorith m="maxima")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3/sec(d*x + c)^(3/2), x)
\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(3/2),x, algorith m="giac")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3/sec(d*x + c)^(3/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^3)/(1/cos(c + d*x))^(3/2),x )
Output:
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^3)/(1/cos(c + d*x))^(3/2), x)
\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{3} b +6 \left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a^{2} b^{2}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) b^{4}+4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) a \,b^{3} \] Input:
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(3/2),x)
Output:
int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**4 + 4*int(sqrt(sec(c + d*x))/ sec(c + d*x),x)*a**3*b + 6*int(sqrt(sec(c + d*x)),x)*a**2*b**2 + int(sqrt( sec(c + d*x))*sec(c + d*x)**2,x)*b**4 + 4*int(sqrt(sec(c + d*x))*sec(c + d *x),x)*a*b**3