\(\int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx\) [40]

Optimal result

Integrand size = 23, antiderivative size = 131 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}-\frac {2 a \cos (c+d x)}{5 d e (e \sin (c+d x))^{5/2}}-\frac {6 a \cos (c+d x)}{5 d e^3 \sqrt {e \sin (c+d x)}}-\frac {6 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d e^4 \sqrt {\sin (c+d x)}} \] Output:

-2/5*b/d/e/(e*sin(d*x+c))^(5/2)-2/5*a*cos(d*x+c)/d/e/(e*sin(d*x+c))^(5/2)- 
6/5*a*cos(d*x+c)/d/e^3/(e*sin(d*x+c))^(1/2)+6/5*a*EllipticE(cos(1/2*c+1/4* 
Pi+1/2*d*x),2^(1/2))*(e*sin(d*x+c))^(1/2)/d/e^4/sin(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\frac {-4 b-7 a \cos (c+d x)+3 a \cos (3 (c+d x))+12 a E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sin ^{\frac {5}{2}}(c+d x)}{10 d e (e \sin (c+d x))^{5/2}} \] Input:

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

Output:

(-4*b - 7*a*Cos[c + d*x] + 3*a*Cos[3*(c + d*x)] + 12*a*EllipticE[(-2*c + P 
i - 2*d*x)/4, 2]*Sin[c + d*x]^(5/2))/(10*d*e*(e*Sin[c + d*x])^(5/2))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 3148, 3042, 3116, 3042, 3116, 3042, 3121, 3042, 3119}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3148

\(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{7/2}}dx-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int \frac {1}{(e \sin (c+d x))^{7/2}}dx-\frac {2 b}{5 d e (e \sin (c+d x))^{5/2}}\)

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3116

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3121

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

Input:

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

Output:

(-2*b)/(5*d*e*(e*Sin[c + d*x])^(5/2)) + a*((-2*Cos[c + d*x])/(5*d*e*(e*Sin 
[c + d*x])^(5/2)) + (3*((-2*Cos[c + d*x])/(d*e*Sqrt[e*Sin[c + d*x]]) - (2* 
EllipticE[(c - Pi/2 + d*x)/2, 2]*Sqrt[e*Sin[c + d*x]])/(d*e^2*Sqrt[Sin[c + 
 d*x]])))/(5*e^2))
 

Defintions of rubi rules used

rule 3042

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

rule 3116

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

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 3121

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

rule 3148

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + 
 Simp[a   Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && 
 (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
 

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.43

method	result	size
default	\(\frac {-\frac {2 b}{5 e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {a \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+6 \sin \left (d x +c \right )^{5}-4 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )\right )}{5 e^{3} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\)	\(187\)
parts	\(\frac {a \left (6 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticE}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2+2 \sin \left (d x +c \right )}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticF}\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )+6 \sin \left (d x +c \right )^{5}-4 \sin \left (d x +c \right )^{3}-2 \sin \left (d x +c \right )\right )}{5 e^{3} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {2 b}{5 d e \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}\)	\(189\)

Input:

int((a+cos(d*x+c)*b)/(e*sin(d*x+c))^(7/2),x,method=_RETURNVERBOSE)
 

Output:

(-2/5*b/e/(e*sin(d*x+c))^(5/2)+1/5*a/e^3*(6*(1-sin(d*x+c))^(1/2)*(2+2*sin( 
d*x+c))^(1/2)*sin(d*x+c)^(7/2)*EllipticE((1-sin(d*x+c))^(1/2),1/2*2^(1/2)) 
-3*(1-sin(d*x+c))^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(7/2)*EllipticF( 
(1-sin(d*x+c))^(1/2),1/2*2^(1/2))+6*sin(d*x+c)^5-4*sin(d*x+c)^3-2*sin(d*x+ 
c))/sin(d*x+c)^3/cos(d*x+c)/(e*sin(d*x+c))^(1/2))/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.27 \[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (i \, a \cos \left (d x + c\right )^{2} - i \, a\right )} \sqrt {-\frac {1}{2} i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, a \cos \left (d x + c\right )^{2} + i \, a\right )} \sqrt {\frac {1}{2} i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (3 \, a \cos \left (d x + c\right )^{3} - 4 \, a \cos \left (d x + c\right ) - b\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{5 \, {\left (d e^{4} \cos \left (d x + c\right )^{2} - d e^{4}\right )} \sin \left (d x + c\right )} \] Input:

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

Output:

-2/5*(3*(I*a*cos(d*x + c)^2 - I*a)*sqrt(-1/2*I*e)*sin(d*x + c)*weierstrass 
Zeta(4, 0, weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c))) + 3*( 
-I*a*cos(d*x + c)^2 + I*a)*sqrt(1/2*I*e)*sin(d*x + c)*weierstrassZeta(4, 0 
, weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c))) + (3*a*cos(d*x 
 + c)^3 - 4*a*cos(d*x + c) - b)*sqrt(e*sin(d*x + c)))/((d*e^4*cos(d*x + c) 
^2 - d*e^4)*sin(d*x + c))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \cos (c+d x)}{(e \sin (c+d x))^{7/2}} \, dx=\frac {\sqrt {e}\, \left (-2 \sqrt {\sin \left (d x +c \right )}\, b +5 \left (\int \frac {\sqrt {\sin \left (d x +c \right )}}{\sin \left (d x +c \right )^{4}}d x \right ) \sin \left (d x +c \right )^{3} a d \right )}{5 \sin \left (d x +c \right )^{3} d \,e^{4}} \] Input:

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

Output:

(sqrt(e)*( - 2*sqrt(sin(c + d*x))*b + 5*int(sqrt(sin(c + d*x))/sin(c + d*x 
)**4,x)*sin(c + d*x)**3*a*d))/(5*sin(c + d*x)**3*d*e**4)