\(\int (a+a \sin (e+f x)) \sqrt [3]{c+d \sin (e+f x)} \, dx\) [675]

Optimal result

Integrand size = 25, antiderivative size = 107 \[ \int (a+a \sin (e+f x)) \sqrt [3]{c+d \sin (e+f x)} \, dx=-\frac {2 \sqrt {2} a \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)}}{f \sqrt {1+\sin (e+f x)} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}} \] Output:

-2*2^(1/2)*a*AppellF1(1/2,-1/3,-1/2,3/2,d*(1-sin(f*x+e))/(c+d),1/2-1/2*sin 
(f*x+e))*cos(f*x+e)*(c+d*sin(f*x+e))^(1/3)/f/(1+sin(f*x+e))^(1/2)/((c+d*si 
n(f*x+e))/(c+d))^(1/3)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1736\) vs. \(2(107)=214\).

Time = 6.61 (sec) , antiderivative size = 1736, normalized size of antiderivative = 16.22 \[ \int (a+a \sin (e+f x)) \sqrt [3]{c+d \sin (e+f x)} \, dx =\text {Too large to display} \] Input:

Integrate[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/3),x]
 

Output:

a*((c*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-2/3, -1/2, -1/2, 1/3, -((Csc 
[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 
 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*C 
os[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2] 
*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[Cot[e 
]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[C 
ot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqr 
t[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[ 
1 + Cot[e]^2] + c*Csc[e])]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e 
]^2]*Sin[e])^(2/3))) - ((3*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[ 
1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) - (Cot[e]*Sin[f*x - A 
rcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[ 
1 + Cot[e]^2]*Sin[e])^(2/3)))/(4*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2 
])^2) + (d*Sec[e]*(1 + Sin[e + f*x])*(-((AppellF1[-2/3, -1/2, -1/2, 1/3, - 
((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*S 
qrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c 
+ d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[ 
e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f*x - ArcTan[ 
Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - Arc 
Tan[Cot[e]]]*Sqrt[1 + Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqr...
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3234, 156, 155}

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.

\(\displaystyle \int (a \sin (e+f x)+a) \sqrt [3]{c+d \sin (e+f x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sin (e+f x)+a) \sqrt [3]{c+d \sin (e+f x)}dx\)

\(\Big \downarrow \) 3234

\(\displaystyle \frac {a \cos (e+f x) \int \frac {\sqrt {\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1}}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {a \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \int \frac {\sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}}{\sqrt {1-\sin (e+f x)}}d\sin (e+f x)}{f \sqrt {1-\sin (e+f x)} \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}\)

\(\Big \downarrow \) 155

\(\displaystyle -\frac {2 \sqrt {2} a \cos (e+f x) \sqrt [3]{c+d \sin (e+f x)} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-\frac {1}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{f \sqrt {\sin (e+f x)+1} \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}\)

Input:

Int[(a + a*Sin[e + f*x])*(c + d*Sin[e + f*x])^(1/3),x]
 

Output:

(-2*Sqrt[2]*a*AppellF1[1/2, -1/2, -1/3, 3/2, (1 - Sin[e + f*x])/2, (d*(1 - 
 Sin[e + f*x]))/(c + d)]*Cos[e + f*x]*(c + d*Sin[e + f*x])^(1/3))/(f*Sqrt[ 
1 + Sin[e + f*x]]*((c + d*Sin[e + f*x])/(c + d))^(1/3))
 

Defintions of rubi rules used

rule 155

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*Simplify[b/(b*c - a*d)]^n* 
Simplify[b/(b*e - a*f)]^p))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/ 
(b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, 
 m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] && GtQ[Sim 
plify[b/(b*c - a*d)], 0] && GtQ[Simplify[b/(b*e - a*f)], 0] &&  !(GtQ[Simpl 
ify[d/(d*a - c*b)], 0] && GtQ[Simplify[d/(d*e - c*f)], 0] && SimplerQ[c + d 
*x, a + b*x]) &&  !(GtQ[Simplify[f/(f*a - e*b)], 0] && GtQ[Simplify[f/(f*c 
- e*d)], 0] && SimplerQ[e + f*x, a + b*x])
 

rule 156

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_)) 
^(p_), x_] :> Simp[(e + f*x)^FracPart[p]/(Simplify[b/(b*e - a*f)]^IntPart[p 
]*(b*((e + f*x)/(b*e - a*f)))^FracPart[p])   Int[(a + b*x)^m*(c + d*x)^n*Si 
mp[b*(e/(b*e - a*f)) + b*f*(x/(b*e - a*f)), x]^p, x], x] /; FreeQ[{a, b, c, 
 d, e, f, m, n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !IntegerQ[p] & 
& GtQ[Simplify[b/(b*c - a*d)], 0] &&  !GtQ[Simplify[b/(b*e - a*f)], 0]
 

rule 3042

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

rule 3234

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

Maple [F]

\[\int \left (a +a \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{\frac {1}{3}}d x\]

Input:

int((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x)
 

Output:

int((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x)
 

Fricas [F]

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

integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x, algorithm="fricas")
 

Output:

integral((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(1/3), x)
 

Sympy [F]

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

integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))**(1/3),x)
 

Output:

a*(Integral((c + d*sin(e + f*x))**(1/3)*sin(e + f*x), x) + Integral((c + d 
*sin(e + f*x))**(1/3), x))
 

Maxima [F]

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

integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x, algorithm="maxima")
 

Output:

integrate((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(1/3), x)
 

Giac [F]

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

integrate((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x, algorithm="giac")
 

Output:

integrate((a*sin(f*x + e) + a)*(d*sin(f*x + e) + c)^(1/3), x)
 

Mupad [F(-1)]

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

int((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(1/3),x)
 

Output:

int((a + a*sin(e + f*x))*(c + d*sin(e + f*x))^(1/3), x)
 

Reduce [F]

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

int((a+a*sin(f*x+e))*(c+d*sin(f*x+e))^(1/3),x)
 

Output:

a*(int((sin(e + f*x)*d + c)**(1/3),x) + int((sin(e + f*x)*d + c)**(1/3)*si 
n(e + f*x),x))