### 3.13 $$\int \frac{1}{(a+a \sin (c+d x))^{4/3}} \, dx$$

Optimal. Leaf size=69 $-\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}$

[Out]

-((Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 3/2, (1 - Sin[c + d*x])/2])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)
*(a + a*Sin[c + d*x])^(1/3)))

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Rubi [A]  time = 0.0336768, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {2652, 2651} $-\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{\sin (c+d x)+1} \sqrt [3]{a \sin (c+d x)+a}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + a*Sin[c + d*x])^(-4/3),x]

[Out]

-((Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 3/2, (1 - Sin[c + d*x])/2])/(2^(5/6)*a*d*(1 + Sin[c + d*x])^(1/6)
*(a + a*Sin[c + d*x])^(1/3)))

Rule 2652

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^IntPart[n]*(a + b*Sin[c + d*x])^FracPart
[n])/(1 + (b*Sin[c + d*x])/a)^FracPart[n], Int[(1 + (b*Sin[c + d*x])/a)^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 2651

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(2^(n + 1/2)*a^(n - 1/2)*b*Cos[c + d*x]*Hy
pergeometric2F1[1/2, 1/2 - n, 3/2, (1*(1 - (b*Sin[c + d*x])/a))/2])/(d*Sqrt[a + b*Sin[c + d*x]]), x] /; FreeQ[
{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \sin (c+d x))^{4/3}} \, dx &=\frac{\sqrt [3]{1+\sin (c+d x)} \int \frac{1}{(1+\sin (c+d x))^{4/3}} \, dx}{a \sqrt [3]{a+a \sin (c+d x)}}\\ &=-\frac{\cos (c+d x) \, _2F_1\left (\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\sin (c+d x))\right )}{2^{5/6} a d \sqrt [6]{1+\sin (c+d x)} \sqrt [3]{a+a \sin (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 0.286556, size = 130, normalized size = 1.88 $-\frac{3 \left (\sqrt{2-2 \sin (c+d x)}-2 (\sin (c+d x)+1) \, _2F_1\left (\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2\left (\frac{1}{4} (2 c+2 d x+\pi )\right )\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{5 d \sqrt{2-2 \sin (c+d x)} (a (\sin (c+d x)+1))^{4/3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sqrt[2 - 2*Sin[c + d*x]] - 2*
Hypergeometric2F1[1/6, 1/2, 7/6, Sin[(2*c + Pi + 2*d*x)/4]^2]*(1 + Sin[c + d*x])))/(5*d*Sqrt[2 - 2*Sin[c + d*x
]]*(a*(1 + Sin[c + d*x]))^(4/3))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*sin(d*x+c))^(4/3),x)

[Out]

int(1/(a+a*sin(d*x+c))^(4/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(-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 (d x + c\right ) + a\right )}^{\frac{2}{3}}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sin{\left (c + d x \right )} + a\right )^{\frac{4}{3}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(d*x+c))**(4/3),x)

[Out]

Integral((a*sin(c + d*x) + a)**(-4/3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{4}{3}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*sin(d*x+c))^(4/3),x, algorithm="giac")

[Out]

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