Integrand size = 33, antiderivative size = 485 \[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 b^4 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^5 d}-\frac {2 (a-b) \sqrt {a+b} \left (3 b^3 (25 A-49 B)+18 a b^2 (A-2 B)+12 a^2 b (2 A-B)-16 a^3 B\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{315 b^4 d}-\frac {2 \left (12 a^2 A b-75 A b^3-8 a^3 B-13 a b^2 B\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^3 d}+\frac {2 \left (9 a A b-6 a^2 B+49 b^2 B\right ) \sec (c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{315 b^2 d}+\frac {2 (9 A b+a B) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{63 b d}+\frac {2 B \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{9 d} \] Output:
-2/315*(a-b)*(a+b)^(1/2)*(24*A*a^3*b+57*A*a*b^3-16*B*a^4-24*B*a^2*b^2+147* B*b^4)*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b ))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/b ^5/d-2/315*(a-b)*(a+b)^(1/2)*(3*b^3*(25*A-49*B)+18*a*b^2*(A-2*B)+12*a^2*b* (2*A-B)-16*B*a^3)*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2), ((a+b)/(a-b))^(1/2))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a- b))^(1/2)/b^4/d-2/315*(12*A*a^2*b-75*A*b^3-8*B*a^3-13*B*a*b^2)*(a+b*sec(d* x+c))^(1/2)*tan(d*x+c)/b^3/d+2/315*(9*A*a*b-6*B*a^2+49*B*b^2)*sec(d*x+c)*( a+b*sec(d*x+c))^(1/2)*tan(d*x+c)/b^2/d+2/63*(9*A*b+B*a)*sec(d*x+c)^2*(a+b* sec(d*x+c))^(1/2)*tan(d*x+c)/b/d+2/9*B*sec(d*x+c)^3*(a+b*sec(d*x+c))^(1/2) *tan(d*x+c)/d
Time = 35.12 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.31 \[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {2 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {a+b \sec (c+d x)} \left (2 (a+b) \left (-24 a^3 A b-57 a A b^3+16 a^4 B+24 a^2 b^2 B-147 b^4 B\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a-b}{a+b}\right )+2 b (a+b) \left (-16 a^3 B+12 a^2 b (2 A+B)-18 a b^2 (A+2 B)+3 b^3 (25 A+49 B)\right ) \sqrt {\frac {\cos (c+d x)}{1+\cos (c+d x)}} \sqrt {\frac {b+a \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {a-b}{a+b}\right )+\left (-24 a^3 A b-57 a A b^3+16 a^4 B+24 a^2 b^2 B-147 b^4 B\right ) \cos (c+d x) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 b^4 d (b+a \cos (c+d x)) \sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )} \sqrt {\sec (c+d x)}}+\frac {\sqrt {a+b \sec (c+d x)} \left (\frac {2 \left (24 a^3 A b+57 a A b^3-16 a^4 B-24 a^2 b^2 B+147 b^4 B\right ) \sin (c+d x)}{315 b^4}+\frac {2 \sec ^3(c+d x) (9 A b \sin (c+d x)+a B \sin (c+d x))}{63 b}+\frac {2 \sec ^2(c+d x) \left (9 a A b \sin (c+d x)-6 a^2 B \sin (c+d x)+49 b^2 B \sin (c+d x)\right )}{315 b^2}+\frac {2 \sec (c+d x) \left (-12 a^2 A b \sin (c+d x)+75 A b^3 \sin (c+d x)+8 a^3 B \sin (c+d x)+13 a b^2 B \sin (c+d x)\right )}{315 b^3}+\frac {2}{9} B \sec ^3(c+d x) \tan (c+d x)\right )}{d} \] Input:
Integrate[Sec[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]
Output:
(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(2*(a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[C os[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Co s[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*( a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/( (a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/( a + b)] + (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B) *Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/( 315*b^4*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]] ) + (Sqrt[a + b*Sec[c + d*x]]*((2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24 *a^2*b^2*B + 147*b^4*B)*Sin[c + d*x])/(315*b^4) + (2*Sec[c + d*x]^3*(9*A*b *Sin[c + d*x] + a*B*Sin[c + d*x]))/(63*b) + (2*Sec[c + d*x]^2*(9*a*A*b*Sin [c + d*x] - 6*a^2*B*Sin[c + d*x] + 49*b^2*B*Sin[c + d*x]))/(315*b^2) + (2* Sec[c + d*x]*(-12*a^2*A*b*Sin[c + d*x] + 75*A*b^3*Sin[c + d*x] + 8*a^3*B*S in[c + d*x] + 13*a*b^2*B*Sin[c + d*x]))/(315*b^3) + (2*B*Sec[c + d*x]^3*Ta n[c + d*x])/9))/d
Time = 2.35 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.04, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4519, 27, 3042, 4590, 27, 3042, 4580, 27, 3042, 4570, 27, 3042, 4493, 3042, 4319, 4492}
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.
