Integrand size = 42, antiderivative size = 450 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (9 a b B+4 a^2 C-8 b^2 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{4 b d}+\frac {\sqrt {a+b} \left (8 b^2 (B-C)+2 a^2 (B+2 C)+3 a b (3 B+8 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{4 d}-\frac {\sqrt {a+b} \left (4 a^2 B+15 b^2 B+20 a b C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{4 d}+\frac {a (7 b B+4 a C) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {a B \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{2 d} \] Output:
1/4*(a-b)*(a+b)^(1/2)*(9*B*a*b+4*C*a^2-8*C*b^2)*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/d+1/4*(a+b)^(1/2)*(8*b^2*(B-C)+2 *a^2*(B+2*C)+3*a*b*(3*B+8*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)/d-1/4*(a+b)^(1/2)*(4*B*a^2+15*B*b^2+20*C*a*b)*cot(d* x+c)*EllipticPi((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((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)/d+1/4 *a*(7*B*b+4*C*a)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d+1/2*a*B*cos(d*x+c)*(a +b*sec(d*x+c))^(3/2)*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(1326\) vs. \(2(450)=900\).
Time = 16.52 (sec) , antiderivative size = 1326, normalized size of antiderivative = 2.95 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(2*b^2*C*Sin[c + d*x] + (a^2*B* Sin[2*(c + d*x)])/4))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^( 5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(9*a^2*b*B*Tan[(c + d*x)/2] + 9*a *b^2*B*Tan[(c + d*x)/2] + 4*a^3*C*Tan[(c + d*x)/2] + 4*a^2*b*C*Tan[(c + d* x)/2] - 8*a*b^2*C*Tan[(c + d*x)/2] - 8*b^3*C*Tan[(c + d*x)/2] - 18*a^2*b*B *Tan[(c + d*x)/2]^3 - 8*a^3*C*Tan[(c + d*x)/2]^3 + 16*a*b^2*C*Tan[(c + d*x )/2]^3 + 9*a^2*b*B*Tan[(c + d*x)/2]^5 - 9*a*b^2*B*Tan[(c + d*x)/2]^5 + 4*a ^3*C*Tan[(c + d*x)/2]^5 - 4*a^2*b*C*Tan[(c + d*x)/2]^5 - 8*a*b^2*C*Tan[(c + d*x)/2]^5 + 8*b^3*C*Tan[(c + d*x)/2]^5 + 8*a^3*B*EllipticPi[-1, ArcSin[T an[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*b^2*B*Ell ipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d *x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 40*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] *Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[( c + d*x)/2]^2)/(a + b)] + 8*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*b^2*B*E llipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^ 2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*T...
Time = 2.17 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4560, 3042, 4513, 27, 3042, 4582, 27, 3042, 4546, 3042, 4409, 3042, 4271, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}-\frac {1}{2} \int -\frac {1}{2} \cos (c+d x) \sqrt {a+b \sec (c+d x)} \left (-b (a B-4 b C) \sec ^2(c+d x)+2 \left (B a^2+4 b C a+2 b^2 B\right ) \sec (c+d x)+a (7 b B+4 a C)\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \cos (c+d x) \sqrt {a+b \sec (c+d x)} \left (-b (a B-4 b C) \sec ^2(c+d x)+2 \left (B a^2+4 b C a+2 b^2 B\right ) \sec (c+d x)+a (7 b B+4 a C)\right )dx+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (-b (a B-4 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (B a^2+4 b C a+2 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (7 b B+4 a C)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{4} \left (\int \frac {-b \left (4 C a^2+9 b B a-8 b^2 C\right ) \sec ^2(c+d x)+2 b \left (B a^2+12 b C a+4 b^2 B\right ) \sec (c+d x)+a \left (4 B a^2+20 b C a+15 b^2 B\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {-b \left (4 C a^2+9 b B a-8 b^2 C\right ) \sec ^2(c+d x)+2 b \left (B a^2+12 b C a+4 b^2 B\right ) \sec (c+d x)+a \left (4 B a^2+20 b C a+15 b^2 B\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \int \frac {-b \left (4 C a^2+9 b B a-8 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left (B a^2+12 b C a+4 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (4 B a^2+20 b C a+15 b^2 B\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (4 B a^2+20 b C a+15 b^2 B\right )+\left (2 b \left (B a^2+12 b C a+4 b^2 B\right )+b \left (4 C a^2+9 b B a-8 b^2 C\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-b \left (4 a^2 C+9 a b B-8 b^2 C\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\int \frac {a \left (4 B a^2+20 b C a+15 b^2 B\right )+\left (2 b \left (B a^2+12 b C a+4 b^2 B\right )+b \left (4 C a^2+9 b B a-8 b^2 C\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (4 a^2 C+9 a b B-8 b^2 C\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-b \left (4 a^2 C+9 a b B-8 b^2 C\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (4 a^2 B+20 a b C+15 b^2 B\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+b \left (2 a^2 (B+2 C)+3 a b (3 B+8 C)+8 b^2 (B-C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (a \left (4 a^2 B+20 a b C+15 b^2 B\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (2 a^2 (B+2 C)+3 a b (3 B+8 C)+8 b^2 (B-C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (4 a^2 C+9 a b B-8 b^2 C\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (b \left (2 a^2 (B+2 C)+3 a b (3 B+8 C)+8 b^2 (B-C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (4 a^2 C+9 a b B-8 b^2 C\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} \left (4 a^2 B+20 a b C+15 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (-b \left (4 a^2 C+9 a b B-8 b^2 C\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (2 a^2 (B+2 C)+3 a b (3 B+8 C)+8 b^2 (B-C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {2 \sqrt {a+b} \left (4 a^2 B+20 a b C+15 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{2} \left (\frac {2 \sqrt {a+b} \left (2 a^2 (B+2 C)+3 a b (3 B+8 C)+8 b^2 (B-C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \left (4 a^2 C+9 a b B-8 b^2 C\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {2 \sqrt {a+b} \left (4 a^2 B+20 a b C+15 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )+\frac {a (4 a C+7 b B) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}\) |
