Integrand size = 42, antiderivative size = 573 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 (a-b) \sqrt {a+b} \left (88 a^4 b B+363 a^2 b^3 B+1617 b^5 B-48 a^5 C-108 a^3 b^2 C+2088 a b^4 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^5 d}-\frac {2 (a-b) \sqrt {a+b} \left (3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)+6 a^2 b^2 (11 B-24 C)+4 a^3 b (22 B-9 C)-48 a^4 C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{3465 b^4 d}+\frac {2 \left (88 a^3 b B+429 a b^3 B-48 a^4 C-144 a^2 b^2 C+675 b^4 C\right ) \sqrt {a+b \sec (c+d x)} \tan (c+d x)}{3465 b^3 d}+\frac {2 \left (88 a^2 b B+539 b^3 B-48 a^3 C-204 a b^2 C\right ) (a+b \sec (c+d x))^{3/2} \tan (c+d x)}{3465 b^3 d}-\frac {2 \left (44 a b B-24 a^2 C-81 b^2 C\right ) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{693 b^3 d}+\frac {2 (11 b B-6 a C) \sec (c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{99 b^2 d}+\frac {2 C \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \tan (c+d x)}{11 b d} \] Output:
-2/3465*(a-b)*(a+b)^(1/2)*(88*B*a^4*b+363*B*a^2*b^3+1617*B*b^5-48*C*a^5-10 8*C*a^3*b^2+2088*C*a*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-2/3465*(a-b)*(a+b)^(1/2)*(3*a*b^3*(143*B-471*C)-3* b^4*(539*B-225*C)+6*a^2*b^2*(11*B-24*C)+4*a^3*b*(22*B-9*C)-48*a^4*C)*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+2/3465* (88*B*a^3*b+429*B*a*b^3-48*C*a^4-144*C*a^2*b^2+675*C*b^4)*(a+b*sec(d*x+c)) ^(1/2)*tan(d*x+c)/b^3/d+2/3465*(88*B*a^2*b+539*B*b^3-48*C*a^3-204*C*a*b^2) *(a+b*sec(d*x+c))^(3/2)*tan(d*x+c)/b^3/d-2/693*(44*B*a*b-24*C*a^2-81*C*b^2 )*(a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b^3/d+2/99*(11*B*b-6*C*a)*sec(d*x+c)*( a+b*sec(d*x+c))^(5/2)*tan(d*x+c)/b^2/d+2/11*C*sec(d*x+c)^2*(a+b*sec(d*x+c) )^(5/2)*tan(d*x+c)/b/d
Leaf count is larger than twice the leaf count of optimal. \(4220\) vs. \(2(573)=1146\).
Time = 24.91 (sec) , antiderivative size = 4220, normalized size of antiderivative = 7.36 \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-2*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sin[c + d*x])/(34 65*b^4) + (2*Sec[c + d*x]^4*(11*b*B*Sin[c + d*x] + 12*a*C*Sin[c + d*x]))/9 9 + (2*Sec[c + d*x]^3*(110*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 81* b^2*C*Sin[c + d*x]))/(693*b) + (2*Sec[c + d*x]^2*(33*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] - 18*a^3*C*Sin[c + d*x] + 606*a*b^2*C*Sin[c + d*x ]))/(3465*b^2) + (2*Sec[c + d*x]*(-44*a^3*b*B*Sin[c + d*x] + 968*a*b^3*B*S in[c + d*x] + 24*a^4*C*Sin[c + d*x] + 57*a^2*b^2*C*Sin[c + d*x] + 675*b^4* C*Sin[c + d*x]))/(3465*b^3) + (2*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d* (b + a*Cos[c + d*x])) + (2*((-11*a^2*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt [Sec[c + d*x]]) - (8*a^4*B)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^2*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16 *a^5*C)/(1155*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (12*a^3*C )/(385*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (232*a*b*C)/(385*S qrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*B*Sqrt[Sec[c + d*x]]) /(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*B*Sqrt[Sec[c + d*x]])/(315*b *Sqrt[b + a*Cos[c + d*x]]) + (13*a*b*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a *Cos[c + d*x]]) - (13*a^2*C*Sqrt[Sec[c + d*x]])/(55*Sqrt[b + a*Cos[c + d*x ]]) + (16*a^6*C*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (32*a^4*C*Sqrt[Sec[c + d*x]])/(1155*b^2*Sqrt[b + a*Cos[c + d*x]]) + (15...
