\(\int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx\) [67]

Optimal result

Integrand size = 18, antiderivative size = 537 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {2 b \sqrt {\frac {1}{c^2}+x^2} \text {arctanh}\left (\frac {\sqrt {d+e x}}{c \sqrt {d} \sqrt {\frac {1}{c^2}+x^2}}\right )}{\sqrt {d} e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {2 b \left (c^2 d^2+e^2-c d \sqrt {c^2 d^2+e^2}\right ) \sqrt {\frac {e^2 \left (1+c^2 x^2\right )}{\left (c^2 d^2+e^2\right ) \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right )^2}} \left (1+\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{\sqrt {c} e^3 \sqrt [4]{c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {b \left (c d-\sqrt {c^2 d^2+e^2}\right )^2 \sqrt {\frac {e^2 \left (1+c^2 x^2\right )}{\left (\sqrt {c^2 d^2+e^2}+c (d+e x)\right )^2}} \left (\sqrt {c^2 d^2+e^2}+c (d+e x)\right ) \operatorname {EllipticPi}\left (\frac {\left (c d+\sqrt {c^2 d^2+e^2}\right )^2}{4 c d \sqrt {c^2 d^2+e^2}},2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (1+\frac {c d}{\sqrt {c^2 d^2+e^2}}\right )\right )}{c^{3/2} d e^3 \sqrt [4]{c^2 d^2+e^2} \sqrt {1+\frac {1}{c^2 x^2}} x} \] Output:

(-2*a-2*b*arccsch(c*x))/e/(e*x+d)^(1/2)+2*b*(1/c^2+x^2)^(1/2)*arctanh((e*x 
+d)^(1/2)/c/d^(1/2)/(1/c^2+x^2)^(1/2))/d^(1/2)/e/(1+1/c^2/x^2)^(1/2)/x+2*b 
*(c^2*d^2+e^2-c*d*(c^2*d^2+e^2)^(1/2))*(e^2*(c^2*x^2+1)/(c^2*d^2+e^2)/(1+c 
*(e*x+d)/(c^2*d^2+e^2)^(1/2))^2)^(1/2)*(1+c*(e*x+d)/(c^2*d^2+e^2)^(1/2))*I 
nverseJacobiAM(2*arctan(c^(1/2)*(e*x+d)^(1/2)/(c^2*d^2+e^2)^(1/4)),1/2*(2+ 
2*c*d/(c^2*d^2+e^2)^(1/2))^(1/2))/c^(1/2)/e^3/(c^2*d^2+e^2)^(1/4)/(1+1/c^2 
/x^2)^(1/2)/x+b*(c*d-(c^2*d^2+e^2)^(1/2))^2*((c^2*x^2+1)*e^2/(c*(e*x+d)+(c 
^2*d^2+e^2)^(1/2))^2)^(1/2)*(c*(e*x+d)+(c^2*d^2+e^2)^(1/2))*EllipticPi(sin 
(2*arctan(c^(1/2)*(e*x+d)^(1/2)/(c^2*d^2+e^2)^(1/4))),1/4*(c*d+(c^2*d^2+e^ 
2)^(1/2))^2/c/d/(c^2*d^2+e^2)^(1/2),1/2*(2+2*c*d/(c^2*d^2+e^2)^(1/2))^(1/2 
))/c^(3/2)/d/e^3/(c^2*d^2+e^2)^(1/4)/(1+1/c^2/x^2)^(1/2)/x
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.48 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.31 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\frac {-2 e \left (1+c^2 x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )+2 b c (i c d+e) \sqrt {2+\frac {2}{c^2 x^2}} x \sqrt {1+i c x} \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )}{e^2 \sqrt {d+e x} \left (1+c^2 x^2\right )} \] Input:

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

Output:

(-2*e*(1 + c^2*x^2)*(a + b*ArcCsch[c*x]) + 2*b*c*(I*c*d + e)*Sqrt[2 + 2/(c 
^2*x^2)]*x*Sqrt[1 + I*c*x]*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*E 
llipticPi[1 + (I*c*d)/e, ArcSin[Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c* 
d + e)/(2*e)])/(e^2*Sqrt[d + e*x]*(1 + c^2*x^2))
 

Rubi [A] (warning: unable to verify)

Time = 1.08 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6844, 1898, 631, 1540, 1416, 2222}

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.

