\(\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx\) [281]

Optimal result

Integrand size = 27, antiderivative size = 278 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}-\frac {7 a e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {7 a e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (1+\cos (c+d x)+\sin (c+d x))} \] Output:

-7/12*a^2*(e*cos(d*x+c))^(5/2)/d/e/(a+a*sin(d*x+c))^(1/2)+7/8*a*e*(e*cos(d 
*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d-1/3*a*(e*cos(d*x+c))^(5/2)*(a+a*sin( 
d*x+c))^(1/2)/d/e-7/8*a*e^(3/2)*arcsinh((e*cos(d*x+c))^(1/2)/e^(1/2))*(1+c 
os(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(1+cos(d*x+c)+sin(d*x+c))+7/8*a* 
e^(3/2)*arctan(e^(1/2)*sin(d*x+c)/(e*cos(d*x+c))^(1/2)/(1+cos(d*x+c))^(1/2 
))*(1+cos(d*x+c))^(1/2)*(a+a*sin(d*x+c))^(1/2)/d/(1+cos(d*x+c)+sin(d*x+c))
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.28 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {8\ 2^{3/4} a (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{5 d e (1+\sin (c+d x))^{7/4}} \] Input:

Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2),x]
 

Output:

(-8*2^(3/4)*a*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-7/4, 5/4, 9/4, (1 
- Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(5*d*e*(1 + Sin[c + d*x])^( 
7/4))
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3157, 3042, 3157, 3042, 3164, 3042, 3156, 3042, 25, 3254, 216, 3312, 63, 222}

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.

\(\displaystyle \int (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}dx\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {7}{6} a \int (e \cos (c+d x))^{3/2} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} a \int (e \cos (c+d x))^{3/2} \sqrt {\sin (c+d x) a+a}dx-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3157

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {\sin (c+d x) a+a}}dx-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3164

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \int \frac {\sqrt {\sin (c+d x) a+a}}{\sqrt {e \cos (c+d x)}}dx}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3156

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\cos (c+d x)+1}}{\sqrt {e \cos (c+d x)}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}+\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )+1}}{\sqrt {e \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3254

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x)}{\cos (c+d x)+1}+1}d\left (-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sqrt {e \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )} \sqrt {\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )+1}}dx}{\sin (c+d x)+\cos (c+d x)+1}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 3312

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {\sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {\cos (c+d x)+1}}d\cos (c+d x)}{d (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 63

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \int \frac {1}{\sqrt {\cos (c+d x)+1}}d\sqrt {e \cos (c+d x)}}{d e (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {7}{6} a \left (\frac {3}{4} a \left (\frac {e^2 \left (\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac {2 \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{d \sqrt {e} (\sin (c+d x)+\cos (c+d x)+1)}\right )}{2 a}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d}\right )-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}\)

Input:

Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2),x]
 

Output:

-1/3*(a*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(d*e) + (7*a*(-1/ 
2*(a*(e*Cos[c + d*x])^(5/2))/(d*e*Sqrt[a + a*Sin[c + d*x]]) + (3*a*((e*Sqr 
t[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) + (e^2*((-2*ArcSinh[Sqrt 
[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) 
/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (2*ArcTan[(Sqrt[e]*Sin[c 
+ d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d* 
x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]) 
)))/(2*a)))/4))/6
 

Defintions of rubi rules used

rule 25

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

rule 63

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2/b   S 
ubst[Int[1/Sqrt[c + d*(x^2/b)], x], x, Sqrt[b*x]], x] /; FreeQ[{b, c, d}, x 
] && GtQ[c, 0]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 3042