Time = 2.95 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.04, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 4560, 3042, 4521, 27, 3042, 4580, 27, 3042, 4570, 27, 3042, 4490, 27, 3042, 4490, 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.
| \(\displaystyle \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \sec ^4(c+d x) (a+b \sec (c+d x))^{3/2} (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^4 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4521 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left ((11 b B-6 a C) \sec ^2(c+d x)+9 b C \sec (c+d x)+4 a C\right )dx}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left ((11 b B-6 a C) \sec ^2(c+d x)+9 b C \sec (c+d x)+4 a C\right )dx}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((11 b B-6 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 b C \csc \left (c+d x+\frac {\pi }{2}\right )+4 a C\right )dx}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4580 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (-\left (\left (-24 C a^2+44 b B a-81 b^2 C\right ) \sec ^2(c+d x)\right )+b (77 b B-6 a C) \sec (c+d x)+2 a (11 b B-6 a C)\right )dx}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (-\left (\left (-24 C a^2+44 b B a-81 b^2 C\right ) \sec ^2(c+d x)\right )+b (77 b B-6 a C) \sec (c+d x)+2 a (11 b B-6 a C)\right )dx}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (\left (24 C a^2-44 b B a+81 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+b (77 b B-6 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a (11 b B-6 a C)\right )dx}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4570 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {1}{2} \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (3 b \left (-12 C a^2+22 b B a-135 b^2 C\right )-\left (-48 C a^3+88 b B a^2-204 b^2 C a+539 b^3 B\right ) \sec (c+d x)\right )dx}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left (3 b \left (-12 C a^2+22 b B a-135 b^2 C\right )-\left (-48 C a^3+88 b B a^2-204 b^2 C a+539 b^3 B\right ) \sec (c+d x)\right )dx}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (-12 C a^2+22 b B a-135 b^2 C\right )+\left (48 C a^3-88 b B a^2+204 b^2 C a-539 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {\frac {-\frac {\frac {2}{5} \int \frac {3}{2} \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (b \left (-12 C a^3+22 b B a^2-471 b^2 C a-539 b^3 B\right )-\left (-48 C a^4+88 b B a^3-144 b^2 C a^2+429 b^3 B a+675 b^4 C\right ) \sec (c+d x)\right )dx-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \int \sec (c+d x) \sqrt {a+b \sec (c+d x)} \left (b \left (-12 C a^3+22 b B a^2-471 b^2 C a-539 b^3 B\right )-\left (-48 C a^4+88 b B a^3-144 b^2 C a^2+429 b^3 B a+675 b^4 C\right ) \sec (c+d x)\right )dx-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \int \csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-12 C a^3+22 b B a^2-471 b^2 C a-539 b^3 B\right )+\left (48 C a^4-88 b B a^3+144 b^2 C a^2-429 b^3 B a-675 b^4 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4490 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \left (\frac {2}{3} \int -\frac {\sec (c+d x) \left (b \left (-12 C a^4+22 b B a^3+1269 b^2 C a^2+2046 b^3 B a+675 b^4 C\right )+\left (-48 C a^5+88 b B a^4-108 b^2 C a^3+363 b^3 B a^2+2088 b^4 C a+1617 b^5 B\right ) \sec (c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \left (-\frac {1}{3} \int \frac {\sec (c+d x) \left (b \left (-12 C a^4+22 b B a^3+1269 b^2 C a^2+2046 b^3 B a+675 b^4 C\right )+\left (-48 C a^5+88 b B a^4-108 b^2 C a^3+363 b^3 B a^2+2088 b^4 C a+1617 b^5 B\right ) \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \left (-\frac {1}{3} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (b \left (-12 C a^4+22 b B a^3+1269 b^2 C a^2+2046 b^3 B a+675 b^4 C\right )+\left (-48 C a^5+88 b B a^4-108 b^2 C a^3+363 b^3 B a^2+2088 b^4 C a+1617 b^5 