Integrand size = 42, antiderivative size = 518 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (16 a^2 B+33 b^2 B+54 a b C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 b d}+\frac {\sqrt {a+b} \left (4 a^2 (4 B+3 C)+3 b^2 (11 B+16 C)+a b (26 B+54 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 d}-\frac {\sqrt {a+b} \left (20 a^2 b B+5 b^3 B+8 a^3 C+30 a b^2 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a d}+\frac {\left (16 a^2 B+33 b^2 B+54 a b C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d}+\frac {a (3 b B+2 a C) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{4 d}+\frac {a B \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{3 d} \]
1/3*a*B*cos(d*x+c)^2*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/24*(a-b)*(16*B* a^2+33*B*b^2+54*C*a*b)*cot(d*x+c)*EllipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^( 1/2),((a+b)/(a-b))^(1/2))*(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+1/24*(4*a^2*(4*B+3*C)+3*b^2*(11*B+16*C)+a*b *(26*B+54*C))*cot(d*x+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+ b)/(a-b))^(1/2))*(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-1/8*(20*B*a^2*b+5*B*b^3+8*C*a^3+30*C*a*b^2)*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) )*(a+b)^(1/2)*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/ 2)/a/d+1/24*(16*B*a^2+33*B*b^2+54*C*a*b)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2) /d+1/4*a*(3*B*b+2*C*a)*cos(d*x+c)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(1530\) vs. \(2(518)=1036\).
Time = 20.36 (sec) , antiderivative size = 1530, normalized size of antiderivative = 2.95 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
(Sqrt[a + b*Sec[c + d*x]]*((a^2*B*Sin[c + d*x])/12 + (a*(13*b*B + 6*a*C)*S in[2*(c + d*x)])/24 + (a^2*B*Sin[3*(c + d*x)])/12))/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*B*Tan[(c + d*x)/2] + 1 6*a^2*b*B*Tan[(c + d*x)/2] + 33*a*b^2*B*Tan[(c + d*x)/2] + 33*b^3*B*Tan[(c + d*x)/2] + 54*a^2*b*C*Tan[(c + d*x)/2] + 54*a*b^2*C*Tan[(c + d*x)/2] - 3 2*a^3*B*Tan[(c + d*x)/2]^3 - 66*a*b^2*B*Tan[(c + d*x)/2]^3 - 108*a^2*b*C*T an[(c + d*x)/2]^3 + 16*a^3*B*Tan[(c + d*x)/2]^5 - 16*a^2*b*B*Tan[(c + d*x) /2]^5 + 33*a*b^2*B*Tan[(c + d*x)/2]^5 - 33*b^3*B*Tan[(c + d*x)/2]^5 + 54*a ^2*b*C*Tan[(c + d*x)/2]^5 - 54*a*b^2*C*Tan[(c + d*x)/2]^5 + 120*a^2*b*B*El lipticPi[-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)] + 30*b^3*B*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)] + 48*a^3*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)] + 180*a*b^2*C*EllipticPi[-1, ArcS in[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)] + 120*a^2*b* 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 + ...
Time = 2.68 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {3042, 4560, 3042, 4513, 27, 3042, 4582, 27, 3042, 4592, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}-\frac {1}{3} \int -\frac {1}{2} \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (a B+6 b C) \sec ^2(c+d x)+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \sec (c+d x)+3 a (3 b B+2 a C)\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \left (b (a B+6 b C) \sec ^2(c+d x)+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \sec (c+d x)+3 a (3 b B+2 a C)\right )dx+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (a B+6 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (3 b B+2 a C)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{2} \int \frac {\cos (c+d x) \left (b \left (6 C a^2+13 b B a+24 b^2 C\right ) \sec ^2(c+d x)+2 \left (6 C a^3+19 b B a^2+36 b^2 C a+12 b^3 B\right ) \sec (c+d x)+a \left (16 B a^2+54 b C a+33 b^2 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {\cos (c+d x) \left (b \left (6 C a^2+13 b B a+24 b^2 C\right ) \sec ^2(c+d x)+2 \left (6 C a^3+19 b B a^2+36 b^2 C a+12 b^3 B\right ) \sec (c+d x)+a \left (16 B a^2+54 b C a+33 b^2 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {b \left (6 C a^2+13 b B a+24 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (6 C a^3+19 b B a^2+36 b^2 C a+12 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (16 B a^2+54 b C a+33 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int -\frac {-a b \left (16 B a^2+54 b C a+33 b^2 B\right ) \sec ^2(c+d x)+2 a b \left (6 C a^2+13 b B a+24 b^2 C\right ) \sec (c+d x)+3 a \left (8 C a^3+20 b B a^2+30 b^2 C a+5 b^3 B\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\int \frac {-a b \left (16 B a^2+54 b C a+33 b^2 B\right ) \sec ^2(c+d x)+2 a b \left (6 C a^2+13 b B a+24 b^2 C\right ) \sec (c+d x)+3 a \left (8 C a^3+20 b B a^2+30 b^2 C a+5 b^3 B\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\int \frac {-a b \left (16 B a^2+54 b C a+33 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (6 C a^2+13 b B a+24 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (8 C a^3+20 b B a^2+30 b^2 C a+5 b^3 B\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\int \frac {3 a \left (8 C