\(\int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx\) [338]

Optimal result

Integrand size = 22, antiderivative size = 283 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx=-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}+\frac {8 a E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}{3 e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}} \] Output:

-2/3*(c*cos(e*x+d)-b*sin(e*x+d))*(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)/e+8/3 
*a*EllipticE(sin(1/2*d+1/2*e*x-1/2*arctan(c,b)),2^(1/2)*((b^2+c^2)^(1/2)/( 
a+(b^2+c^2)^(1/2)))^(1/2))*(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)/e/((a+b*cos 
(e*x+d)+c*sin(e*x+d))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/3*(a^2-b^2-c^2)*Inverse 
JacobiAM(1/2*d+1/2*e*x-1/2*arctan(c,b),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^ 
2)^(1/2)))^(1/2))*((a+b*cos(e*x+d)+c*sin(e*x+d))/(a+(b^2+c^2)^(1/2)))^(1/2 
)/e/(a+b*cos(e*x+d)+c*sin(e*x+d))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.31 (sec) , antiderivative size = 2190, normalized size of antiderivative = 7.74 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(((8*a*b)/(3*c) - (2*c*Cos[d + e*x])/3 + (2*b*Sin[d + e*x])/3)*Sqrt[a + b* 
Cos[d + e*x] + c*Sin[d + e*x]])/e + (2*a^2*AppellF1[1/2, 1/2, 1/2, 3/2, -( 
(a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(1 
 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + 
ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c))]*Sec[d 
 + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 + c^2)/c 
^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sqrt[a + c* 
Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b^2 + c^2) 
/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + c*Sqrt[( 
b^2 + c^2)/c^2])])/(Sqrt[1 + b^2/c^2]*c*e) + (2*b^2*AppellF1[1/2, 1/2, 1/2 
, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sqrt[1 + b^ 
2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/c^2]*c*Sin[d 
 + e*x + ArcTan[b/c]])/(Sqrt[1 + b^2/c^2]*(-1 - a/(Sqrt[1 + b^2/c^2]*c))*c 
))]*Sec[d + e*x + ArcTan[b/c]]*Sqrt[(c*Sqrt[(b^2 + c^2)/c^2] - c*Sqrt[(b^2 
 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(a + c*Sqrt[(b^2 + c^2)/c^2])]*Sq 
rt[a + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]]]*Sqrt[(c*Sqrt[(b 
^2 + c^2)/c^2] + c*Sqrt[(b^2 + c^2)/c^2]*Sin[d + e*x + ArcTan[b/c]])/(-a + 
 c*Sqrt[(b^2 + c^2)/c^2])])/(3*Sqrt[1 + b^2/c^2]*c*e) + (2*c*AppellF1[1/2, 
 1/2, 1/2, 3/2, -((a + Sqrt[1 + b^2/c^2]*c*Sin[d + e*x + ArcTan[b/c]])/(Sq 
rt[1 + b^2/c^2]*(1 - a/(Sqrt[1 + b^2/c^2]*c))*c)), -((a + Sqrt[1 + b^2/...
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3599, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3599

\(\displaystyle \frac {2}{3} \int \frac {3 a^2+4 b \cos (d+e x) a+4 c \sin (d+e x) a+b^2+c^2}{2 \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {3 a^2+4 b \cos (d+e x) a+4 c \sin (d+e x) a+b^2+c^2}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \int \frac {3 a^2+4 b \cos (d+e x) a+4 c \sin (d+e x) a+b^2+c^2}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 3628

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3598

\(\displaystyle \frac {1}{3} \left (\frac {4 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{3} \left (\frac {8 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\left (a^2-b^2-c^2\right ) \int \frac {1}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}dx\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 3606

\(\displaystyle \frac {1}{3} \left (\frac {8 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {\left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (d+e x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{3} \left (\frac {8 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {\left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (d+e x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{3} \left (\frac {8 a \sqrt {a+b \cos (d+e x)+c \sin (d+e x)} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) \sqrt {\frac {a+b \cos (d+e x)+c \sin (d+e x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{e \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}\right )-\frac {2 (c \cos (d+e x)-b \sin (d+e x)) \sqrt {a+b \cos (d+e x)+c \sin (d+e x)}}{3 e}\)

Input:

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

Output:

(-2*(c*Cos[d + e*x] - b*Sin[d + e*x])*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + 
e*x]])/(3*e) + ((8*a*EllipticE[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c 
^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e* 
Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]) - (2*(a 
^2 - b^2 - c^2)*EllipticF[(d + e*x - ArcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/ 
(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqr 
t[b^2 + c^2])])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]))/3
 

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 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 3598

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ 
)]], x_Symbol] :> Simp[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + 
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]   Int[Sqrt[a/(a + S 
qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc 
Tan[b, c]]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] 
 && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]
 

rule 3599

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

rule 3606

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

rule 3628

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

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1489\) vs. \(2(264)=528\).