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4493 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left ((a-b) \left (-48 a^4 C+a^3 b (88 B-36 C)+6 a^2 b^2 (11 B-24 C)+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-\left (-48 a^5 C+88 a^4 b B-108 a^3 b^2 C+363 a^2 b^3 B+2088 a b^4 C+1617 b^5 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx\right )-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left ((a-b) \left (-48 a^4 C+a^3 b (88 B-36 C)+6 a^2 b^2 (11 B-24 C)+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (-48 a^5 C+88 a^4 b B-108 a^3 b^2 C+363 a^2 b^3 B+2088 a b^4 C+1617 b^5 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {\frac {-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 (a-b) \sqrt {a+b} \left (-48 a^4 C+a^3 b (88 B-36 C)+6 a^2 b^2 (11 B-24 C)+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}-\left (-48 a^5 C+88 a^4 b B-108 a^3 b^2 C+363 a^2 b^3 B+2088 a b^4 C+1617 b^5 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {\frac {-\frac {2 \left (-24 a^2 C+44 a b B-81 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}-\frac {\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 (a-b) \sqrt {a+b} \left (-48 a^4 C+a^3 b (88 B-36 C)+6 a^2 b^2 (11 B-24 C)+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 (a-b) \sqrt {a+b} \left (-48 a^5 C+88 a^4 b B-108 a^3 b^2 C+363 a^2 b^3 B+2088 a b^4 C+1617 b^5 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}\right )-\frac {2 \left (-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right ) \tan (c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\right )-\frac {2 \left (-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right ) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}}{7 b}}{9 b}+\frac {2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}}{11 b}+\frac {2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}\) |
Input:
Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(2*C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*b*d) + (( 2*(11*b*B - 6*a*C)*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/( 9*b*d) + ((-2*(44*a*b*B - 24*a^2*C - 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)* Tan[c + d*x])/(7*b*d) - ((-2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 204*a*b^ 2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d) + (3*(((2*(a - b)*Sqrt [a + b]*(88*a^4*b*B + 363*a^2*b^3*B + 1617*b^5*B - 48*a^5*C - 108*a^3*b^2* C + 2088*a*b^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/S qrt[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*a*b^ 3*(143*B - 471*C) - 3*b^4*(539*B - 225*C) + a^3*b*(88*B - 36*C) + 6*a^2*b^ 2*(11*B - 24*C) - 48*a^4*C)*Cot[c + d*x]*EllipticF[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*d))/3 - (2*(88*a^3*b*B + 42 9*a*b^3*B - 48*a^4*C - 144*a^2*b^2*C + 675*b^4*C)*Sqrt[a + b*Sec[c + d*x]] *Tan[c + d*x])/(3*d)))/5)/(7*b))/(9*b))/(11*b)
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_))^(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[1/(m + 1) Int[Csc[e + f*x]* (a + b*Csc[e + f*x])^(m - 1)*Simp[b*B*m + a*A*(m + 1) + (a*B*m + A*b*(m + 1 ))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, A, B, e, f}, x] && NeQ[A*b - a* B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 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^ 2*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 2)/(b*f* (m + n))), x] + Simp[d^2/(b*(m + n)) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 2)*Simp[a*B*(n - 2) + B*b*(m + n - 1)*Csc[e + f*x] + (A*b*(m + n) - a*B*(n - 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B , m}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && NeQ[m + n, 0] && !IGtQ[m, 1]
Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_. )*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.) *(x_)]*(d_.))^(n_.), x_Symbol] :> Simp[1/b^2 Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(2992\) vs. \(2(531)=1062\).
