3.1.66 \(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx\) [66]

3.1.66.1 Optimal result

Integrand size = 18, antiderivative size = 119 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]

output

-2*(a+b*arccsc(c*x))/e/(e*x+d)^(1/2)+4*b*EllipticPi(1/2*(-c*x+1)^(1/2)*2^( 
1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^( 
1/2)/c/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
 

3.1.66.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\frac {-2 \left (-1+c^2 x^2\right ) \left (a+b \csc ^{-1}(c x)\right )+4 b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x} \left (-1+c^2 x^2\right )} \]

input

Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^(3/2),x]
 

output

(-2*(-1 + c^2*x^2)*(a + b*ArcCsc[c*x]) + 4*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqr 
t[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - 
 c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(e*Sqrt[d + e*x]*(-1 + c^2*x^2))
 

3.1.66.3 Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5750, 1898, 633, 632, 186, 413, 412}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5750

\(\displaystyle -\frac {2 b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c e}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 1898

\(\displaystyle -\frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 633

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 632

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {4 b \sqrt {1-c^2 x^2} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {4 b \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {4 b \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

input

Int[(a + b*ArcCsc[c*x])/(d + e*x)^(3/2),x]
 

output

(-2*(a + b*ArcCsc[c*x]))/(e*Sqrt[d + e*x]) + (4*b*Sqrt[1 - c^2*x^2]*Sqrt[1 
 - (e*(1 - c*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], ( 
2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e/c - (e*(1 - c*x)) 
/c])
 

3.1.66.3.1 Defintions of rubi rules used

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 632

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

rule 633

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

rule 1898

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ 
(q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( 
c + a*x^(2*n))^FracPart[p])   Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n 
))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !I 
ntegerQ[p] &&  !IntegerQ[q] && PosQ[n]
 

rule 5750

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ 
(c*e*(m + 1))   Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / 
; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

3.1.66.4 Maple [A] (verified)

Time = 3.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.81

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {c}{c d -e}}}\right )}{e}\)	\(215\)
default	\(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {c}{c d -e}}}\right )}{e}\)	\(215\)
parts	\(-\frac {2 a}{\sqrt {e x +d}\, e}+\frac {2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {c}{c d -e}}}\right )}{e}\)	\(219\)

input

int((a+b*arccsc(c*x))/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
 

output

2/e*(-a/(e*x+d)^(1/2)+b*(-1/(e*x+d)^(1/2)*arccsc(c*x)+2/c/((c^2*(e*x+d)^2- 
2*c^2*d*(e*x+d)+c^2*d^2-e^2)/c^2/e^2/x^2)^(1/2)/x/d/(c/(c*d-e))^(1/2)*((-c 
*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*Elliptic 
Pi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d 
-e))^(1/2))))
 

3.1.66.5 Fricas [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

input

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

output

integral(sqrt(e*x + d)*(b*arccsc(c*x) + a)/(e^2*x^2 + 2*d*e*x + d^2), x)
 

3.1.66.6 Sympy [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]

input

integrate((a+b*acsc(c*x))/(e*x+d)**(3/2),x)
 

output

Integral((a + b*acsc(c*x))/(d + e*x)**(3/2), x)
 

3.1.66.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor 
e details)
 

3.1.66.8 Giac [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

input

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

output

integrate((b*arccsc(c*x) + a)/(e*x + d)^(3/2), x)
 

3.1.66.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]

input

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

output

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