\(\int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx\) [573]

Optimal result

Integrand size = 33, antiderivative size = 316 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {\left (2 a^2 A-3 A b^2+a b B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{a^2 \left (a^2-b^2\right ) d}+\frac {(A b-a B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a \left (a^2-b^2\right ) d}-\frac {\left (5 a^2 A b-3 A b^3-3 a^3 B+a b^2 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{a^2 (a-b) (a+b)^2 d}+\frac {\left (2 a^2 A-3 A b^2+a b B\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac {b (A b-a B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d (b+a \sec (c+d x))} \] Output:

-(2*A*a^2-3*A*b^2+B*a*b)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^( 
1/2))*sec(d*x+c)^(1/2)/a^2/(a^2-b^2)/d+(A*b-B*a)*cos(d*x+c)^(1/2)*InverseJ 
acobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/a/(a^2-b^2)/d-(5*A*a^2*b-3 
*A*b^3-3*B*a^3+B*a*b^2)*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*b 
/(a+b),2^(1/2))*sec(d*x+c)^(1/2)/a^2/(a-b)/(a+b)^2/d+(2*A*a^2-3*A*b^2+B*a* 
b)*sec(d*x+c)^(1/2)*sin(d*x+c)/a^2/(a^2-b^2)/d+b*(A*b-B*a)*sec(d*x+c)^(3/2 
)*sin(d*x+c)/a/(a^2-b^2)/d/(b+a*sec(d*x+c))
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(681\) vs. \(2(316)=632\).

Time = 7.32 (sec) , antiderivative size = 681, normalized size of antiderivative = 2.16 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=-\frac {\frac {2 \left (10 a^2 A b-9 A b^3-4 a^3 B+3 a b^2 B\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (4 a^3 A-8 a A b^2+4 a^2 b B\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (2 a^2 A b-3 A b^3+a b^2 B\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{4 a^2 (a-b) (a+b) d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {\left (2 a^2 A-3 A b^2+a b B\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right )}+\frac {A b^2 \sin (c+d x)-a b B \sin (c+d x)}{a \left (a^2-b^2\right ) (a+b \cos (c+d x))}\right )}{d} \] Input:

Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2 
,x]
 

Output:

-1/4*((2*(10*a^2*A*b - 9*A*b^3 - 4*a^3*B + 3*a*b^2*B)*Cos[c + d*x]^2*(Elli 
pticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec 
[c + d*x]]], -1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d* 
x])/(a*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3*A - 8*a*A*b^ 
2 + 4*a^2*b*B)*Cos[c + d*x]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]] 
], -1]*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + 
 b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((2*a^2*A*b - 3*A*b^3 + a*b^2*B)* 
Cos[2*(c + d*x)]*(b + a*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a 
*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - S 
ec[c + d*x]^2] + 2*(2*a - b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*S 
qrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcS 
in[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 
2*b^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x] 
]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - 
 Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a^2*(a - b)*(a 
 + b)*d) + (Sqrt[Sec[c + d*x]]*(((2*a^2*A - 3*A*b^2 + a*b*B)*Sin[c + d*x]) 
/(a^2*(a^2 - b^2)) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x])/(a*(a^2 - b 
^2)*(a + b*Cos[c + d*x]))))/d
 

Rubi [A] (verified)

Time = 2.40 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.97, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3439, 3042, 4517, 27, 3042, 4590, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 3284}

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.

\(\displaystyle \int \frac {\sec ^{\frac {3}{2}}(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 3439

\(\displaystyle \int \frac {\sec ^{\frac {5}{2}}(c+d x) (A \sec (c+d x)+B)}{(a \sec (c+d x)+b)^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (A \csc \left (c+d x+\frac {\pi }{2}\right )+B\right )}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+b\right )^2}dx\)

\(\Big \downarrow \) 4517

\(\displaystyle \frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}-\frac {\int -\frac {\sqrt {\sec (c+d x)} \left (\left (2 A a^2+b B a-3 A b^2\right ) \sec ^2(c+d x)-2 a (A b-a B) \sec (c+d x)+b (A b-a B)\right )}{2 (b+a \sec (c+d x))}dx}{a \left (a^2-b^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {\sec (c+d x)} \left (\left (2 A a^2+b B a-3 A b^2\right ) \sec ^2(c+d x)-2 a (A b-a B) \sec (c+d x)+b (A b-a B)\right )}{b+a \sec (c+d x)}dx}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (\left (2 A a^2+b B a-3 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-2 a (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )+b (A b-a B)\right )}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 4590

\(\displaystyle \frac {\frac {2 \int -\frac {\left (-2 B a^3+4 A b a^2+b^2 B a-3 A b^3\right ) \sec ^2(c+d x)+2 a \left (A a^2+b B a-2 A b^2\right ) \sec (c+d x)+b \left (2 A a^2+b B a-3 A b^2\right )}{2 \sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}+\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\int \frac {\left (-2 B a^3+4 A b a^2+b^2 B a-3 A b^3\right ) \sec ^2(c+d x)+2 a \left (A a^2+b B a-2 A b^2\right ) \sec (c+d x)+b \left (2 A a^2+b B a-3 A b^2\right )}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\int \frac {\left (-2 B a^3+4 A b a^2+b^2 B a-3 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (A a^2+b B a-2 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+b \left (2 A a^2+b B a-3 A b^2\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 4594