Time = 0.73 (sec) , antiderivative size = 1490, normalized size of antiderivative = 5.27

method	result	size
default	\(\text {Expression too large to display}\)	\(1490\)

Input:

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

Output:

(-(-b^2*sin(e*x+d-arctan(-b,c))-c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+c^2)^(1 
/2))*cos(e*x+d-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)*(2*a^2*(1/(b^2+c^2)^ 
(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2) 
))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)) 
)^(1/2)*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c))-1)/(a+(b^2+c^2)^(1/2))) 
^(1/2)/(-(-b^2*sin(e*x+d-arctan(-b,c))-c^2*sin(e*x+d-arctan(-b,c))-a*(b^2+ 
c^2)^(1/2))*cos(e*x+d-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)*EllipticF(((( 
b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a- 
(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+(b^2+c^2)*(-2/3/(b^2+c^2)^(1 
/2)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)*cos(e*x+d-arctan(-b,c))^2 
)^(1/2)+2/3*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b, 
c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(e*x+d-arctan(-b,c))+1)*(b^2+c^2)^( 
1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*(-(b^2+c^2)^(1/2)*(sin(e*x+d-arctan(-b,c) 
)-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+ 
a)*cos(e*x+d-arctan(-b,c))^2)^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(e*x+d- 
arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2 
+c^2)^(1/2)))^(1/2))-4/3/(b^2+c^2)^(1/2)*a*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+ 
c^2)^(1/2)*sin(e*x+d-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(e*x 
+d-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*(-(b^2+c^2 
)^(1/2)*(sin(e*x+d-arctan(-b,c))-1)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 1481, normalized size of antiderivative = 5.23 \[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/9*((-I*a^2*b - 3*I*b^3 - 3*I*b*c^2 - 3*c^3 - (a^2 + 3*b^2)*c)*sqrt(1/2* 
b + 1/2*I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I* 
b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8 
*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 
6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 
+ 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3* 
(I*b^2 + I*c^2)*sin(e*x + d))/(b^2 + c^2)) + (I*a^2*b + 3*I*b^3 + 3*I*b*c^ 
2 - 3*c^3 - (a^2 + 3*b^2)*c)*sqrt(1/2*b - 1/2*I*c)*weierstrassPInverse(4/3 
*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3 
)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9* 
I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a 
^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I 
*a*c + 3*(b^2 + c^2)*cos(e*x + d) - 3*(-I*b^2 - I*c^2)*sin(e*x + d))/(b^2 
+ c^2)) + 12*(I*a*b^2 + I*a*c^2)*sqrt(1/2*b + 1/2*I*c)*weierstrassZeta(4/3 
*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3 
)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9* 
I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a 
^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrassPInve 
rse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b 
- 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (a+b \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)*cos(d + e*x)*b**2*c**2 + 6 
*sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)*sin(d + e*x)*b**3*c + 4*sqrt(co 
s(d + e*x)*b + sin(d + e*x)*c + a)*sin(d + e*x)*b*c**3 + 12*sqrt(cos(d + e 
*x)*b + sin(d + e*x)*c + a)*a*b**3 + 4*sqrt(cos(d + e*x)*b + sin(d + e*x)* 
c + a)*a*b*c**2 + 12*int(sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)/(2*cos( 
d + e*x)*b**3 + cos(d + e*x)*b*c**2 + 2*sin(d + e*x)*b**2*c + sin(d + e*x) 
*c**3 + 2*a*b**2 + a*c**2),x)*a**2*b**4*c*e + 12*int(sqrt(cos(d + e*x)*b + 
 sin(d + e*x)*c + a)/(2*cos(d + e*x)*b**3 + cos(d + e*x)*b*c**2 + 2*sin(d 
+ e*x)*b**2*c + sin(d + e*x)*c**3 + 2*a*b**2 + a*c**2),x)*a**2*b**2*c**3*e 
 + 3*int(sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)/(2*cos(d + e*x)*b**3 + 
cos(d + e*x)*b*c**2 + 2*sin(d + e*x)*b**2*c + sin(d + e*x)*c**3 + 2*a*b**2 
 + a*c**2),x)*a**2*c**5*e + 6*int((sqrt(cos(d + e*x)*b + sin(d + e*x)*c + 
a)*sin(d + e*x)**2)/(2*cos(d + e*x)*b**3 + cos(d + e*x)*b*c**2 + 2*sin(d + 
 e*x)*b**2*c + sin(d + e*x)*c**3 + 2*a*b**2 + a*c**2),x)*b**6*c*e + 15*int 
((sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)*sin(d + e*x)**2)/(2*cos(d + e* 
x)*b**3 + cos(d + e*x)*b*c**2 + 2*sin(d + e*x)*b**2*c + sin(d + e*x)*c**3 
+ 2*a*b**2 + a*c**2),x)*b**4*c**3*e + 12*int((sqrt(cos(d + e*x)*b + sin(d 
+ e*x)*c + a)*sin(d + e*x)**2)/(2*cos(d + e*x)*b**3 + cos(d + e*x)*b*c**2 
+ 2*sin(d + e*x)*b**2*c + sin(d + e*x)*c**3 + 2*a*b**2 + a*c**2),x)*b**2*c 
**5*e + 3*int((sqrt(cos(d + e*x)*b + sin(d + e*x)*c + a)*sin(d + e*x)**...