\(\int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx\) [546]

Optimal result

Integrand size = 14, antiderivative size = 282 \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}} \] Output:

2/15*(23*a^2+9*b^2)*(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^2-b^2)^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-16/15* 
a*((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))/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^(1/2)-2/5*b*sin(d*x+c)/(a^2-b 
^2)/d/(a+b*cos(d*x+c))^(5/2)-16/15*a*b*sin(d*x+c)/(a^2-b^2)^2/d/(a+b*cos(d 
*x+c))^(3/2)-2/15*b*(23*a^2+9*b^2)*sin(d*x+c)/(a^2-b^2)^3/d/(a+b*cos(d*x+c 
))^(1/2)
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.67 \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+8 a (-a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (34 a^4-5 a^2 b^2+3 b^4+2 a b \left (27 a^2+5 b^2\right ) \cos (c+d x)+b^2 \left (23 a^2+9 b^2\right ) \cos ^2(c+d x)\right ) \sin (c+d x)}{\left (-a^2+b^2\right )^3}\right )}{15 d (a+b \cos (c+d x))^{5/2}} \] Input:

Integrate[(a + b*Cos[c + d*x])^(-7/2),x]
 

Output:

(2*((((a + b*Cos[c + d*x])/(a + b))^(5/2)*((23*a^2 + 9*b^2)*EllipticE[(c + 
 d*x)/2, (2*b)/(a + b)] + 8*a*(-a + b)*EllipticF[(c + d*x)/2, (2*b)/(a + b 
)]))/(a - b)^3 + (b*(34*a^4 - 5*a^2*b^2 + 3*b^4 + 2*a*b*(27*a^2 + 5*b^2)*C 
os[c + d*x] + b^2*(23*a^2 + 9*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/(-a^2 + b 
^2)^3))/(15*d*(a + b*Cos[c + d*x])^(5/2))
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3042, 3143, 27, 3042, 3233, 27, 3042, 3233, 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.

\(\displaystyle \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\)

\(\Big \downarrow \) 3143

\(\displaystyle -\frac {2 \int -\frac {5 a-3 b \cos (c+d x)}{2 (a+b \cos (c+d x))^{5/2}}dx}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {5 a-3 b \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}}dx}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {2 \int -\frac {3 \left (5 a^2+3 b^2\right )-8 a b \cos (c+d x)}{2 (a+b \cos (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 b^2\right )-8 a b \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {3 \left (5 a^2+3 b^2\right )-8 a b \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {-\frac {2 \int -\frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\int \frac {a \left (15 a^2+17 b^2\right )+b \left (23 a^2+9 b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cos (c+d x)}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {\left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-8 a \left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {8 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {\frac {\frac {\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}}{a^2-b^2}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}}{3 \left (a^2-b^2\right )}-\frac {16 a b \sin (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}}{5 \left (a^2-b^2\right )}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}\)

Input:

Int[(a + b*Cos[c + d*x])^(-7/2),x]
 

Output:

(-2*b*Sin[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(5/2)) + ((-16*a 
*b*Sin[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(3/2)) + (((2*(23*a 
^2 + 9*b^2)*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)]) - (16*a*(a^2 - b^2)*Sqrt[(a + b*C 
os[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b 
*Cos[c + d*x]]))/(a^2 - b^2) - (2*b*(23*a^2 + 9*b^2)*Sin[c + d*x])/((a^2 - 
 b^2)*d*Sqrt[a + b*Cos[c + d*x]]))/(3*(a^2 - b^2)))/(5*(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 3143

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

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 3233

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(615\) vs. \(2(267)=534\).

Time = 3.06 (sec) , antiderivative size = 616, normalized size of antiderivative = 2.18

method	result	size
default	\(-\frac {\sqrt {-\left (-2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{3}}+\frac {8 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}{15 b \left (a -b \right )^{2} \left (a +b \right )^{2} \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\frac {a -b}{2 b}\right )^{2}}+\frac {4 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (23 a^{2}+9 b^{2}\right )}{15 \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {-\left (-2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{\left (15 a^{5}+15 a^{4} b -30 a^{3} b^{2}-30 a^{2} b^{3}+15 a \,b^{4}+15 b^{5}\right ) \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}-\frac {2 \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 b \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a -b}{a -b}}\, \left (\operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-\operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )\right )}{15 \left (a -b \right )^{2} \left (a +b \right )^{3} \sqrt {-2 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (a +b \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}\, d}\)	\(616\)

