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

Optimal. Leaf size=112 \[ -\frac{2 \sqrt{2} a \cos (e+f x) \sqrt [3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left (\frac{1}{2};-\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f (c+d) \sqrt{\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}} \]

[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))

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Rubi [A]  time = 0.0963662, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2755, 139, 138} \[ -\frac{2 \sqrt{2} a \cos (e+f x) \sqrt [3]{\frac{c+d \sin (e+f x)}{c+d}} F_1\left (\frac{1}{2};-\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sin (e+f x)),\frac{d (1-\sin (e+f x))}{c+d}\right )}{f (c+d) \sqrt{\sin (e+f x)+1} \sqrt [3]{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[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 2755

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*x)/c])/Sqrt
[1 - (d*x)/c], 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]

Rule 139

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 138

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[((a + b*x)
^(m + 1)*AppellF1[m + 1, -n, -p, m + 2, -((d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*(m + 1
)*(b/(b*c - a*d))^n*(b/(b*e - a*f))^p), 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])

Rubi steps

\begin{align*} \int \frac{a+a \sin (e+f x)}{(c+d \sin (e+f x))^{4/3}} \, dx &=\frac{(a \cos (e+f x)) \operatorname{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 ) \operatorname{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 F_1\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]  time = 6.40232, size = 942, normalized size = 8.41 \[ a \left (\frac{(c+d \sin (e+f x))^{2/3} \left (\frac{3 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}-\frac{3 \csc (e) \sec (e)}{d (c+d) f}\right ) (\sin (e+f x)+1)}{\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) \left (-\frac{F_1\left (-\frac{1}{3};-\frac{1}{2},-\frac{1}{2};\frac{2}{3};-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (1-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}\right )},-\frac{\csc (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{d \sqrt{\cot ^2(e)+1} \left (-\frac{c \csc (e)}{d \sqrt{\cot ^2(e)+1}}-1\right )}\right ) \cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1} \sqrt{\frac{\cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} d+\sqrt{\cot ^2(e)+1} d}{d \sqrt{\cot ^2(e)+1}-c \csc (e)}} \sqrt{\frac{d \sqrt{\cot ^2(e)+1}-d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1}}{\sqrt{\cot ^2(e)+1} d+c \csc (e)}} \sqrt [3]{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}-\frac{\frac{3 d \sin (e) \left (c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)\right )}{2 \left (d^2 \cos ^2(e)+d^2 \sin ^2(e)\right )}-\frac{\cot (e) \sin \left (f x-\tan ^{-1}(\cot (e))\right )}{\sqrt{\cot ^2(e)+1}}}{\sqrt [3]{c+d \cos \left (f x-\tan ^{-1}(\cot (e))\right ) \sqrt{\cot ^2(e)+1} \sin (e)}}\right ) (\sin (e+f x)+1)}{(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 F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (1-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}\right )},-\frac{\sec (e) \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )}{d \sqrt{\tan ^2(e)+1} \left (-\frac{c \sec (e)}{d \sqrt{\tan ^2(e)+1}}-1\right )}\right ) \sec (e) \sec \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\frac{d \sqrt{\tan ^2(e)+1}-d \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}}{\sqrt{\tan ^2(e)+1} d+c \sec (e)}} \sqrt{\frac{\sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1} d+\sqrt{\tan ^2(e)+1} d}{d \sqrt{\tan ^2(e)+1}-c \sec (e)}} \left (c+d \cos (e) \sin \left (f x+\tan ^{-1}(\tan (e))\right ) \sqrt{\tan ^2(e)+1}\right )^{2/3} (\sin (e+f x)+1)}{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{\tan ^2(e)+1}}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*(((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]))

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Maple [F]  time = 0.151, size = 0, normalized size = 0. \begin{align*} \int{(a+a\sin \left ( fx+e \right ) ) \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a \sin \left (f x + e\right ) + a\right )}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{2}{3}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)