Integrand size = 42, antiderivative size = 520 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(a-b) \sqrt {a+b} \left (16 a^2 B+3 b^2 B+30 a b C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a b d}+\frac {\sqrt {a+b} \left (16 a^2 B+14 a b B+3 b^2 B+12 a^2 C+30 a b C\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a d}-\frac {\sqrt {a+b} \left (12 a^2 b B-b^3 B+8 a^3 C+6 a b^2 C\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^2 d}+\frac {\left (16 a^2 B+3 b^2 B+30 a b C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a d}+\frac {(7 b B+6 a C) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 d}+\frac {a B \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d} \] Output:
1/24*(a-b)*(a+b)^(1/2)*(16*B*a^2+3*B*b^2+30*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))*(b*(1-sec(d*x+c))/(a +b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a/b/d+1/24*(a+b)^(1/2)*(16*B*a^ 2+14*B*a*b+3*B*b^2+12*C*a^2+30*C*a*b)*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)/a/d-1/8*(a+b)^(1/2)*(12*B*a^2*b-B*b^3+8*C*a^ 3+6*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))*(b*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c)) /(a-b))^(1/2)/a^2/d+1/24*(16*B*a^2+3*B*b^2+30*C*a*b)*(a+b*sec(d*x+c))^(1/2 )*sin(d*x+c)/a/d+1/12*(7*B*b+6*C*a)*cos(d*x+c)*(a+b*sec(d*x+c))^(1/2)*sin( d*x+c)/d+1/3*a*B*cos(d*x+c)^2*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d
Leaf count is larger than twice the leaf count of optimal. \(1516\) vs. \(2(520)=1040\).
Time = 15.82 (sec) , antiderivative size = 1516, normalized size of antiderivative = 2.92 \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(Sqrt[a + b*Sec[c + d*x]]*((a*B*Sin[c + d*x])/12 + ((7*b*B + 6*a*C)*Sin[2* (c + d*x)])/24 + (a*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] + 16*a^2*b *B*Tan[(c + d*x)/2] + 3*a*b^2*B*Tan[(c + d*x)/2] + 3*b^3*B*Tan[(c + d*x)/2 ] + 30*a^2*b*C*Tan[(c + d*x)/2] + 30*a*b^2*C*Tan[(c + d*x)/2] - 32*a^3*B*T an[(c + d*x)/2]^3 - 6*a*b^2*B*Tan[(c + d*x)/2]^3 - 60*a^2*b*C*Tan[(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 + 3*a *b^2*B*Tan[(c + d*x)/2]^5 - 3*b^3*B*Tan[(c + d*x)/2]^5 + 30*a^2*b*C*Tan[(c + d*x)/2]^5 - 30*a*b^2*C*Tan[(c + d*x)/2]^5 + 72*a^2*b*B*EllipticPi[-1, A rcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqr t[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*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)] + 36*a*b^2*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)] + 72*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 + b*Tan[(c + d*x)/2...
Time = 2.69 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.00, 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, 4592, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}-\frac {1}{3} \int -\frac {\cos ^2(c+d x) \left (3 b (a B+2 b C) \sec ^2(c+d x)+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \sec (c+d x)+a (7 b B+6 a C)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int \frac {\cos ^2(c+d x) \left (3 b (a B+2 b C) \sec ^2(c+d x)+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \sec (c+d x)+a (7 b B+6 a C)\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {3 b (a B+2 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (2 B a^2+6 b C a+3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (7 b B+6 a C)}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{6} \left (\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}-\frac {\int -\frac {\cos (c+d x) \left (a b (7 b B+6 a C) \sec ^2(c+d x)+2 a \left (6 C a^2+13 b B a+12 b^2 C\right ) \sec (c+d x)+a \left (16 B a^2+30 b C a+3 b^2 B\right )\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\cos (c+d x) \left (a b (7 b B+6 a C) \sec ^2(c+d x)+2 a \left (6 C a^2+13 b B a+12 b^2 C\right ) \sec (c+d x)+a \left (16 B a^2+30 b C a+3 b^2 B\right )\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\int \frac {a b (7 b B+6 a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (6 C a^2+13 b B a+12 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (16 B a^2+30 b C a+3 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}-\frac {\int -\frac {2 b (7 b B+6 a C) \sec (c+d x) a^2-b \left (16 B a^2+30 b C a+3 b^2 B\right ) \sec ^2(c+d x) a+3 \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {2 b (7 b B+6 a C) \sec (c+d x) a^2-b \left (16 B a^2+30 b C a+3 b^2 B\right ) \sec ^2(c+d x) a+3 \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {2 b (7 b B+6 a C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2-b \left (16 B a^2+30 b C a+3 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right ) a}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4546 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {3 a \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right )+\left (2 b (7 b B+6 a C) a^2+b \left (16 B a^2+30 b C a+3 b^2 B\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-a b \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {3 a \left (8 C a^3+12 b B a^2+6 b^2 C a-b^3 B\right )+\left (2 b (7 b B+6 a C) a^2+b \left (16 B a^2+30 b C a+3 b^2 B\right ) a\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {-a b \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+a b \left (16 a^2 B+12 a^2 C+14 a b B+30 a b C+3 b^2 B\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+3 a \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {a b \left (16 a^2 B+12 a^2 C+14 a b B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {a b \left (16 a^2 B+12 a^2 C+14 a b B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {-a b \left (16 a^2 B+30 a b C+3 b^2 B\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sqrt {a+b} \left (16 a^2 B+12 a^2 C+14 a b B+30 a b C+3 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 B+30 a b C+3 b^2 B\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d}+\frac {\frac {2 a \sqrt {a+b} \left (16 a^2 B+12 a^2 C+14 a b B+30 a b C+3 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}+\frac {2 a (a-b) \sqrt {a+b} \left (16 a^2 B+30 a b C+3 b^2 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \left (8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}}{4 a}+\frac {(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 d}\right )+\frac {a B \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
Input:
Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(a*B*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (((7*b* B + 6*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 + 3*b^2*B + 30*a*b*C)*Cot[c + d*x]*Ellip ticE[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]*(16*a^2*B + 14*a*b*B + 3*b^2*B + 12*a^2*C + 30*a*b* 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 - (6*Sqrt[a + b]*(12*a^2*b*B - b^3*B + 8*a^3*C + 6*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 + 3*b^2*B + 30*a* b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d)/(4*a))/6
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 + 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. \(1779\) vs. \(2(475)=950\).
Time = 11.99 (sec) , antiderivative size = 1780, normalized size of antiderivative = 3.42
Input:
int(cos(d*x+c)^4*(a+b*sec(d*x+c))^(3/2)*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
Output:
1/24/d/a*((-72*cos(d*x+c)^2-144*cos(d*x+c)-72)*B*(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))+(6*cos(d*x+c)^2+12*cos(d*x+ c)+6)*B*(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)*b^3*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/ 2))+(-48*cos(d*x+c)^2-96*cos(d*x+c)-48)*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*EllipticPi(-csc(d*x+ c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+(-36*cos(d*x+c)^2-72*cos(d*x+c)-36)* 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*b^2*EllipticPi(-csc(d*x+c)+cot(d*x+c),-1,((a-b)/(a+b))^(1/2))+ (-16*cos(d*x+c)^2-32*cos(d*x+c)-16)*B*(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)+co t(d*x+c),((a-b)/(a+b))^(1/2))+(-16*cos(d*x+c)^2-32*cos(d*x+c)-16)*B*(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))+(-3*cos(d*x+ c)^2-6*cos(d*x+c)-3)*B*(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*b^2*EllipticE(-csc(d*x+c)+cot(d*x+c),((a- b)/(a+b))^(1/2))+(-3*cos(d*x+c)^2-6*cos(d*x+c)-3)*B*(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)*b^3*EllipticE( -csc(d*x+c)+cot(d*x+c),((a-b)/(a+b))^(1/2))+(-30*cos(d*x+c)^2-60*cos(d*...
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cos(d*x+c)^4*(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*cos(d*x + c)^4*sec(d*x + c)^3 + B*a*cos(d*x + c)^4*sec(d*x + c) + (C*a + B*b)*cos(d*x + c)^4*sec(d*x + c)^2)*sqrt(b*sec(d*x + c) + a), x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**4*(a+b*sec(d*x+c))**(3/2)*(B*sec(d*x+c)+C*sec(d*x+c) **2),x)
Output:
Timed out
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cos(d*x+c)^4*(a+b*sec(d*x+c))^(3/2)*(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)^(3/2)*c os(d*x + c)^4, x)
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right )\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cos(d*x+c)^4*(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)*c os(d*x + c)^4, x)
Timed out. \[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^4\,\left (\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^4*(B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + b/cos(c + d*x) )^(3/2),x)
Output:
int(cos(c + d*x)^4*(B/cos(c + d*x) + C/cos(c + d*x)^2)*(a + b/cos(c + d*x) )^(3/2), x)
\[ \int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) a c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )^{2}d x \right ) b^{2}+\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sec \left (d x +c \right )d x \right ) a b \] Input:
int(cos(d*x+c)^4*(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)*cos(c + d*x)**4*sec(c + d*x)**3,x)*b*c + int( sqrt(sec(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x)**2,x)*a*c + int(sqrt (sec(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x)**2,x)*b**2 + int(sqrt(se c(c + d*x)*b + a)*cos(c + d*x)**4*sec(c + d*x),x)*a*b