a^3+20 b B a^2+30 b^2 C a+5 b^3 B\right )+\left (a b \left (16 B a^2+54 b C a+33 b^2 B\right )+2 a b \left (6 C a^2+13 b B a+24 b^2 C\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-a b \left (16 a^2 B+54 a b C+33 b^2 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\int \frac {3 a \left (8 C a^3+20 b B a^2+30 b^2 C a+5 b^3 B\right )+\left (a b \left (16 B a^2+54 b C a+33 b^2 B\right )+2 a b \left (6 C a^2+13 b B a+24 b^2 C\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (16 a^2 B+54 a b C+33 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {-a b \left (16 a^2 B+54 a b C+33 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a b \left (4 a^2 (4 B+3 C)+a b (26 B+54 C)+3 b^2 (11 B+16 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 a \left (8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {a b \left (4 a^2 (4 B+3 C)+a b (26 B+54 C)+3 b^2 (11 B+16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (16 a^2 B+54 a b C+33 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a \left (8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {a b \left (4 a^2 (4 B+3 C)+a b (26 B+54 C)+3 b^2 (11 B+16 C)\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (16 a^2 B+54 a b C+33 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {-a b \left (16 a^2 B+54 a b C+33 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sqrt {a+b} \left (4 a^2 (4 B+3 C)+a b (26 B+54 C)+3 b^2 (11 B+16 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {\left (16 a^2 B+54 a b C+33 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}+\frac {\frac {2 a \sqrt {a+b} \left (4 a^2 (4 B+3 C)+a b (26 B+54 C)+3 b^2 (11 B+16 C)\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 a (a-b) \sqrt {a+b} \left (16 a^2 B+54 a b C+33 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \left (8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}\right )+\frac {3 a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}\) |
(a*B*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + ((3*a *(3*b*B + 2*a*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (((2*a*(a - b)*Sqrt[a + b]*(16*a^2*B + 33*b^2*B + 54*a*b*C)*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*d) + (2*a*Sqrt[a + b]*(4*a^2*(4*B + 3*C) + 3*b^2*(11*B + 16*C) + a *b*(26*B + 54*C))*Cot[c + d*x]*EllipticF[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))])/d - (6*Sqrt[a + b]*(20*a^2*b*B + 5*b^3*B + 8*a^3*C + 30*a*b^2*C)*Cot[c + d*x]*EllipticPi[(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) + ((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/4)/6
3.9.35.3.1 Defintions of rubi rules used
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]
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 + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*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] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(4780\) vs. \(2(473)=946\).
Time = 169.18 (sec) , antiderivative size = 4781, normalized size of antiderivative = 9.23
int(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
1/24/d*(34*B*a^2*b*cos(d*x+c)^3*sin(d*x+c)+34*B*a^2*b*cos(d*x+c)^2*sin(d*x +c)+59*B*a*b^2*cos(d*x+c)^2*sin(d*x+c)+66*C*a^2*b*cos(d*x+c)^2*sin(d*x+c)+ 96*B*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*co s(d*x+c))/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*b^3*cos( d*x+c)-16*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot( d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a ^2*b-33*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticE(cot(d* x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*a*b ^2+48*B*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(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^3*c os(d*x+c)^2+76*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF (cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1 /2)*a^2*b-26*B*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF(c ot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2 )*a*b^2+8*B*a^3*cos(d*x+c)^4*sin(d*x+c)+8*B*a^3*cos(d*x+c)^3*sin(d*x+c)+12 *C*a^3*cos(d*x+c)^3*sin(d*x+c)+16*B*a^3*cos(d*x+c)^2*sin(d*x+c)+33*B*b^3*c os(d*x+c)*sin(d*x+c)-24*C*EllipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1 /2))*(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*cos(d*x+c)-108*C*EllipticE(cot(d*x+c)-csc(d*x+c),((a-b) /(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+...
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="fricas")
integral((C*b^2*cos(d*x + c)^4*sec(d*x + c)^4 + B*a^2*cos(d*x + c)^4*sec(d *x + c) + (2*C*a*b + B*b^2)*cos(d*x + c)^4*sec(d*x + c)^3 + (C*a^2 + 2*B*a *b)*cos(d*x + c)^4*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="maxima")
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{4} \,d x } \]
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(5/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="giac")
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \]