\(\int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx\) [670]

Optimal result

Integrand size = 25, antiderivative size = 111 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=-\frac {6 \sqrt {2} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right ) \cos (e+f x) \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}}}{(c+d) f \sqrt {1+\sin (e+f x)} \sqrt [3]{c+d \sin (e+f x)}} \]

[Out]

-2*a*AppellF1(1/2,4/3,-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))/(c+d))^
(1/3)*2^(1/2)/(c+d)/f/(c+d*sin(f*x+e))^(1/3)/(1+sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2834, 144, 143} \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=-\frac {2 \sqrt {2} a \cos (e+f x) \sqrt [3]{\frac {c+d \sin (e+f x)}{c+d}} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},\frac {4}{3},\frac {3}{2},\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{f (c+d) \sqrt {\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}} \]

[In]

Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(4/3),x]

[Out]

(-2*Sqrt[2]*a*AppellF1[1/2, -1/2, 4/3, 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])/(c + d))^(1/3))/((c + d)*f*Sqrt[1 + Sin[e + f*x]]*(c + d*Sin[e + f*x])^(1/3))

Rule 143

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n*(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] &&  !Inte
gerQ[n] &&  !IntegerQ[p] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !(GtQ[d/(d*a - c*b), 0] && GtQ[
d/(d*e - c*f), 0] && SimplerQ[c + d*x, a + b*x]) &&  !(GtQ[f/(f*a - e*b), 0] && GtQ[f/(f*c - e*d), 0] && Simpl
erQ[e + f*x, a + b*x])

Rule 144

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

Rule 2834

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(C
os[e + f*x]/(f*Sqrt[1 + Sin[e + f*x]]*Sqrt[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] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b
^2, 0] &&  !IntegerQ[2*m] && EqQ[c^2 - d^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(a \cos (e+f x)) \text {Subst}\left (\int \frac {\sqrt {1+x}}{\sqrt {1-x} (c+d x)^{4/3}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}} \\ & = \frac {\left (a \cos (e+f x) \sqrt [3]{-\frac {c+d \sin (e+f x)}{-c-d}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{\sqrt {1-x} \left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^{4/3}} \, dx,x,\sin (e+f x)\right )}{(c+d) f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)} \sqrt [3]{c+d \sin (e+f x)}} \\ & = -\frac {2 \sqrt {2} a \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},\frac {4}{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]{\frac {c+d \sin (e+f x)}{c+d}}}{(c+d) f \sqrt {1+\sin (e+f x)} \sqrt [3]{c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(942\) vs. \(2(111)=222\).

Time = 6.68 (sec) , antiderivative size = 942, normalized size of antiderivative = 8.49 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=3 \left (\frac {(1+\sin (e+f x)) (c+d \sin (e+f x))^{2/3} \left (-\frac {3 \csc (e) \sec (e)}{d (c+d) f}+\frac {3 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}\right )}{\left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}-\frac {2 \sec (e) (1+\sin (e+f x)) \left (-\frac {\operatorname {AppellF1}\left (-\frac {1}{3},-\frac {1}{2},-\frac {1}{2},\frac {2}{3},-\frac {\csc (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin (f x-\arctan (\cot (e)))}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt [3]{c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {3 d \sin (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{2 \left (d^2 \cos ^2(e)+d^2 \sin ^2(e)\right )}-\frac {\cot (e) \sin (f x-\arctan (\cot (e)))}{\sqrt {1+\cot ^2(e)}}}{\sqrt [3]{c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{(c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {3 \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {\sec (e) \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec (f x+\arctan (\tan (e))) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )^{2/3}}{2 d (c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {1+\tan ^2(e)}}\right ) \]

[In]

Integrate[(3 + 3*Sin[e + f*x])/(c + d*Sin[e + f*x])^(4/3),x]

[Out]

3*(((1 + Sin[e + f*x])*(c + d*Sin[e + f*x])^(2/3)*((-3*Csc[e]*Sec[e])/(d*(c + d)*f) + (3*Csc[e]*(c*Cos[e] + d*
Sin[f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (2*Sec[e]*(1 + S
in[e + f*x])*(-((AppellF1[-1/3, -1/2, -1/2, 2/3, -((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*Cos[f*x - ArcTan[C
ot[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]*S
in[f*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[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])]*Sqrt[(d*Sqrt[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])^
(1/3))) - ((3*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(2*(d^2*Cos[e]^2 + d^2*Sin
[e]^2)) - (Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot
[e]^2]*Sin[e])^(1/3)))/((c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (3*AppellF1[2/3, 1/2, 1/2, 5/
3, -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e]
)/(d*Sqrt[1 + Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1
+ Tan[e]^2]*(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sq
rt[(d*Sqrt[1 + Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*
Sqrt[(d*Sqrt[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^
2])]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])^(2/3))/(2*d*(c + d)*f*(Cos[e/2 + (f*x)/2] + S
in[e/2 + (f*x)/2])^2*Sqrt[1 + Tan[e]^2]))

Maple [F]

\[\int \frac {a +a \sin \left (f x +e \right )}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {4}{3}}}d x\]

[In]

int((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(4/3),x)

[Out]

int((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(4/3),x)

Fricas [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(4/3),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(4/3),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(4/3),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(4/3), x)

Giac [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {4}{3}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(4/3),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(4/3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{4/3}} \,d x \]

[In]

int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(4/3),x)

[Out]

int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(4/3), x)