Input:

int(1/(a+cos(d*x+c)*b)^(7/2),x,method=_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)*(1/10/b^2/( 
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/b*(a-b))^3+8/15*a/b/(a-b)^2/(a+ 
b)^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)/(cos(1/2*d*x+1/2*c)^2+1/2/b*(a-b))^2+4/15*sin(1/2*d*x+1/2*c)^2* 
b/(a-b)^3/(a+b)^3*cos(1/2*d*x+1/2*c)*(23*a^2+9*b^2)/(-(-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*(15*a^2-8*a*b+9*b^2)/(15*a^5+15* 
a^4*b-30*a^3*b^2-30*a^2*b^3+15*a*b^4+15*b^5)*(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))-2/15*(23*a^2+9*b^2)/(a-b)^2/(a+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)*(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))))/sin(1/2*d*x+1/2*c 
)/(-2*sin(1/2*d*x+1/2*c)^2*b+a+b)^(1/2)/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 941, normalized size of antiderivative = 3.34 \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*cos(d*x+c))^(7/2),x, algorithm="fricas")
 

Output:

2/45*(sqrt(1/2)*(I*a^6 - 33*I*a^4*b^2 + (I*a^3*b^3 - 33*I*a*b^5)*cos(d*x + 
 c)^3 + 3*(I*a^4*b^2 - 33*I*a^2*b^4)*cos(d*x + c)^2 + 3*(I*a^5*b - 33*I*a^ 
3*b^3)*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)*(-I*a^6 + 33*I*a^4*b^2 + (-I*a^3*b^3 + 33*I*a*b^5)*cos 
(d*x + c)^3 + 3*(-I*a^4*b^2 + 33*I*a^2*b^4)*cos(d*x + c)^2 + 3*(-I*a^5*b + 
 33*I*a^3*b^3)*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) + 3*sqrt(1/2)*(23*I*a^5*b + 9*I*a^3*b^3 + (23*I*a^2*b^4 + 
9*I*b^6)*cos(d*x + c)^3 + 3*(23*I*a^3*b^3 + 9*I*a*b^5)*cos(d*x + c)^2 + 3* 
(23*I*a^4*b^2 + 9*I*a^2*b^4)*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)*(-23*I*a^5*b - 9*I*a^3*b^3 + (-23 
*I*a^2*b^4 - 9*I*b^6)*cos(d*x + c)^3 + 3*(-23*I*a^3*b^3 - 9*I*a*b^5)*cos(d 
*x + c)^2 + 3*(-23*I*a^4*b^2 - 9*I*a^2*b^4)*cos(d*x + c))*sqrt(b)*weierstr 
assZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassP 
Inverse(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*(34*a^4*b^2 - 5*a^2*b^4 + 3* 
b^6 + (23*a^2*b^4 + 9*b^6)*cos(d*x + c)^2 + 2*(27*a^3*b^3 + 5*a*b^5)*co...
 

Sympy [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int \frac {1}{\left (a + b \cos {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(a+b*cos(d*x+c))**(7/2),x)
 

Output:

Integral((a + b*cos(c + d*x))**(-7/2), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*cos(d*x+c))^(7/2),x, algorithm="maxima")
 

Output:

integrate((b*cos(d*x + c) + a)^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int { \frac {1}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(a+b*cos(d*x+c))^(7/2),x, algorithm="giac")
 

Output:

integrate((b*cos(d*x + c) + a)^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \] Input:

int(1/(a + b*cos(c + d*x))^(7/2),x)
 

Output:

int(1/(a + b*cos(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx=\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}}{\cos \left (d x +c \right )^{4} b^{4}+4 \cos \left (d x +c \right )^{3} a \,b^{3}+6 \cos \left (d x +c \right )^{2} a^{2} b^{2}+4 \cos \left (d x +c \right ) a^{3} b +a^{4}}d x \] Input:

int(1/(a+b*cos(d*x+c))^(7/2),x)
 

Output:

int(sqrt(cos(c + d*x)*b + a)/(cos(c + d*x)**4*b**4 + 4*cos(c + d*x)**3*a*b 
**3 + 6*cos(c + d*x)**2*a**2*b**2 + 4*cos(c + d*x)*a**3*b + a**4),x)