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {\int \frac {b^2 \left (2 A a^2+b B a-3 A b^2\right )-a b^2 (A b-a B) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {\int \frac {b^2 \left (2 A a^2+b B a-3 A b^2\right )-a b^2 (A b-a B) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-a b^2 (A b-a B) \int \sqrt {\sec (c+d x)}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-a b^2 (A b-a B) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx-a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {\frac {2 b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b^2}+\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{b+a \csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\frac {2 b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 4336

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx+\frac {\frac {2 b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {\frac {2 b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}}{a}}{2 a \left (a^2-b^2\right )}+\frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {b (A b-a B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)}+\frac {\frac {2 \left (2 a^2 A+a b B-3 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{a d}-\frac {\frac {\frac {2 b^2 \left (2 a^2 A+a b B-3 A b^2\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}-\frac {2 a b^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}}{b^2}+\frac {2 \left (-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{d (a+b)}}{a}}{2 a \left (a^2-b^2\right )}\)

Input:

Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^(3/2))/(a + b*Cos[c + d*x])^2,x]
 

Output:

(b*(A*b - a*B)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Se 
c[c + d*x])) + (-((((2*b^2*(2*a^2*A - 3*A*b^2 + a*b*B)*Sqrt[Cos[c + d*x]]* 
EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (2*a*b^2*(A*b - a*B)*Sqr 
t[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d)/b^2 + (2* 
(5*a^2*A*b - 3*A*b^3 - 3*a^3*B + a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticPi[(2 
*b)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a + b)*d))/a) + (2*(2*a 
^2*A - 3*A*b^2 + a*b*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d))/(2*a*(a^2 
- b^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* 
(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3284

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) 
 + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 
2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 
0] && GtQ[c + d, 0]
 

rule 3439

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* 
(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim 
p[g^(m + n)   Int[(g*Csc[e + f*x])^(p - m - n)*(b + a*Csc[e + f*x])^m*(d + 
c*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c 
- a*d, 0] &&  !IntegerQ[p] && IntegerQ[m] && IntegerQ[n]
 

rule 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4336

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]   Int[ 
1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4517

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*d^2*( 
A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 
 2)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[d/(b*(m + 1)*(a^2 - b^2))   Int[( 
a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^(n - 2)*Simp[a*d*(A*b - a*B)*( 
n - 2) + b*d*(A*b - a*B)*(m + 1)*Csc[e + f*x] - (a*A*b*d*(m + n) - d*B*(a^2 
*(n - 1) + b^2*(m + 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[m, -1] && GtQ[ 
n, 1]
 

rule 4590

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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 
)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) 
   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( 
A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*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] && GtQ[n, 0]
 

rule 4594

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2)   Int[(d*Csc[e + 
f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2   Int[(a*A - (A*b - a 
*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, 
B, C}, x] && NeQ[a^2 - b^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(855\) vs. \(2(305)=610\).

Time = 7.31 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.71

method	result	size
default	\(\text {Expression too large to display}\)	\(856\)

Input:

int((A+B*cos(d*x+c))*sec(d*x+c)^(3/2)/(a+cos(d*x+c)*b)^2,x,method=_RETURNV 
ERBOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A/a^2/sin(1/ 
2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 
*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(sin(1/2*d 
*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+ 
1/2*c),2^(1/2)))-2*(A*b-B*a)/a*(-b^2/a/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*si 
n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*b*cos(1/2*d*x+1/2*c)^2+a 
-b)-1/2/a/(a+b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^( 
1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/ 
2*d*x+1/2*c),2^(1/2))-1/2*b/(a^2-b^2)/a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*c 
os(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2 
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1/2*b/(a^2-b^2)/a*(sin(1/2*d 
*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c 
)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3*a/ 
(a^2-b^2)/(-2*a*b+2*b^2)*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/ 
2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellip 
ticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2))+1/a/(a^2-b^2)/(-2*a*b+2*b^2)* 
b^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin 
(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c 
),-2*b/(a-b),2^(1/2)))+4*A*b^2/a^2/(-2*a*b+2*b^2)*(sin(1/2*d*x+1/2*c)^2)^( 
1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^2,x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)**(3/2)/(a+b*cos(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^2,x, algorith 
m="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}} \,d x } \] Input:

integrate((A+B*cos(d*x+c))*sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^2,x, algorith 
m="giac")
 

Output:

integrate((B*cos(d*x + c) + A)*sec(d*x + c)^(3/2)/(b*cos(d*x + c) + a)^2, 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2} \,d x \] Input:

int(((A + B*cos(c + d*x))*(1/cos(c + d*x))^(3/2))/(a + b*cos(c + d*x))^2,x 
)
 

Output:

int(((A + B*cos(c + d*x))*(1/cos(c + d*x))^(3/2))/(a + b*cos(c + d*x))^2, 
x)
 

Reduce [F]

\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx=\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )}{\cos \left (d x +c \right ) b +a}d x \] Input:

int((A+B*cos(d*x+c))*sec(d*x+c)^(3/2)/(a+b*cos(d*x+c))^2,x)
 

Output:

int((sqrt(sec(c + d*x))*sec(c + d*x))/(cos(c + d*x)*b + a),x)