3.246 \(\int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=75 \[ \frac{2 \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right ),2\right )}{39 \sqrt [4]{13} d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}} \]

[Out]

(2*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(39*13^(1/4)*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(39*d*(2*C
os[c + d*x] + 3*Sin[c + d*x])^(3/2))

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Rubi [A]  time = 0.0421979, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3076, 3077, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )\right |2\right )}{39 \sqrt [4]{13} d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-5/2),x]

[Out]

(2*EllipticF[(c + d*x - ArcTan[3/2])/2, 2])/(39*13^(1/4)*d) - (2*(3*Cos[c + d*x] - 2*Sin[c + d*x]))/(39*d*(2*C
os[c + d*x] + 3*Sin[c + d*x])^(3/2))

Rule 3076

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[((b*Cos[c + d*x] -
a*Sin[c + d*x])*(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 1))/(d*(n + 1)*(a^2 + b^2)), x] + Dist[(n + 2)/((n + 1
)*(a^2 + b^2)), Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && LtQ[n, -1] && NeQ[n, -2]

Rule 3077

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(a^2 + b^2)^(n/2),
Int[Cos[c + d*x - ArcTan[a, b]]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] &&  !(GeQ[n, 1] || LeQ[n, -1]) && GtQ[
a^2 + b^2, 0]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{1}{(2 \cos (c+d x)+3 \sin (c+d x))^{5/2}} \, dx &=-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (2 \cos (c+d x)+3 \sin (c+d x))^{3/2}}+\frac{1}{39} \int \frac{1}{\sqrt{2 \cos (c+d x)+3 \sin (c+d x)}} \, dx\\ &=-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (2 \cos (c+d x)+3 \sin (c+d x))^{3/2}}+\frac{\int \frac{1}{\sqrt{\cos \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )}} \, dx}{39 \sqrt [4]{13}}\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (c+d x-\tan ^{-1}\left (\frac{3}{2}\right )\right )\right |2\right )}{39 \sqrt [4]{13} d}-\frac{2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (2 \cos (c+d x)+3 \sin (c+d x))^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.741956, size = 157, normalized size = 2.09 \[ \frac{\sqrt{2} 13^{3/4} \sqrt{\sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )+1} (3 \sin (c+d x)+2 \cos (c+d x))^{3/2} \sec \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right ) \sqrt{2 \sin \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )+\cos \left (2 \left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )\right )-1} \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (c+d x+\tan ^{-1}\left (\frac{2}{3}\right )\right )\right )+52 \sin (c+d x)-78 \cos (c+d x)}{507 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(2*Cos[c + d*x] + 3*Sin[c + d*x])^(-5/2),x]

[Out]

(-78*Cos[c + d*x] + 52*Sin[c + d*x] + Sqrt[2]*13^(3/4)*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[c + d*x + ArcT
an[2/3]]^2]*Sec[c + d*x + ArcTan[2/3]]*(2*Cos[c + d*x] + 3*Sin[c + d*x])^(3/2)*Sqrt[1 + Sin[c + d*x + ArcTan[2
/3]]]*Sqrt[-1 + Cos[2*(c + d*x + ArcTan[2/3])] + 2*Sin[c + d*x + ArcTan[2/3]]])/(507*d*(2*Cos[c + d*x] + 3*Sin
[c + d*x])^(3/2))

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Maple [A]  time = 1.101, size = 118, normalized size = 1.6 \begin{align*}{\frac{1}{39\,\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) \cos \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) d} \left ( \sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }\sqrt{-2\,\sin \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) +2}\sqrt{-\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }{\it EllipticF} \left ( \sqrt{1+\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) \sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) -2\, \left ( \cos \left ( dx+c+\arctan \left ( 2/3 \right ) \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{\sqrt{13}\sin \left ( dx+c+\arctan \left ({\frac{2}{3}} \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*cos(d*x+c)+3*sin(d*x+c))^(5/2),x)

[Out]

1/39/sin(d*x+c+arctan(2/3))*((1+sin(d*x+c+arctan(2/3)))^(1/2)*(-2*sin(d*x+c+arctan(2/3))+2)^(1/2)*(-sin(d*x+c+
arctan(2/3)))^(1/2)*EllipticF((1+sin(d*x+c+arctan(2/3)))^(1/2),1/2*2^(1/2))*sin(d*x+c+arctan(2/3))-2*cos(d*x+c
+arctan(2/3))^2)/cos(d*x+c+arctan(2/3))/(13^(1/2)*sin(d*x+c+arctan(2/3)))^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((2*cos(d*x + c) + 3*sin(d*x + c))^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{46 \, \cos \left (d x + c\right )^{3} - 9 \,{\left (\cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(2*cos(d*x + c) + 3*sin(d*x + c))/(46*cos(d*x + c)^3 - 9*(cos(d*x + c)^2 + 3)*sin(d*x + c) - 54*
cos(d*x + c)), 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(1/(2*cos(d*x+c)+3*sin(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*cos(d*x+c)+3*sin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((2*cos(d*x + c) + 3*sin(d*x + c))^(-5/2), x)