Input:
Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(a*B*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d) + (((2*(a - b)*Sqrt[a + b]*(9*a*b*B + 4*a^2*C - 8*b^2*C)*Cot[c + d*x]*EllipticE[Arc Sin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - S ec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2 *Sqrt[a + b]*(8*b^2*(B - C) + 2*a^2*(B + 2*C) + 3*a*b*(3*B + 8*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))])/d - (2*Sqrt[a + b]*(4*a^2*B + 15*b^2*B + 20*a*b*C)*Cot[c + d*x]*E llipticPi[(a + b)/a, 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))])/d)/2 + (a*(7*b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d *x])/d)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) *((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
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_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ .) + (a_)], x_Symbol] :> Simp[c Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && 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_)]*(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[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C )*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C 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, C}, x] && NeQ[a^2 - b^2, 0]
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[((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[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1828\) vs. \(2(409)=818\).
Time = 65.53 (sec) , antiderivative size = 1829, normalized size of antiderivative = 4.06
Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
1/4/d*(8*(-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)*a^3*EllipticPi(-csc(d*x+ c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+30*(-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)*a*b^2*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+40 *(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+ b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*EllipticPi(-csc(d*x+c)+cot (d*x+c),-1,((a-b)/(a+b))^(1/2))+9*(-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) *a^2*b*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+9*(-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)*a*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b )/(a+b))^(1/2))+4*(-cos(d*x+c)^2-2*cos(d*x+c)-1)*C*(cos(d*x+c)/(cos(d*x+c) +1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*EllipticE(- csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+4*(-cos(d*x+c)^2-2*cos(d*x+c)-1 )*C*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c )+1))^(1/2)*a^2*b*EllipticE(-csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+8* (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*b^2*EllipticE(-csc(d*x+c)+cot(d* x+c),((a-b)/(a+b))^(1/2))+8*(cos(d*x+c)^2+2*cos(d*x+c)+1)*C*(1/(a+b)*(b...
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="fricas")
Output:
integral((C*b^2*cos(d*x + c)^3*sec(d*x + c)^4 + B*a^2*cos(d*x + c)^3*sec(d *x + c) + (2*C*a*b + B*b^2)*cos(d*x + c)^3*sec(d*x + c)^3 + (C*a^2 + 2*B*a *b)*cos(d*x + c)^3*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(a+b*sec(d*x+c))**(5/2)*(B*sec(d*x+c)+C*sec(d*x+c) **2),x)
Output:
Timed out
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c))*(b*sec(d*x + c) + a)^(5/2)*c os(d*x + c)^3, x)
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3} \,d x } \] Input:
integrate(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(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)^(5/2)*c os(d*x + c)^3, x)
Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \] Input:
int(cos(c + d*x)^3*(B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + b/cos(c + d*x) )^(5/2),x)
Output:
int(cos(c + d*x)^3*(B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + b/cos(c + d*x) )^(5/2), x)
\[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{4}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3}d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{2}d x \right ) a \,b^{2}+\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \sec \left (d x +c \right )d x \right ) a^{2} b \] Input:
int(cos(d*x+c)^3*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**4,x)*b**2*c + 2 *int(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*a*b*c + i nt(sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**3,x)*b**3 + int( sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**2,x)*a**2*c + 2*int (sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x)**2,x)*a*b**2 + int( sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**3*sec(c + d*x),x)*a**2*b