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

rule 3156

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[cos[(e_.) + (f_.)*(x_)] 
*(g_.)], x_Symbol] :> Simp[a*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x 
]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x]))   Int[Sqrt[1 + Cos[e + f*x]]/Sqrt 
[g*Cos[e + f*x]], x], x] + Simp[b*Sqrt[1 + Cos[e + f*x]]*(Sqrt[a + b*Sin[e 
+ f*x]]/(a + a*Cos[e + f*x] + b*Sin[e + f*x]))   Int[Sin[e + f*x]/(Sqrt[g*C 
os[e + f*x]]*Sqrt[1 + Cos[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, g}, x] & 
& EqQ[a^2 - b^2, 0]
 

rule 3157

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

rule 3164

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(3/2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ 
.)*(x_)]], x_Symbol] :> Simp[g*Sqrt[g*Cos[e + f*x]]*(Sqrt[a + b*Sin[e + f*x 
]]/(b*f)), x] + Simp[g^2/(2*a)   Int[Sqrt[a + b*Sin[e + f*x]]/Sqrt[g*Cos[e 
+ f*x]], x], x] /; FreeQ[{a, b, e, f, g}, x] && EqQ[a^2 - b^2, 0]
 

rule 3254

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

rule 3312

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( 
c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f)   Su 
bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 10.96 (sec) , antiderivative size = 796, normalized size of antiderivative = 2.86

method	result	size
default	\(\text {Expression too large to display}\)	\(796\)

Input:

int((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

-1/48/d/(2*2^(1/2)+3)^(1/2)/(1+2^(1/2))*a/e*(84*(-2+(-cos(1/2*d*x+1/2*c)-1 
)*2^(1/2)-2*cos(1/2*d*x+1/2*c))*(-2*(2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)-2* 
cos(1/2*d*x+1/2*c)+1)/(cos(1/2*d*x+1/2*c)+1))^(1/2)*((2^(1/2)*cos(1/2*d*x+ 
1/2*c)-2^(1/2)+2*cos(1/2*d*x+1/2*c)-1)/(cos(1/2*d*x+1/2*c)+1))^(1/2)*e^2*E 
llipticPi((2*2^(1/2)+3)^(1/2)*(csc(1/2*d*x+1/2*c)-cot(1/2*d*x+1/2*c)),-1/( 
2*2^(1/2)+3),(-2*2^(1/2)+3)^(1/2)/(2*2^(1/2)+3)^(1/2))+21*(4+3*(cos(1/2*d* 
x+1/2*c)+1)*2^(1/2)+4*cos(1/2*d*x+1/2*c))*(e*(2*cos(1/2*d*x+1/2*c)^2-1)/(c 
os(1/2*d*x+1/2*c)+1)^2)^(1/2)*e^(3/2)*arctanh(2^(1/2)*e^(1/2)*cos(1/2*d*x+ 
1/2*c)/(cos(1/2*d*x+1/2*c)+1)/(e*(2*cos(1/2*d*x+1/2*c)^2-1)/(cos(1/2*d*x+1 
/2*c)+1)^2)^(1/2))+42*(2+(cos(1/2*d*x+1/2*c)+1)*2^(1/2)+2*cos(1/2*d*x+1/2* 
c))*EllipticF((1+2^(1/2))*(csc(1/2*d*x+1/2*c)-cot(1/2*d*x+1/2*c)),-2*2^(1/ 
2)+3)*((2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)+2*cos(1/2*d*x+1/2*c)-1)/(cos(1/ 
2*d*x+1/2*c)+1))^(1/2)*e^2*(-2*(2^(1/2)*cos(1/2*d*x+1/2*c)-2^(1/2)-2*cos(1 
/2*d*x+1/2*c)+1)/(cos(1/2*d*x+1/2*c)+1))^(1/2)+2*(2*cos(1/2*d*x+1/2*c)^2-1 
)*(2*((32*cos(1/2*d*x+1/2*c)^4-60*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2* 
c)+cos(1/2*d*x+1/2*c)*(32*cos(1/2*d*x+1/2*c)^4-4*cos(1/2*d*x+1/2*c)^2-27)) 
*2^(1/2)+3*(32*cos(1/2*d*x+1/2*c)^4-60*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x 
+1/2*c)+3*cos(1/2*d*x+1/2*c)*(32*cos(1/2*d*x+1/2*c)^4-4*cos(1/2*d*x+1/2*c) 
^2-27))*e^2)*(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)*((1+2*cos(1/2*d*x+1/2*c) 
*sin(1/2*d*x+1/2*c))*a)^(1/2)/(2*cos(1/2*d*x+1/2*c)^3+2*sin(1/2*d*x+1/2...
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.36 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\frac {84 \, \sqrt {2} \sqrt {a e} a e \arctan \left (\frac {\sqrt {2} \sqrt {a e} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sin \left (d x + c\right )}{a e \cos \left (d x + c\right )^{2} + a e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )}\right ) - 21 \, \sqrt {2} \sqrt {a e} a e \log \left (\frac {2 \, a e \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a e} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 1\right )} + 3 \, a e \cos \left (d x + c\right ) + a e + {\left (2 \, a e \cos \left (d x + c\right ) + a e\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1}\right ) + 21 \, \sqrt {2} \sqrt {a e} a e \log \left (\frac {2 \, a e \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a e} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) + 1\right )} + 3 \, a e \cos \left (d x + c\right ) + a e + {\left (2 \, a e \cos \left (d x + c\right ) + a e\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1}\right ) - 8 \, {\left (8 \, a e \cos \left (d x + c\right )^{2} - 14 \, a e \sin \left (d x + c\right ) - 7 \, a e\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{192 \, d} \] Input:

integrate((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^(3/2),x, algorithm="fricas 
")
 

Output:

1/192*(84*sqrt(2)*sqrt(a*e)*a*e*arctan(sqrt(2)*sqrt(a*e)*sqrt(e*cos(d*x + 
c))*sqrt(a*sin(d*x + c) + a)*sin(d*x + c)/(a*e*cos(d*x + c)^2 + a*e*cos(d* 
x + c)*sin(d*x + c) + a*e*cos(d*x + c))) - 21*sqrt(2)*sqrt(a*e)*a*e*log((2 
*a*e*cos(d*x + c)^2 + 2*sqrt(2)*sqrt(a*e)*sqrt(e*cos(d*x + c))*sqrt(a*sin( 
d*x + c) + a)*(cos(d*x + c) + 1) + 3*a*e*cos(d*x + c) + a*e + (2*a*e*cos(d 
*x + c) + a*e)*sin(d*x + c))/(cos(d*x + c) + sin(d*x + c) + 1)) + 21*sqrt( 
2)*sqrt(a*e)*a*e*log((2*a*e*cos(d*x + c)^2 - 2*sqrt(2)*sqrt(a*e)*sqrt(e*co 
s(d*x + c))*sqrt(a*sin(d*x + c) + a)*(cos(d*x + c) + 1) + 3*a*e*cos(d*x + 
c) + a*e + (2*a*e*cos(d*x + c) + a*e)*sin(d*x + c))/(cos(d*x + c) + sin(d* 
x + c) + 1)) - 8*(8*a*e*cos(d*x + c)^2 - 14*a*e*sin(d*x + c) - 7*a*e)*sqrt 
(e*cos(d*x + c))*sqrt(a*sin(d*x + c) + a))/d
 

Sympy [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate((e*cos(d*x+c))**(3/2)*(a+a*sin(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^(3/2),x, algorithm="maxima 
")
 

Output:

integrate((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^(3/2), x)
 

Giac [F]

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

integrate((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

integrate((e*cos(d*x + c))^(3/2)*(a*sin(d*x + c) + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:

int((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2),x)
 

Output:

int((e*cos(c + d*x))^(3/2)*(a + a*sin(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\sqrt {e}\, \sqrt {a}\, a e \left (\int \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )d x +\int \sqrt {\sin \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) \] Input:

int((e*cos(d*x+c))^(3/2)*(a+a*sin(d*x+c))^(3/2),x)
 

Output:

sqrt(e)*sqrt(a)*a*e*(int(sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + 
 d*x)*sin(c + d*x),x) + int(sqrt(sin(c + d*x) + 1)*sqrt(cos(c + d*x))*cos( 
c + d*x),x))