\(\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx\) [536]

Optimal result

Integrand size = 23, antiderivative size = 124 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx=-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^4 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \] Output:

-2/9*b*(e*cos(d*x+c))^(9/2)/d/e+10/21*a*e^4*cos(d*x+c)^(1/2)*InverseJacobi 
AM(1/2*d*x+1/2*c,2^(1/2))/d/(e*cos(d*x+c))^(1/2)+10/21*a*e^3*(e*cos(d*x+c) 
)^(1/2)*sin(d*x+c)/d+2/7*a*e*(e*cos(d*x+c))^(5/2)*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 1.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.84 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx=\frac {e^3 \sqrt {e \cos (c+d x)} \left (120 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} (-21 b-28 b \cos (2 (c+d x))-7 b \cos (4 (c+d x))+138 a \sin (c+d x)+18 a \sin (3 (c+d x)))\right )}{252 d \sqrt {\cos (c+d x)}} \] Input:

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

Output:

(e^3*Sqrt[e*Cos[c + d*x]]*(120*a*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + 
d*x]]*(-21*b - 28*b*Cos[2*(c + d*x)] - 7*b*Cos[4*(c + d*x)] + 138*a*Sin[c 
+ d*x] + 18*a*Sin[3*(c + d*x)])))/(252*d*Sqrt[Cos[c + d*x]])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.05, 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, 3115, 3042, 3115, 3042, 3121, 3042, 3120}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3148

\(\displaystyle a \int (e \cos (c+d x))^{7/2}dx-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}dx-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3115

\(\displaystyle a \left (\frac {5}{7} e^2 \int (e \cos (c+d x))^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {5}{7} e^2 \int \left (e \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3115

\(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \cos (c+d x)}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {1}{3} e^2 \int \frac {1}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3121

\(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {e^2 \sqrt {\cos (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

\(\Big \downarrow \) 3120

\(\displaystyle a \left (\frac {5}{7} e^2 \left (\frac {2 e^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}\right )+\frac {2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}\right )-\frac {2 b (e \cos (c+d x))^{9/2}}{9 d e}\)

Input:

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

Output:

(-2*b*(e*Cos[c + d*x])^(9/2))/(9*d*e) + a*((2*e*(e*Cos[c + d*x])^(5/2)*Sin 
[c + d*x])/(7*d) + (5*e^2*((2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2 
, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x]) 
/(3*d)))/7)
 

Defintions of rubi rules used

rule 3042

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

rule 3115

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

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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(230\) vs. \(2(107)=214\).

Time = 2.62 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.86

method	result	size
parts	\(-\frac {2 a \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, e^{4} \left (48 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-120 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+128 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-72 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+16 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{21 \sqrt {-e \left (2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\, d}-\frac {2 b \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}{9 d e}\)	\(231\)
default	\(-\frac {2 e^{4} \left (-224 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{11} b +144 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+560 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-216 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-560 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+168 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+280 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-48 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a -70 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+7 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} e +e}\, d}\)	\(259\)

Input:

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

Output:

-2/21*a*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^4*(48* 
cos(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-72* 
cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c) 
^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*cos(1/2*d*x+1/2*c))/( 
-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c) 
/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d-2/9*b*(e*cos(d*x+c))^(9/2)/d/e
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93 \[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {\frac {1}{2}} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {\frac {1}{2}} a e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (7 \, b e^{3} \cos \left (d x + c\right )^{4} - 3 \, {\left (3 \, a e^{3} \cos \left (d x + c\right )^{2} + 5 \, a e^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}\right )}}{63 \, d} \] Input:

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

Output:

-2/63*(15*I*sqrt(1/2)*a*e^(7/2)*weierstrassPInverse(-4, 0, cos(d*x + c) + 
I*sin(d*x + c)) - 15*I*sqrt(1/2)*a*e^(7/2)*weierstrassPInverse(-4, 0, cos( 
d*x + c) - I*sin(d*x + c)) + (7*b*e^3*cos(d*x + c)^4 - 3*(3*a*e^3*cos(d*x 
+ c)^2 + 5*a*e^3)*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx=\frac {\sqrt {e}\, e^{3} \left (-2 \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4} b +9 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a d \right )}{9 d} \] Input:

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

Output:

(sqrt(e)*e**3*( - 2*sqrt(cos(c + d*x))*cos(c + d*x)**4*b + 9*int(sqrt(cos( 
c + d*x))*cos(c + d*x)**3,x)*a*d))/(9*d)