\(\int \frac {\cos (c+d x) (B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^{5/2}} \, dx\) [848]

Optimal result

Integrand size = 40, antiderivative size = 331 \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=-\frac {2 \left (2 a^3 b B-6 a b^3 B-8 a^4 C+15 a^2 b^2 C-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (2 a^2 b B-3 b^3 B-8 a^3 C+9 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{3 b^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {2 a \left (2 a^2 b B-6 b^3 B-5 a^3 C+9 a b^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}} \] Output:

-2/3*(2*B*a^3*b-6*B*a*b^3-8*C*a^4+15*C*a^2*b^2-3*C*b^4)*(a+b*cos(d*x+c))^( 
1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^3/(a^2-b^2)^2 
/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/3*(2*B*a^2*b-3*B*b^3-8*C*a^3+9*C*a*b^2 
)*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(b/ 
(a+b))^(1/2))/b^3/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)-2/3*a^2*(B*b-C*a)*sin 
(d*x+c)/b^2/(a^2-b^2)/d/(a+b*cos(d*x+c))^(3/2)+2/3*a*(2*B*a^2*b-6*B*b^3-5* 
C*a^3+9*C*a*b^2)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 3.13 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.83 \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{3/2} \left (b^2 \left (a^2 b B+3 b^3 B+2 a^3 C-6 a b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (-2 a^3 b B+6 a b^3 B+8 a^4 C-15 a^2 b^2 C+3 b^4 C\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )}{(a-b)^2 (a+b)}-\frac {a b \left (a \left (-a^2 b B+5 b^3 B+4 a^3 C-8 a b^2 C\right )+b \left (-2 a^2 b B+6 b^3 B+5 a^3 C-9 a b^2 C\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2}\right )}{3 b^3 d (a+b \cos (c+d x))^{3/2}} \] Input:

Integrate[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c 
+ d*x])^(5/2),x]
 

Output:

(2*((((a + b*Cos[c + d*x])/(a + b))^(3/2)*(b^2*(a^2*b*B + 3*b^3*B + 2*a^3* 
C - 6*a*b^2*C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (-2*a^3*b*B + 6*a*b 
^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*((a + b)*EllipticE[(c + d*x)/2, ( 
2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])))/((a - b)^2*(a + 
 b)) - (a*b*(a*(-(a^2*b*B) + 5*b^3*B + 4*a^3*C - 8*a*b^2*C) + b*(-2*a^2*b* 
B + 6*b^3*B + 5*a^3*C - 9*a*b^2*C)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2) 
^2))/(3*b^3*d*(a + b*Cos[c + d*x])^(3/2))
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.425, Rules used = {3042, 3508, 3042, 3467, 27, 3042, 3500, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

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.

\(\displaystyle \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3508

\(\displaystyle \int \frac {\cos ^2(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^{5/2}}dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3467

\(\displaystyle \frac {2 \int \frac {3 b \left (a^2-b^2\right ) C \cos ^2(c+d x)+\left (2 a^2-3 b^2\right ) (b B-a C) \cos (c+d x)+3 a b (b B-a C)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 b \left (a^2-b^2\right ) C \cos ^2(c+d x)+\left (2 a^2-3 b^2\right ) (b B-a C) \cos (c+d x)+3 a b (b B-a C)}{(a+b \cos (c+d x))^{3/2}}dx}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3500

\(\displaystyle \frac {\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {b^2 \left (2 C a^3+b B a^2-6 b^2 C a+3 b^3 B\right )-b \left (-8 C a^4+2 b B a^3+15 b^2 C a^2-6 b^3 B a-3 b^4 C\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {b^2 \left (2 C a^3+b B a^2-6 b^2 C a+3 b^3 B\right )-b \left (-8 C a^4+2 b B a^3+15 b^2 C a^2-6 b^3 B a-3 b^4 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {b^2 \left (2 C a^3+b B a^2-6 b^2 C a+3 b^3 B\right )-b \left (-8 C a^4+2 b B a^3+15 b^2 C a^2-6 b^3 B a-3 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx-\left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}+\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {2 a \left (-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\frac {2 \left (a^2-b^2\right ) \left (-8 a^3 C+2 a^2 b B+9 a b^2 C-3 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{b \left (a^2-b^2\right )}}{3 b^2 \left (a^2-b^2\right )}-\frac {2 a^2 (b B-a C) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}\)

Input:

Int[(Cos[c + d*x]*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x] 
)^(5/2),x]
 

Output:

(-2*a^2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Cos[c + d*x] 
)^(3/2)) + (((-2*(2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C 
)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[ 
(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(2*a^2*b*B - 3*b^3*B - 8*a 
^3*C + 9*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2 
, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]))/(b*(a^2 - b^2)) + (2*a*(2* 
a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Sin[c + d*x])/((a^2 - b^2)*d*Sqrt 
[a + b*Cos[c + d*x]]))/(3*b^2*(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 3132

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

rule 3134

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

rule 3140

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

rule 3142

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

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*Sin[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]
 

rule 3467

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2*((A_.) + (B_.)*sin[(e_.) + (f 
_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[ 
(B*c - A*d)*(b*c - a*d)^2*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(f*d^2 
*(n + 1)*(c^2 - d^2))), x] - Simp[1/(d^2*(n + 1)*(c^2 - d^2))   Int[(c + d* 
Sin[e + f*x])^(n + 1)*Simp[d*(n + 1)*(B*(b*c - a*d)^2 - A*d*(a^2*c + b^2*c 
- 2*a*b*d)) - ((B*c - A*d)*(a^2*d^2*(n + 2) + b^2*(c^2 + d^2*(n + 1))) + 2* 
a*b*d*(A*c*d*(n + 2) - B*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b^2*B*d*(n + 
1)*(c^2 - d^2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[ 
n, -1]
 

rule 3500

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 
 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* 
(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A 
*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, 
B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
 

rule 3508

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2   Int[(a + b*Sin[e + f*x])^(m 
+ 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ 
[{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - 
a*b*B + a^2*C, 0]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(953\) vs. \(2(320)=640\).

