Integrand size = 33, antiderivative size = 509 \[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\frac {(a-b) \sqrt {a+b} \left (16 a^2 A-3 A b^2+6 a b B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a^2 b d}+\frac {\sqrt {a+b} (2 a+b) (8 a A-3 A b+6 a B) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a^2 d}-\frac {\sqrt {a+b} \left (4 a^2 A b+A b^3+8 a^3 B-2 a b^2 B\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^3 d}+\frac {\left (16 a^2 A-3 A b^2+6 a b B\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 a^2 d}+\frac {(A b+6 a B) \cos (c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{12 a d}+\frac {A \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 d} \]
1/24*(a-b)*(16*A*a^2-3*A*b^2+6*B*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)/a^2/b/d+1/24*(2*a+b)*(8*A*a-3*A* b+6*B*a)*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)/a^2/d-1/8*(4*A*a^2*b+A*b^3+8*B*a^3-2*B*a*b^2)*cot(d*x+c)*Elli pticPi((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^ 3/d+1/24*(16*A*a^2-3*A*b^2+6*B*a*b)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/ d+1/12*(A*b+6*B*a)*cos(d*x+c)*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a/d+1/3*A* cos(d*x+c)^2*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d
Leaf count is larger than twice the leaf count of optimal. \(1548\) vs. \(2(509)=1018\).
Time = 19.10 (sec) , antiderivative size = 1548, normalized size of antiderivative = 3.04 \[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx =\text {Too large to display} \]
(Sqrt[a + b*Sec[c + d*x]]*((A*Sin[c + d*x])/12 + ((A*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a) + (A*Sin[3*(c + d*x)])/12))/d - (Sqrt[a + b*Sec[c + d*x]]*S qrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b* Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] - 3*A*b^3*Tan[(c + d*x)/2] + 6*a^2*b*B*Tan[(c + d*x)/2] + 6*a*b^2*B*Tan[(c + d*x)/2] - 32* a^3*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 12*a^2*b*B*Tan[( c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^ 5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x)/2]^5 + 6*a^2*b*B* Tan[(c + d*x)/2]^5 - 6*a*b^2*B*Tan[(c + d*x)/2]^5 + 24*a^2*A*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)] + 6* A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - T an[(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*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)] - 12*a*b^2*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*T an[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a^2*A*b*EllipticPi [-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt...
Time = 2.48 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4520, 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.
\(\displaystyle \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
\(\Big \downarrow \) 4520 |
\(\displaystyle \frac {1}{3} \int \frac {\cos ^2(c+d x) \left (3 A b \sec ^2(c+d x)+2 (2 a A+3 b B) \sec (c+d x)+A b+6 a B\right )}{2 \sqrt {a+b \sec (c+d x)}}dx+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int \frac {\cos ^2(c+d x) \left (3 A b \sec ^2(c+d x)+2 (2 a A+3 b B) \sec (c+d x)+A b+6 a B\right )}{\sqrt {a+b \sec (c+d x)}}dx+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \int \frac {3 A b \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 (2 a A+3 b B) \csc \left (c+d x+\frac {\pi }{2}\right )+A b+6 a B}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \frac {1}{6} \left (\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}-\frac {\int -\frac {\cos (c+d x) \left (16 A a^2+6 b B a+2 (7 A b+6 a B) \sec (c+d x) a-3 A b^2+b (A b+6 a B) \sec ^2(c+d x)\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{2 a}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\cos (c+d x) \left (16 A a^2+6 b B a+2 (7 A b+6 a B) \sec (c+d x) a-3 A b^2+b (A b+6 a B) \sec ^2(c+d x)\right )}{\sqrt {a+b \sec (c+d x)}}dx}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {16 A a^2+6 b B a+2 (7 A b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a-3 A b^2+b (A b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4592 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}-\frac {\int -\frac {-b \left (16 A a^2+6 b B a-3 A b^2\right ) \sec ^2(c+d x)+2 a b (A b+6 a B) \sec (c+d x)+3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {-b \left (16 A a^2+6 b B a-3 A b^2\right ) \sec ^2(c+d x)+2 a b (A b+6 a B) \sec (c+d x)+3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {-b \left (16 A a^2+6 b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (A b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4546 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+\left (2 a b (A b+6 