3.465 \(\int \frac{\csc ^{\frac{3}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2}} \, dx\)
Optimal. Leaf size=240 \[ -\frac{2 \csc ^{\frac{3}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left (a^2-b^2+c^2\right ) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \]
[Out]
(-2*Csc[d + e*x]^(3/2)*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*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sqrt[(b +
c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*Csc[d + e*x]^(3/2)*(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))
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Rubi [A] time = 0.212266, 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 =
{3168, 3128, 3119, 2653} \[ -\frac{2 \csc ^{\frac{3}{2}}(d+e x) (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left (a^2-b^2+c^2\right ) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[Csc[d + e*x]^(3/2)/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2),x]
[Out]
(-2*Csc[d + e*x]^(3/2)*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*Sin[d + e*x])^2)/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)*Sqrt[(b +
c*Cos[d + e*x] + a*Sin[d + e*x])/(b + Sqrt[a^2 + c^2])]) - (2*Csc[d + e*x]^(3/2)*(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))
Rule 3168
Int[csc[(d_.) + (e_.)*(x_)]^(n_.)*((a_.) + csc[(d_.) + (e_.)*(x_)]*(b_.) + cot[(d_.) + (e_.)*(x_)]*(c_.))^(m_)
, x_Symbol] :> Dist[(Csc[d + e*x]^n*(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n)/(a + b*Csc[d + e*x] + c*Cot[d + e
*x])^n, Int[1/(b + a*Sin[d + e*x] + c*Cos[d + e*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + n, 0] &&
!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{\csc ^{\frac{3}{2}}(d+e x)}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2}} \, dx &=\frac{\left (\csc ^{\frac{3}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^{3/2}\right ) \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}}\\ &=-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (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}}-\frac{\left (\csc ^{\frac{3}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^{3/2}\right ) \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}}\\ &=-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (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}}-\frac{\left (\csc ^{\frac{3}{2}}(d+e x) (b+c \cos (d+e x)+a \sin (d+e x))^2\right ) \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} \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}\\ &=-\frac{2 \csc ^{\frac{3}{2}}(d+e x) 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} \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}-\frac{2 \csc ^{\frac{3}{2}}(d+e x) (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}}\\ \end{align*}
Mathematica [C] time = 6.41498, size = 1732, normalized size = 7.22 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Csc[d + e*x]^(3/2)/(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2),x]
[Out]
(Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^2*((-2*(a^2 + c^2))/(a*c*(a^2 - b^2 + c^2)) + (2*(a*
b + a^2*Sin[d + e*x] + c^2*Sin[d + e*x]))/(c*(a^2 - b^2 + c^2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x]))))/(e*(a
+ c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)) - (a*Csc[d + e*x]^(3/2)*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*
(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2
]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]
*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)
/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b + c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^
2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x
- ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]])
)/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x
- ArcTan[a/c]]]))/((a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2)) - (c^2*Csc[d + e*x]^(3/2)
*(b + c*Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*(-((a*AppellF1[-1/2, -1/2, -1/2, 1/2, -((b + Sqrt[1 + a^2/c^2]*c*
Cos[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(1 - b/(Sqrt[1 + a^2/c^2]*c))*c)), -((b + Sqrt[1 + a^2/c^2]*c*C
os[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]*(-1 - b/(Sqrt[1 + a^2/c^2]*c))*c))]*Sin[d + e*x - ArcTan[a/c]])/
(Sqrt[1 + a^2/c^2]*c*Sqrt[(c*Sqrt[(a^2 + c^2)/c^2] - c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(b +
c*Sqrt[(a^2 + c^2)/c^2])]*Sqrt[b + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]]]*Sqrt[(c*Sqrt[(a^2 + c^2
)/c^2] + c*Sqrt[(a^2 + c^2)/c^2]*Cos[d + e*x - ArcTan[a/c]])/(-b + c*Sqrt[(a^2 + c^2)/c^2])])) - ((2*c*(b + Sq
rt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]))/(a^2 + c^2) - (a*Sin[d + e*x - ArcTan[a/c]])/(Sqrt[1 + a^2/c^2]
*c))/Sqrt[b + Sqrt[1 + a^2/c^2]*c*Cos[d + e*x - ArcTan[a/c]]]))/(a*(a^2 - b^2 + c^2)*e*(a + c*Cot[d + e*x] + b
*Csc[d + e*x])^(3/2)) - (2*b*AppellF1[1/2, 1/2, 1/2, 3/2, -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]
])/(a*Sqrt[1 + c^2/a^2]*(1 - b/(a*Sqrt[1 + c^2/a^2])))), -((b + a*Sqrt[1 + c^2/a^2]*Sin[d + e*x + ArcTan[c/a]]
)/(a*Sqrt[1 + c^2/a^2]*(-1 - b/(a*Sqrt[1 + c^2/a^2]))))]*Csc[d + e*x]^(3/2)*Sec[d + e*x + ArcTan[c/a]]*(b + c*
Cos[d + e*x] + a*Sin[d + e*x])^(3/2)*Sqrt[(a*Sqrt[(a^2 + c^2)/a^2] - a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + Arc
Tan[c/a]])/(b + a*Sqrt[(a^2 + c^2)/a^2])]*Sqrt[b + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]]]*Sqrt[(a
*Sqrt[(a^2 + c^2)/a^2] + a*Sqrt[(a^2 + c^2)/a^2]*Sin[d + e*x + ArcTan[c/a]])/(-b + a*Sqrt[(a^2 + c^2)/a^2])])/
(a*(a^2 - b^2 + c^2)*Sqrt[1 + c^2/a^2]*e*(a + c*Cot[d + e*x] + b*Csc[d + e*x])^(3/2))
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Maple [C] time = 0.556, size = 12477, 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(csc(e*x+d)^(3/2)/(a+c*cot(e*x+d)+b*csc(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{\csc \left (e x + d\right )^{\frac{3}{2}}}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(e*x+d)^(3/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2),x, algorithm="maxima")
[Out]
integrate(csc(e*x + d)^(3/2)/(c*cot(e*x + d) + b*csc(e*x + d) + a)^(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} \csc \left (e x + d\right )^{\frac{3}{2}}}{c^{2} \cot \left (e x + d\right )^{2} + b^{2} \csc \left (e x + d\right )^{2} + 2 \, a c \cot \left (e x + d\right ) + a^{2} + 2 \,{\left (b c \cot \left (e x + d\right ) + a b\right )} \csc \left (e x + d\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(e*x+d)^(3/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2),x, algorithm="fricas")
[Out]
integral(sqrt(c*cot(e*x + d) + b*csc(e*x + d) + a)*csc(e*x + d)^(3/2)/(c^2*cot(e*x + d)^2 + b^2*csc(e*x + d)^2
+ 2*a*c*cot(e*x + d) + a^2 + 2*(b*c*cot(e*x + d) + a*b)*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(csc(e*x+d)**(3/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))**(3/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(e*x+d)^(3/2)/(a+c*cot(e*x+d)+b*csc(e*x+d))^(3/2),x, algorithm="giac")
[Out]
sage0*x