Time = 8.86 (sec) , antiderivative size = 954, normalized size of antiderivative = 2.88

method	result	size
default	\(\text {Expression too large to display}\)	\(954\)
parts	\(\text {Expression too large to display}\)	\(1763\)

Input:

int(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x,meth 
od=_RETURNVERBOSE)
 

Output:

-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/b^3/(-2* 
b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2* 
c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*(B*b*Ellip 
ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-3*C*a*EllipticF(cos(1/2*d*x+1/ 
2*c),(-2*b/(a-b))^(1/2))+C*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) 
)*a-C*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b)+2*a^2*(B*b-C*a)/ 
b^3*(1/6/b/(a-b)/(a+b)*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b) 
*sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2+1/2*(a-b)/b)^2+8/3*b*si 
n(1/2*d*x+1/2*c)^2/(a-b)^2/(a+b)^2*cos(1/2*d*x+1/2*c)*a/(-(-2*b*cos(1/2*d* 
x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+(3*a-b)/(3*a^3+3*a^2*b-3*a*b^2 
-3*b^3)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b) 
)^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellip 
ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-4/3*a/(a-b)/(a+b)^2*(sin(1/2*d 
*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin( 
1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(EllipticF(cos(1/2*d*x+ 
1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2) 
)))+2*a*(2*B*b-3*C*a)/b^3/sin(1/2*d*x+1/2*c)^2/(2*b*sin(1/2*d*x+1/2*c)^2-a 
-b)/(a^2-b^2)*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2) 
*(2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^2*b+(-2*b/(a-b)*sin(1/2*d*x+1/2* 
c)^2+(a+b)/(a-b))^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 1159, normalized size of antiderivative = 3.50 \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2), 
x, algorithm="fricas")
 

Output:

-2/9*(sqrt(1/2)*(-16*I*C*a^7 + 4*I*B*a^6*b + 36*I*C*a^5*b^2 - 9*I*B*a^4*b^ 
3 - 24*I*C*a^3*b^4 + 9*I*B*a^2*b^5 + (-16*I*C*a^5*b^2 + 4*I*B*a^4*b^3 + 36 
*I*C*a^3*b^4 - 9*I*B*a^2*b^5 - 24*I*C*a*b^6 + 9*I*B*b^7)*cos(d*x + c)^2 + 
2*(-16*I*C*a^6*b + 4*I*B*a^5*b^2 + 36*I*C*a^4*b^3 - 9*I*B*a^3*b^4 - 24*I*C 
*a^2*b^5 + 9*I*B*a*b^6)*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a 
^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I* 
b*sin(d*x + c) + 2*a)/b) + sqrt(1/2)*(16*I*C*a^7 - 4*I*B*a^6*b - 36*I*C*a^ 
5*b^2 + 9*I*B*a^4*b^3 + 24*I*C*a^3*b^4 - 9*I*B*a^2*b^5 + (16*I*C*a^5*b^2 - 
 4*I*B*a^4*b^3 - 36*I*C*a^3*b^4 + 9*I*B*a^2*b^5 + 24*I*C*a*b^6 - 9*I*B*b^7 
)*cos(d*x + c)^2 + 2*(16*I*C*a^6*b - 4*I*B*a^5*b^2 - 36*I*C*a^4*b^3 + 9*I* 
B*a^3*b^4 + 24*I*C*a^2*b^5 - 9*I*B*a*b^6)*cos(d*x + c))*sqrt(b)*weierstras 
sPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*c 
os(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(-8*I*C*a^6*b + 2 
*I*B*a^5*b^2 + 15*I*C*a^4*b^3 - 6*I*B*a^3*b^4 - 3*I*C*a^2*b^5 + (-8*I*C*a^ 
4*b^3 + 2*I*B*a^3*b^4 + 15*I*C*a^2*b^5 - 6*I*B*a*b^6 - 3*I*C*b^7)*cos(d*x 
+ c)^2 + 2*(-8*I*C*a^5*b^2 + 2*I*B*a^4*b^3 + 15*I*C*a^3*b^4 - 6*I*B*a^2*b^ 
5 - 3*I*C*a*b^6)*cos(d*x + c))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2) 
/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2) 
/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + 
 c) + 2*a)/b)) + 3*sqrt(1/2)*(8*I*C*a^6*b - 2*I*B*a^5*b^2 - 15*I*C*a^4*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**(5/2 
),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2), 
x, algorithm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*cos(d*x + c)/(b*cos(d*x + c) 
 + a)^(5/2), x)
 

Giac [F]

\[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2), 
x, algorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*cos(d*x + c)/(b*cos(d*x + c) 
 + a)^(5/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:

int((cos(c + d*x)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x) 
)^(5/2),x)
 

Output:

int((cos(c + d*x)*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d*x) 
)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b \] Input:

int(cos(d*x+c)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(5/2),x)
 

Output:

int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3)/(cos(c + d*x)**3*b**3 + 3*c 
os(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*c + int((sqrt(cos 
(c + d*x)*b + a)*cos(c + d*x)**2)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)** 
2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*b