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

\(\Big \downarrow \) 6844

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

\(\Big \downarrow \) 1898

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

\(\Big \downarrow \) 631

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

\(\Big \downarrow \) 1540

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

\(\Big \downarrow \) 1416

\(\displaystyle \frac {4 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {d^2}{e^2}-\frac {2 (d+e x) d}{e^2}+\frac {(d+e x)^2}{e^2}+\frac {1}{c^2}}}d\sqrt {d+e x}}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

\(\Big \downarrow \) 2222

\(\displaystyle \frac {4 b \sqrt {\frac {1}{c^2}+x^2} \left (\frac {\left (c^2 d^2+e^2\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c d+\sqrt {c^2 d^2+e^2}\right )^2}{4 c d \sqrt {c^2 d^2+e^2}},2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {\sqrt {c^2 d^2+e^2} d}{2 c \left (\frac {d^2}{e^2}+\frac {1}{c^2}\right ) e^2}+\frac {1}{2}\right )}{4 \sqrt {c} d \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}+\frac {c \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{c \sqrt {d} \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )}{2 \sqrt {d}}\right )}{e^2}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}{\left (\frac {1}{c^2}+\frac {d^2}{e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {1}{c^2}+\frac {d^2}{e^2}-\frac {2 d (d+e x)}{e^2}+\frac {(d+e x)^2}{e^2}}}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e \sqrt {d+e x}}\)

Input:

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

Output:

(-2*(a + b*ArcCsch[c*x]))/(e*Sqrt[d + e*x]) + (4*b*Sqrt[c^(-2) + x^2]*(-1/ 
2*(Sqrt[c]*(c^2*d^2 + e^2)^(1/4)*(c*d - Sqrt[c^2*d^2 + e^2])*(1 + (c*(d + 
e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + 
(d + e*x)^2/e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2 
])^2)]*EllipticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], 
(1 + (c*d)/Sqrt[c^2*d^2 + e^2])/2])/(e^2*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + 
 e*x))/e^2 + (d + e*x)^2/e^2]) + ((c^2*d^2 + e^2)*(1 - (c*d)/Sqrt[c^2*d^2 
+ e^2])*((c*(1 + (c*d)/Sqrt[c^2*d^2 + e^2])*ArcTanh[Sqrt[d + e*x]/(c*Sqrt[ 
d]*Sqrt[c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])])/(2*Sq 
rt[d]) + ((c^2*d^2 + e^2)^(1/4)*(1 - (c*d)/Sqrt[c^2*d^2 + e^2])*(1 + (c*(d 
 + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt[(c^(-2) + d^2/e^2 - (2*d*(d + e*x))/e^2 
 + (d + e*x)^2/e^2)/((c^(-2) + d^2/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + 
e^2])^2)]*EllipticPi[(c*d + Sqrt[c^2*d^2 + e^2])^2/(4*c*d*Sqrt[c^2*d^2 + e 
^2]), 2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], 1/2 + (d*Sq 
rt[c^2*d^2 + e^2])/(2*c*(c^(-2) + d^2/e^2)*e^2)])/(4*Sqrt[c]*d*Sqrt[c^(-2) 
 + d^2/e^2 - (2*d*(d + e*x))/e^2 + (d + e*x)^2/e^2])))/e^2))/(c*e*Sqrt[1 + 
 1/(c^2*x^2)]*x)
 

Defintions of rubi rules used

rule 631

Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : 
> Simp[-2   Subst[Int[1/((c - x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^ 
2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && 
PosQ[b/a]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1540

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2)   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

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 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

rule 6844

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsch[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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.23 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\)	\(328\)
default	\(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\)	\(328\)
parts	\(-\frac {2 a}{\sqrt {e x +d}\, e}+\frac {2 b \left (-\frac {\operatorname {arccsch}\left (c x \right )}{\sqrt {e x +d}}+\frac {2 \sqrt {-\frac {i c \left (e x +d \right ) e +c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c \left (e x +d \right ) e -c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x d \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\)	\(330\)

Input:

int((a+b*arccsch(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)*arccsch(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/((I*e+c*d)*c/(c^2*d^2 
+e^2))^(1/2)*(-(I*c*(e*x+d)*e+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1 
/2)*((I*c*(e*x+d)*e-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*Ellipt 
icPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^ 
2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1 
/2))))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(2*c^2*integrate(x/((c^2*e*x^2 + e)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^ 
2*e*x^2 + e)*sqrt(e*x + d)), x) + 2*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + 
 d)*e) + integrate(((e*log(c) - 2*e)*c^2*x^2 - 2*c^2*d*x + e*log(c) + (c^2 
*e*x^2 + e)*log(x))/((c^2*e^2*x^3 + c^2*d*e*x^2 + e^2*x + d*e)*sqrt(e*x + 
d)), x))*b - 2*a/(sqrt(e*x + d)*e)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {e x +d}\, \left (\int \frac {\mathit {acsch} \left (c x \right )}{\sqrt {e x +d}\, d +\sqrt {e x +d}\, e x}d x \right ) b e -2 a}{\sqrt {e x +d}\, e} \] Input:

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

Output:

(sqrt(d + e*x)*int(acsch(c*x)/(sqrt(d + e*x)*d + sqrt(d + e*x)*e*x),x)*b*e 
 - 2*a)/(sqrt(d + e*x)*e)