a B)+b \left (16 A a^2+6 b B a-3 A b^2\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx-b \left (16 a^2 A+6 a b B-3 A b^2\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {3 \left (8 B a^3+4 A b a^2-2 b^2 B a+A b^3\right )+\left (2 a b (A b+6 a B)+b \left (16 A a^2+6 b B a-3 A b^2\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (16 a^2 A+6 a b B-3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4409 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {-b \left (16 a^2 A+6 a b B-3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx+b (2 a+b) (8 a A+6 a B-3 A b) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {-b \left (16 a^2 A+6 a b B-3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b (2 a+b) (8 a A+6 a B-3 A b) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4271 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {-b \left (16 a^2 A+6 a b B-3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b (2 a+b) (8 a A+6 a B-3 A b) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4319 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {-b \left (16 a^2 A+6 a b B-3 A b^2\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 \sqrt {a+b} (2 a+b) (8 a A+6 a B-3 A b) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}+\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
\(\Big \downarrow \) 4492 |
\(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 A+6 a b B-3 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{a d}+\frac {\frac {2 (a-b) \sqrt {a+b} \left (16 a^2 A+6 a b B-3 A b^2\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}-\frac {6 \sqrt {a+b} \left (8 a^3 B+4 a^2 A b-2 a b^2 B+A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}+\frac {2 \sqrt {a+b} (2 a+b) (8 a A+6 a B-3 A b) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}}{2 a}}{4 a}+\frac {(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sec (c+d x)}}{2 a d}\right )+\frac {A \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \sec (c+d x)}}{3 d}\) |
(A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (((A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d) + (((2* (a - b)*Sqrt[a + b]*(16*a^2*A - 3*A*b^2 + 6*a*b*B)*Cot[c + d*x]*EllipticE[ ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(2*a + b)*(8*a*A - 3*A*b + 6*a*B)*Cot[c + d*x]*EllipticF[A rcSin[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*S qrt[a + b]*(4*a^2*A*b + A*b^3 + 8*a^3*B - 2*a*b^2*B)*Cot[c + d*x]*Elliptic Pi[(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))])/(a*d))/(2*a) + ((16*a^2*A - 3*A*b^2 + 6*a*b*B)*Sqrt[a + b*Sec[c + d *x]]*Sin[c + d*x])/(a*d))/(4*a))/6
3.4.55.3.1 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*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*A*(n + 1))*Csc[e + f*x] - A*b*(m + n + 1)*Csc[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ [a^2 - b^2, 0] && LtQ[0, m, 1] && 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_.) + 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. \(4000\) vs. \(2(464)=928\).
Time = 9.22 (sec) , antiderivative size = 4001, normalized size of antiderivative = 7.86
1/24/d/a^2*(-24*A*(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)*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b) )^(1/2))*a^2*b*cos(d*x+c)^2+12*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)*EllipticPi(cot(d*x+c)-csc(d*x+c), -1,((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c)^2+56*A*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)*a^2*b*cos(d*x+c)+28*A*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)*a^2*b*cos(d*x+c)^2+24*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 )*EllipticPi(cot(d*x+c)-csc(d*x+c),-1,((a-b)/(a+b))^(1/2))*a*b^2*cos(d*x+c )-16*A*EllipticE(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)*a^3+3* A*EllipticE(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^3-6*A*Ell ipticPi(cot(d*x+c)-csc(d*x+c),-1,((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^3-48*B*Ell ipticPi(cot(d*x+c)-csc(d*x+c),-1,((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)*a^3+24*B*Ell ipticF(cot(d*x+c)-csc(d*x+c),((a-b)/(a+b))^(1/2))*(1/(a+b)*(b+a*cos(d*x...
\[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3} \,d x } \]
\[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int \left (A + B \sec {\left (c + d x \right )}\right ) \sqrt {a + b \sec {\left (c + d x \right )}} \cos ^{3}{\left (c + d x \right )}\, dx \]
\[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3} \,d x } \]
\[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} \sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(c+d x) \sqrt {a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]