3.470 \(\int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)} \, dx\)

Optimal. Leaf size=240 \[ -\frac{2 (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac{3}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac{3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \]

[Out]

(-2*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*S
in[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*Sqrt[(b +
c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d +
e*x] - c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2))

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Rubi [A]  time = 0.205311, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {3164, 3128, 3119, 2653} \[ -\frac{2 (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac{3}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac{3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)),x]

[Out]

(-2*EllipticE[(d + e*x - ArcTan[c, a])/2, (2*Sqrt[a^2 + c^2])/(b + Sqrt[a^2 + c^2])]*(b + c*Cos[d + e*x] + a*S
in[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)*Sqrt[(b +
c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*(b + c*Cos[d + e*x] + a*Sin[d + e*x])*(a*Cos[d +
e*x] - c*Sin[d + e*x]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2))

Rule 3164

Int[((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(n_)*sin[(d_.) + (e_.)*(x_)]^(n_),
x_Symbol] :> Dist[(Sin[d + e*x]^n*(a + b*Csc[d + e*x] + c*Cot[d + e*x])^n)/(b + a*Sin[d + e*x] + c*Cos[d + e*
x])^n, Int[(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] &&  !IntegerQ[n]

Rule 3128

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-3/2), x_Symbol] :> Simp[(2*(c*Cos
[d + e*x] - b*Sin[d + e*x]))/(e*(a^2 - b^2 - c^2)*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]), x] + Dist[1/(a^2
- b^2 - c^2), Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 -
b^2 - c^2, 0]

Rule 3119

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]*Cos[d + e*x - ArcTan[b, c]])/(a + Sqrt[b^2 + c^2])], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 2653

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*Sqrt[a + b]*EllipticE[(1*(c - Pi/2 + d*x)
)/2, (2*b)/(a + b)])/d, x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)} \, dx &=\frac{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \int \frac{1}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}} \, dx}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)}\\ &=-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)}-\frac{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \int \sqrt{b+c \cos (d+e x)+a \sin (d+e x)} \, dx}{\left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)}\\ &=-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)}-\frac{(b+c \cos (d+e x)+a \sin (d+e x))^2 \int \sqrt{\frac{b}{b+\sqrt{a^2+c^2}}+\frac{\sqrt{a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt{a^2+c^2}}} \, dx}{\left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x) \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}\\ &=-\frac{2 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x) \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)}\\ \end{align*}

Mathematica [F]  time = 21.461, size = 0, normalized size = 0. \[ \int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac{3}{2}}(d+e x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)),x]

[Out]

Integrate[1/((a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sin[d + e*x]^(3/2)), x]

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Maple [C]  time = 0.433, size = 12467, normalized size = 52. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2)/sin(e*x+d)^(3/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(3/2)*sin(e*x + d)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(c*cot(e*x + d) + b*csc(e*x + d) + a)*sqrt(sin(e*x + d))/(a^2*cos(e*x + d)^2 + (c^2*cos(e*x + d)
^2 - c^2)*cot(e*x + d)^2 + (b^2*cos(e*x + d)^2 - b^2)*csc(e*x + d)^2 - a^2 + 2*(a*c*cos(e*x + d)^2 - a*c)*cot(
e*x + d) + 2*(a*b*cos(e*x + d)^2 - a*b + (b*c*cos(e*x + d)^2 - b*c)*cot(e*x + d))*csc(e*x + d)), 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/(a+c*cot(e*x+d)+b*csc(e*x+d))**(3/2)/sin(e*x+d)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((c*cot(e*x + d) + b*csc(e*x + d) + a)^(3/2)*sin(e*x + d)^(3/2)), x)