| \(\displaystyle \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^4 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4519 |
| \(\displaystyle \frac {2}{9} \int \frac {\sec ^3(c+d x) \left ((9 A b+a B) \sec ^2(c+d x)+(9 a A+7 b B) \sec (c+d x)+6 a B\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {\sec ^3(c+d x) \left ((9 A b+a B) \sec ^2(c+d x)+(9 a A+7 b B) \sec (c+d x)+6 a B\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left ((9 A b+a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+7 b B) \csc \left (c+d x+\frac {\pi }{2}\right )+6 a B\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4590 |
| \(\displaystyle \frac {1}{9} \left (\frac {2 \int \frac {\sec ^2(c+d x) \left (\left (-6 B a^2+9 A b a+49 b^2 B\right ) \sec ^2(c+d x)+b (45 A b+47 a B) \sec (c+d x)+4 a (9 A b+a B)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\int \frac {\sec ^2(c+d x) \left (\left (-6 B a^2+9 A b a+49 b^2 B\right ) \sec ^2(c+d x)+b (45 A b+47 a B) \sec (c+d x)+4 a (9 A b+a B)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-6 B a^2+9 A b a+49 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (45 A b+47 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+4 a (9 A b+a B)\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {2 \int \frac {\sec (c+d x) \left (-3 \left (-8 B a^3+12 A b a^2-13 b^2 B a-75 A b^3\right ) \sec ^2(c+d x)+b \left (2 B a^2+207 A b a+147 b^2 B\right ) \sec (c+d x)+2 a \left (-6 B a^2+9 A b a+49 b^2 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\int \frac {\sec (c+d x) \left (-3 \left (-8 B a^3+12 A b a^2-13 b^2 B a-75 A b^3\right ) \sec ^2(c+d x)+b \left (2 B a^2+207 A b a+147 b^2 B\right ) \sec (c+d x)+2 a \left (-6 B a^2+9 A b a+49 b^2 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (-3 \left (-8 B a^3+12 A b a^2-13 b^2 B a-75 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b \left (2 B a^2+207 A b a+147 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-6 B a^2+9 A b a+49 b^2 B\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {2 \int \frac {3 \sec (c+d x) \left (b \left (-4 B a^3+6 A b a^2+111 b^2 B a+75 A b^3\right )+\left (-16 B a^4+24 A b a^3-24 b^2 B a^2+57 A b^3 a+147 b^4 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{3 b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\sec (c+d x) \left (b \left (-4 B a^3+6 A b a^2+111 b^2 B a+75 A b^3\right )+\left (-16 B a^4+24 A b a^3-24 b^2 B a^2+57 A b^3 a+147 b^4 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (-4 B a^3+6 A b a^2+111 b^2 B a+75 A b^3\right )+\left (-16 B a^4+24 A b a^3-24 b^2 B a^2+57 A b^3 a+147 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx-(a-b) \left (-16 a^3 B+12 a^2 b (2 A-B)+18 a b^2 (A-2 B)+3 b^3 (25 A-49 B)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-(a-b) \left (-16 a^3 B+12 a^2 b (2 A-B)+18 a b^2 (A-2 B)+3 b^3 (25 A-49 B)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {\frac {\left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \left (-16 a^3 B+12 a^2 b (2 A-B)+18 a b^2 (A-2 B)+3 b^3 (25 A-49 B)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}}{b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}+\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{9} \left (\frac {\frac {2 \left (-6 a^2 B+9 a A b+49 b^2 B\right ) \tan (c+d x) \sec (c+d x) \sqrt {a+b \sec (c+d x)}}{5 b d}+\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} \left (-16 a^3 B+12 a^2 b (2 A-B)+18 a b^2 (A-2 B)+3 b^3 (25 A-49 B)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\frac {2 (a-b) \sqrt {a+b} \left (-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 b^4 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}}{b}-\frac {2 \left (-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{b d}}{5 b}}{7 b}+\frac {2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{7 b d}\right )+\frac {2 B \tan (c+d x) \sec ^3(c+d x) \sqrt {a+b \sec (c+d x)}}{9 d}\) |
Input:
Int[Sec[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]
Output:
(2*B*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + ((2*(9* A*b + a*B)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d) + ((2*(9*a*A*b - 6*a^2*B + 49*b^2*B)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]* Tan[c + d*x])/(5*b*d) + (((-2*(a - b)*Sqrt[a + b]*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt [a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d *x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*A - 49*B) + 18*a*b^2*(A - 2*B) + 12*a^2*b*(2*A - B) - 16*a^3*B)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqr t[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b *(1 + Sec[c + d*x]))/(a - b))])/(b*d))/b - (2*(12*a^2*A*b - 75*A*b^3 - 8*a ^3*B - 13*a*b^2*B)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(b*d))/(5*b))/(7 *b))/9
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*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt [a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(A - B) Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[B Int[Csc[e + f*x]*((1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A, B} , x] && NeQ[a^2 - b^2, 0] && NeQ[A^2 - B^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[(-B)*d* Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^(n - 1)/(f*(m + n))), x] + Simp[d/(m + n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n - 1)*Simp[a*B*(n - 1) + (b*B*(m + n - 1) + a*A*(m + n))*Csc[e + f*x] + (a* B*m + A*b*(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] && LtQ[0, m, 1] && GtQ[n, 0 ]
Int[csc[(e_.) + (f_.)*(x_)]*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e _.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_S ymbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 2) )), x] + Simp[1/(b*(m + 2)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[ b*A*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[ (e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x _Symbol] :> Simp[(-C)*Csc[e + f*x]*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2) + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B* (m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] & & NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
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 _))^(m_), x_Symbol] :> Simp[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 )*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*Csc [e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2418\) vs. \(2(447)=894\).