Time = 114.43 (sec) , antiderivative size = 2993, normalized size of antiderivative = 5.22
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2993\) |
| parts | \(\text {Expression too large to display}\) | \(2999\) |
Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
-2/3465/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)*(48*(cos(d*x+c)^2+2*cos(d*x+c)+1)*C*(1/(a+b)*(b+a*cos(d*x+c))/(cos (d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a^6*EllipticE(-csc(d*x +c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+44*sin(d*x+c)*(cos(d*x+c)-1)*B*a^4*b^2 -88*B*a^5*b*cos(d*x+c)*sin(d*x+c)+45*(-15*cos(d*x+c)^5-15*cos(d*x+c)^4-9*c os(d*x+c)^3-9*cos(d*x+c)^2-7*cos(d*x+c)-7)*C*b^6*tan(d*x+c)*sec(d*x+c)^4+7 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^6*ta n(d*x+c)*sec(d*x+c)^3+6*(-1+18*cos(d*x+c)^2-cos(d*x+c))*a^4*b^2*C*tan(d*x+ c)+11*(-33*cos(d*x+c)^2+cos(d*x+c)+1)*a^3*b^3*B*tan(d*x+c)+24*sin(d*x+c)*( 1-cos(d*x+c))*a^5*C*b+3*(-225*cos(d*x+c)^5-921*cos(d*x+c)^4-337*cos(d*x+c) ^3-337*cos(d*x+c)^2-245*cos(d*x+c)-245)*C*a*b^5*tan(d*x+c)*sec(d*x+c)^3+11 *(-88*cos(d*x+c)^3-121*cos(d*x+c)^2-53*cos(d*x+c)-53)*B*a^2*b^4*tan(d*x+c) *sec(d*x+c)+11*(-147*cos(d*x+c)^4-137*cos(d*x+c)^3-137*cos(d*x+c)^2-85*cos (d*x+c)-85)*a*b^5*B*tan(d*x+c)*sec(d*x+c)^2+1617*(-cos(d*x+c)^2-2*cos(d*x+ c)-1)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d *x+c)+1))^(1/2)*b^6*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+ 1617*(cos(d*x+c)^2+2*cos(d*x+c)+1)*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c) +1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^6*EllipticF(-csc(d*x+c)+cot (d*x+c),((a-b)/(a+b))^(1/2))+675*(cos(d*x+c)^2+2*cos(d*x+c)+1)*C*(1/(a+b)* (b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2...
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="fricas")
Output:
integral((C*b*sec(d*x + c)^6 + B*a*sec(d*x + c)^4 + (C*a + B*b)*sec(d*x + c)^5)*sqrt(b*sec(d*x + c) + a), x)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (B + C \sec {\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \sec ^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(sec(d*x+c)**3*(a+b*sec(d*x+c))**(3/2)*(B*sec(d*x+c)+C*sec(d*x+c) **2),x)
Output:
Integral((B + C*sec(c + d*x))*(a + b*sec(c + d*x))**(3/2)*sec(c + d*x)**4, x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="maxima")
Output:
Timed out
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{3} \,d x } \] Input:
integrate(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(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))*(b*sec(d*x + c) + a)^(3/2)*s ec(d*x + c)^3, x)
Timed out. \[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \frac {\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \] Input:
int(((B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2))/cos(c + d*x)^3,x)
Output:
int(((B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + b/cos(c + d*x))^(3/2))/cos(c + d*x)^3, x)
\[ \int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{6}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{5}d x \right ) b^{2}+\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a b \] Input:
int(sec(d*x+c)^3*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**6,x)*b*c + int(sqrt(sec(c + d*x )*b + a)*sec(c + d*x)**5,x)*a*c + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x )**5,x)*b**2 + int(sqrt(sec(c + d*x)*b + a)*sec(c + d*x)**4,x)*a*b