3.4.26 \(\int \frac {1}{(e \cos (c+d x))^{4/3} \sqrt {a+a \sin (c+d x)}} \, dx\) [326]

Optimal. Leaf size=75 \[ \frac {3 \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{2/3}}{2^{2/3} d e \sqrt [3]{e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \]

[Out]

3/2*hypergeom([-1/6, 5/3],[5/6],1/2-1/2*sin(d*x+c))*(1+sin(d*x+c))^(2/3)*2^(1/3)/d/e/(e*cos(d*x+c))^(1/3)/(a+a
*sin(d*x+c))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2768, 72, 71} \begin {gather*} \frac {3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-\sin (c+d x))\right )}{2^{2/3} d e \sqrt {a \sin (c+d x)+a} \sqrt [3]{e \cos (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*Cos[c + d*x])^(4/3)*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

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

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*(b*((c + d*x)/(b*c - a*d)))^FracPart[n]), Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c -
a*d)), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 2768

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[a^2*(
(g*Cos[e + f*x])^(p + 1)/(f*g*(a + b*Sin[e + f*x])^((p + 1)/2)*(a - b*Sin[e + f*x])^((p + 1)/2))), Subst[Int[(
a + b*x)^(m + (p - 1)/2)*(a - b*x)^((p - 1)/2), x], x, Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&
EqQ[a^2 - b^2, 0] &&  !IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {1}{(e \cos (c+d x))^{4/3} \sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\left (a^2 \sqrt [6]{a-a \sin (c+d x)} \sqrt [6]{a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{(a-a x)^{7/6} (a+a x)^{5/3}} \, dx,x,\sin (c+d x)\right )}{d e \sqrt [3]{e \cos (c+d x)}}\\ &=\frac {\left (a \sqrt [6]{a-a \sin (c+d x)} \left (\frac {a+a \sin (c+d x)}{a}\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\left (\frac {1}{2}+\frac {x}{2}\right )^{5/3} (a-a x)^{7/6}} \, dx,x,\sin (c+d x)\right )}{2\ 2^{2/3} d e \sqrt [3]{e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}\\ &=\frac {3 \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{2/3}}{2^{2/3} d e \sqrt [3]{e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.10, size = 75, normalized size = 1.00 \begin {gather*} \frac {3 \, _2F_1\left (-\frac {1}{6},\frac {5}{3};\frac {5}{6};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{2/3}}{2^{2/3} d e \sqrt [3]{e \cos (c+d x)} \sqrt {a (1+\sin (c+d x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*Cos[c + d*x])^(4/3)*Sqrt[a + a*Sin[c + d*x]]),x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \cos \left (d x +c \right )\right )^{\frac {4}{3}} \sqrt {a +a \sin \left (d x +c \right )}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*cos(d*x+c))^(4/3)/(a+a*sin(d*x+c))^(1/2),x)

[Out]

int(1/(e*cos(d*x+c))^(4/3)/(a+a*sin(d*x+c))^(1/2),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(4/3)/(a+a*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

e^(-4/3)*integrate(1/(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)^(4/3)), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(4/3)/(a+a*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sin(d*x + c) + a)*cos(d*x + c)^(2/3)/(a*cos(d*x + c)^2*e^(4/3)*sin(d*x + c) + a*cos(d*x + c)^2
*e^(4/3)), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))**(4/3)/(a+a*sin(d*x+c))**(1/2),x)

[Out]

Integral(1/(sqrt(a*(sin(c + d*x) + 1))*(e*cos(c + d*x))**(4/3)), x)

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*cos(d*x+c))^(4/3)/(a+a*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{4/3}\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*cos(c + d*x))^(4/3)*(a + a*sin(c + d*x))^(1/2)),x)

[Out]

int(1/((e*cos(c + d*x))^(4/3)*(a + a*sin(c + d*x))^(1/2)), x)

________________________________________________________________________________________