\(\int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx\) [310]

Optimal result

Integrand size = 33, antiderivative size = 375 \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=-\frac {\left (16 a^2 A+3 A b^2+30 a b B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{24 a d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (16 a^2 A+17 A b^2+42 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{24 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (12 a^2 A b-A b^3+8 a^3 B+6 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{8 a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (16 a^2 A+3 A b^2+30 a b B\right ) \sqrt {a+b \cos (c+d x)} \tan (c+d x)}{24 a d}+\frac {(7 A b+6 a B) \sqrt {a+b \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{12 d}+\frac {a A \sqrt {a+b \cos (c+d x)} \sec ^2(c+d x) \tan (c+d x)}{3 d} \] Output:

-1/24*(16*A*a^2+3*A*b^2+30*B*a*b)*(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))/a/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+1/ 
24*(16*A*a^2+17*A*b^2+42*B*a*b)*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJaco 
biAM(1/2*d*x+1/2*c,2^(1/2)*(b/(a+b))^(1/2))/d/(a+b*cos(d*x+c))^(1/2)+1/8*( 
12*A*a^2*b-A*b^3+8*B*a^3+6*B*a*b^2)*((a+b*cos(d*x+c))/(a+b))^(1/2)*Ellipti 
cPi(sin(1/2*d*x+1/2*c),2,2^(1/2)*(b/(a+b))^(1/2))/a/d/(a+b*cos(d*x+c))^(1/ 
2)+1/24*(16*A*a^2+3*A*b^2+30*B*a*b)*(a+b*cos(d*x+c))^(1/2)*tan(d*x+c)/a/d+ 
1/12*(7*A*b+6*B*a)*(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)*tan(d*x+c)/d+1/3*a*A* 
(a+b*cos(d*x+c))^(1/2)*sec(d*x+c)^2*tan(d*x+c)/d
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.19 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.69 \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\frac {\frac {2 \left (28 a A b^2+24 a^2 b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (56 a^2 A b-9 A b^3+48 a^3 B+6 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-16 a^2 A b-3 A b^3-30 a b^2 B\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{96 a d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{12} \sec ^2(c+d x) (7 A b \sin (c+d x)+6 a B \sin (c+d x))+\frac {\sec (c+d x) \left (16 a^2 A \sin (c+d x)+3 A b^2 \sin (c+d x)+30 a b B \sin (c+d x)\right )}{24 a}+\frac {1}{3} a A \sec ^2(c+d x) \tan (c+d x)\right )}{d} \] Input:

Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x 
]
 

Output:

((2*(28*a*A*b^2 + 24*a^2*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF 
[(c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] + (2*(56*a^2*A*b - 
9*A*b^3 + 48*a^3*B + 6*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipti 
cPi[2, (c + d*x)/2, (2*b)/(a + b)])/Sqrt[a + b*Cos[c + d*x]] - ((2*I)*(-16 
*a^2*A*b - 3*A*b^3 - 30*a*b^2*B)*Sqrt[(b - b*Cos[c + d*x])/(a + b)]*Sqrt[- 
((b + b*Cos[c + d*x])/(a - b))]*Cos[2*(c + d*x)]*(2*a*(a - b)*EllipticE[I* 
ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)] + 
b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Cos[c + d*x]]], 
(a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*S 
qrt[a + b*Cos[c + d*x]]], (a + b)/(a - b)]))*Sin[c + d*x])/(a*Sqrt[-(a + b 
)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Cos[c + d* 
x]) + (a + b*Cos[c + d*x])^2)/b^2)]*(2*a^2 - b^2 - 4*a*(a + b*Cos[c + d*x] 
) + 2*(a + b*Cos[c + d*x])^2)))/(96*a*d) + (Sqrt[a + b*Cos[c + d*x]]*((Sec 
[c + d*x]^2*(7*A*b*Sin[c + d*x] + 6*a*B*Sin[c + d*x]))/12 + (Sec[c + d*x]* 
(16*a^2*A*Sin[c + d*x] + 3*A*b^2*Sin[c + d*x] + 30*a*b*B*Sin[c + d*x]))/(2 
4*a) + (a*A*Sec[c + d*x]^2*Tan[c + d*x])/3))/d
 

Rubi [A] (verified)

Time = 3.37 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.04, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 3468, 27, 3042, 3534, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 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.

\(\displaystyle \int \sec ^4(c+d x) (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3468