Time = 74.63 (sec) , antiderivative size = 2419, normalized size of antiderivative = 4.99
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2419\) |
| parts | \(\text {Expression too large to display}\) | \(2433\) |
Input:
int(sec(d*x+c)^4*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x,method=_RETURNV ERBOSE)
Output:
-2/315/d/b^4*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a+a*cos(d*x+c)+b*cos(d*x +c)+b)*(7*(-21*cos(d*x+c)^4-7*cos(d*x+c)^3-7*cos(d*x+c)^2-5*cos(d*x+c)-5)* B*b^5*tan(d*x+c)*sec(d*x+c)^3+8*sin(d*x+c)*(1-cos(d*x+c))*B*a^4*b+12*sin(d *x+c)*(cos(d*x+c)-1)*A*a^3*b^2+15*(-5*cos(d*x+c)^3-5*cos(d*x+c)^2-3*cos(d* x+c)-3)*A*b^5*tan(d*x+c)*sec(d*x+c)^2+3*(-19*cos(d*x+c)^2+cos(d*x+c)+1)*A* a^2*b^3*tan(d*x+c)+2*(12*cos(d*x+c)^2-cos(d*x+c)-1)*B*a^3*b^2*tan(d*x+c)-2 4*A*a^4*b*cos(d*x+c)*sin(d*x+c)+3*(-25*cos(d*x+c)^3-44*cos(d*x+c)^2-18*cos (d*x+c)-18)*A*a*b^4*tan(d*x+c)*sec(d*x+c)+147*(cos(d*x+c)^2+2*cos(d*x+c)+1 )*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x +c)))^(1/2)*b^5*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+16*( cos(d*x+c)^2+2*cos(d*x+c)+1)*B*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^( 1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^5*EllipticE(-csc(d*x+c)+cot(d*x+c ),((a-b)/(a+b))^(1/2))+147*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*B*(1/(a+b)*(b+a* cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*b^5*El lipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+75*(cos(d*x+c)^2+2*cos (d*x+c)+1)*A*(1/(a+b)*(b+a*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/( 1+cos(d*x+c)))^(1/2)*b^5*EllipticF(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1 /2))+(-147*cos(d*x+c)^4-62*cos(d*x+c)^3-62*cos(d*x+c)^2-40*cos(d*x+c)-40)* B*a*b^4*tan(d*x+c)*sec(d*x+c)^2+(-13*cos(d*x+c)^3+11*cos(d*x+c)^2+cos(d*x+ c)+1)*B*a^2*b^3*tan(d*x+c)*sec(d*x+c)+111*(cos(d*x+c)^2+2*cos(d*x+c)+1)...
\[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \] Input:
integrate(sec(d*x+c)^4*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x, algorith m="fricas")
Output:
integral((B*sec(d*x + c)^5 + A*sec(d*x + c)^4)*sqrt(b*sec(d*x + c) + a), x )
\[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \sqrt {a + b \sec {\left (c + d x \right )}} \sec ^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**4*(a+b*sec(d*x+c))**(1/2)*(A+B*sec(d*x+c)),x)
Output:
Integral((A + B*sec(c + d*x))*sqrt(a + b*sec(c + d*x))*sec(c + d*x)**4, x)
\[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \] Input:
integrate(sec(d*x+c)^4*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x, algorith m="maxima")
Output:
integrate((B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^4, x)
\[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4} \,d x } \] Input:
integrate(sec(d*x+c)^4*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x, algorith m="giac")
Output:
integrate((B*sec(d*x + c) + A)*sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^4, x)
Timed out. \[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{{\cos \left (c+d\,x\right )}^4} \,d x \] Input:
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(1/2))/cos(c + d*x)^4,x)
Output:
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(1/2))/cos(c + d*x)^4, x)
\[ \int \sec ^4(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a \] Input:
int(sec(d*x+c)^4*(a+b*sec(d*x+c))^(1/2)*(A+B*sec(d*x+c)),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**5,x)*b + int(sqrt(sec(c + d*x)* b + a)*sec(c + d*x)**4,x)*a