\(\displaystyle \frac {1}{3} \int \frac {\left (3 b (a A+2 b B) \cos ^2(c+d x)+2 \left (2 A a^2+6 b B a+3 A b^2\right ) \cos (c+d x)+a (7 A b+6 a B)\right ) \sec ^3(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \int \frac {\left (3 b (a A+2 b B) \cos ^2(c+d x)+2 \left (2 A a^2+6 b B a+3 A b^2\right ) \cos (c+d x)+a (7 A b+6 a B)\right ) \sec ^3(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \int \frac {3 b (a A+2 b B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (2 A a^2+6 b B a+3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (7 A b+6 a B)}{\sin \left (c+d x+\frac {\pi }{2}\right )^3 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (a b (7 A b+6 a B) \cos ^2(c+d x)+2 a \left (6 B a^2+13 A b a+12 b^2 B\right ) \cos (c+d x)+a \left (16 A a^2+30 b B a+3 A b^2\right )\right ) \sec ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {\left (a b (7 A b+6 a B) \cos ^2(c+d x)+2 a \left (6 B a^2+13 A b a+12 b^2 B\right ) \cos (c+d x)+a \left (16 A a^2+30 b B a+3 A b^2\right )\right ) \sec ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\int \frac {a b (7 A b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a \left (6 B a^2+13 A b a+12 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (16 A a^2+30 b B a+3 A b^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (2 b (7 A b+6 a B) \cos (c+d x) a^2-b \left (16 A a^2+30 b B a+3 A b^2\right ) \cos ^2(c+d x) a+3 \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a\right ) \sec (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {\left (2 b (7 A b+6 a B) \cos (c+d x) a^2-b \left (16 A a^2+30 b B a+3 A b^2\right ) \cos ^2(c+d x) a+3 \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\int \frac {2 b (7 A b+6 a B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2-b \left (16 A a^2+30 b B a+3 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {1}{6} \left (\frac {\frac {-a \left (16 a^2 A+30 a b B+3 A b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx-\frac {\int -\frac {\left (b \left (16 A a^2+42 b B a+17 A b^2\right ) \cos (c+d x) a^2+3 b \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {\left (b \left (16 A a^2+42 b B a+17 A b^2\right ) \cos (c+d x) a^2+3 b \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-a \left (16 a^2 A+30 a b B+3 A b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 A a^2+42 b B a+17 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-a \left (16 a^2 A+30 a b B+3 A b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 A a^2+42 b B a+17 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 A a^2+42 b B a+17 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\int \frac {b \left (16 A a^2+42 b B a+17 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2+3 b \left (8 B a^3+12 A b a^2+6 b^2 B a-A b^3\right ) a}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx+3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\frac {\frac {3 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}+\frac {2 a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}+\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {1}{6} \left (\frac {\frac {\left (16 a^2 A+30 a b B+3 A b^2\right ) \tan (c+d x) \sqrt {a+b \cos (c+d x)}}{d}+\frac {\frac {\frac {2 a^2 b \left (16 a^2 A+42 a b B+17 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}+\frac {6 a b \left (8 a^3 B+12 a^2 A b+6 a b^2 B-A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {2 a \left (16 a^2 A+30 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}}{2 a}}{4 a}+\frac {(6 a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {a A \tan (c+d x) \sec ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\)

Input:

Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x])*Sec[c + d*x]^4,x]
 

Output:

(a*A*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (((7*A* 
b + 6*a*B)*Sqrt[a + b*Cos[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(2*d) + ((( 
-2*a*(16*a^2*A + 3*A*b^2 + 30*a*b*B)*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*(16*a^2*A + 17*A*b^2 + 42*a*b*B)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*El 
lipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]]) + (6*a*b 
*(12*a^2*A*b - A*b^3 + 8*a^3*B + 6*a*b^2*B)*Sqrt[(a + b*Cos[c + d*x])/(a + 
 b)]*EllipticPi[2, (c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x] 
]))/b)/(2*a) + ((16*a^2*A + 3*A*b^2 + 30*a*b*B)*Sqrt[a + b*Cos[c + d*x]]*T 
an[c + d*x])/d)/(4*a))/6
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 3286

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

rule 3468

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si 
mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c 
 + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 
1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n 
+ 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a 
*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) 
 - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - 
a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*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] && GtQ[m, 1] && LtQ[n, -1]
 

rule 3481

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

rule 3534

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3538

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2326\) vs. \(2(359)=718\).

Time = 18.17 (sec) , antiderivative size = 2327, normalized size of antiderivative = 6.21

method	result	size
default	\(\text {Expression too large to display}\)	\(2327\)
parts	\(\text {Expression too large to display}\)	\(2727\)

Input:

int((a+cos(d*x+c)*b)^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x,method=_RETURNV 
ERBOSE)
 

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*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)*EllipticPi(cos 
(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))+2*a*(2*A*b+B*a)*(-1/2*cos(1/2*d*x+1/ 
2*c)/a*(-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)^2-1)^2+3/4*b/a^2*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)/(2*cos(1/2*d*x+1/2*c)^2-1)-1/8*b/a* 
(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))+3/8/a*(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)*b*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^( 
1/2))-3/8*b^2/a^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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/2*(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)*EllipticPi(cos(1/2*d*x+1/2*c) 
,2,(-2*b/(a-b))^(1/2))-3/8/a^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)*EllipticPi(cos(1/2*d*x+1/2*c),2,(-2*b/(a-b))^(1/2))*b...
 

Fricas [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{4} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x, algorith 
m="maxima")
 

Output:

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/2)*sec(d*x + c)^4, 
x)
 

Giac [F]

\[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\int { {\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{4} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x, algorith 
m="giac")
 

Output:

integrate((B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/2)*sec(d*x + c)^4, 
x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^4} \,d x \] Input:

int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2))/cos(c + d*x)^4,x)
 

Output:

int(((A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(3/2))/cos(c + d*x)^4, x)
 

Reduce [F]

\[ \int (a+b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^4(c+d x) \, dx=2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{4}d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{4}d x \right ) b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sec \left (d x +c \right )^{4}d x \right ) a^{2} \] Input:

int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c))*sec(d*x+c)^4,x)
 

Output:

2*int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)*sec(c + d*x)**4,x)*a*b + int(s 
qrt(cos(c + d*x)*b + a)*cos(c + d*x)**2*sec(c + d*x)**4,x)*b**2 + int(sqrt 
(cos(c + d*x)*b + a)*sec(c + d*x)**4,x)*a**2