3.195 \(\int \frac {c e+d e x}{(a+b \cosh ^{-1}(c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=266 \[ \frac {8 \sqrt {2 \pi } e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 \sqrt {2 \pi } e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {32 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]

[Out]

4/15*e/b^2/d/(a+b*arccosh(d*x+c))^(3/2)-8/15*e*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+c))^(3/2)+8/15*e*exp(2*a/b)*er
f(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d+8/15*e*erfi(2^(1/2)*(a+b*arccosh(d*x+
c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d/exp(2*a/b)-2/5*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+
b*arccosh(d*x+c))^(5/2)-32/15*e*(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b^3/d/(a+b*arccosh(d*x+c))^(1/2)

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Rubi [A]  time = 0.78, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5866, 12, 5668, 5775, 5666, 3307, 2180, 2204, 2205, 5676} \[ \frac {8 \sqrt {2 \pi } e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 \sqrt {2 \pi } e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {32 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(-2*e*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (4*e)/(15*b^2*d
*(a + b*ArcCosh[c + d*x])^(3/2)) - (8*e*(c + d*x)^2)/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (32*e*Sqrt[-1
+ c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (8*e*E^((2*a)/b)*Sqrt[2*Pi]
*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (8*e*Sqrt[2*Pi]*Erfi[(Sqrt[2]*Sqrt[a +
b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((2*a)/b))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5668

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5866

Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

\begin {align*} \int \frac {c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e x}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac {(2 e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac {(4 e) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {(16 e) \operatorname {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {32 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(32 e) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {32 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(16 e) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {(16 e) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {32 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(32 e) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {(32 e) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {32 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {8 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}\\ \end {align*}

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Mathematica [B]  time = 4.77, size = 916, normalized size = 3.44 \[ \frac {e \left (8 c \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x)^2 b^{5/2}+4 c (c+d x) \cosh ^{-1}(c+d x) b^{5/2}-4 \cosh ^{-1}(c+d x) \cosh \left (2 \cosh ^{-1}(c+d x)\right ) b^{5/2}-16 \cosh ^{-1}(c+d x)^2 \sinh \left (2 \cosh ^{-1}(c+d x)\right ) b^{5/2}-3 \sinh \left (2 \cosh ^{-1}(c+d x)\right ) b^{5/2}+4 a c (c+d x) b^{3/2}+16 a c \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \cosh ^{-1}(c+d x) b^{3/2}-4 a \cosh \left (2 \cosh ^{-1}(c+d x)\right ) b^{3/2}-32 a \cosh ^{-1}(c+d x) \sinh \left (2 \cosh ^{-1}(c+d x)\right ) b^{3/2}+8 a^2 c \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \sqrt {b}-2 c e^{-\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (-2 a+b-2 b \cosh ^{-1}(c+d x)+2 e^{\frac {a}{b}+\cosh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sqrt {b}-2 c e^{-\frac {a}{b}} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 b \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2}+e^{\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (2 a+b+2 b \cosh ^{-1}(c+d x)\right )\right ) \sqrt {b}-16 a^2 \sinh \left (2 \cosh ^{-1}(c+d x)\right ) \sqrt {b}-4 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac {a}{b}\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac {2 a}{b}\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-4 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-4 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+4 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+8 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-8 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )\right )}{15 b^{7/2} d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c*e + d*e*x)/(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(e*(4*a*b^(3/2)*c*(c + d*x) + 8*a^2*Sqrt[b]*c*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x) + 4*b^(5/2)*c*(
c + d*x)*ArcCosh[c + d*x] + 16*a*b^(3/2)*c*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x] +
8*b^(5/2)*c*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*ArcCosh[c + d*x]^2 - 4*a*b^(3/2)*Cosh[2*ArcCosh[
c + d*x]] - 4*b^(5/2)*ArcCosh[c + d*x]*Cosh[2*ArcCosh[c + d*x]] - 4*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*
Cosh[a/b]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] + 8*Sqrt[2*Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Cosh[(2*a)/b
]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]] - 4*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Cosh[a/b]*
Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]] + 8*Sqrt[2*Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Cosh[(2*a)/b]*Erfi[(S
qrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]] - (2*Sqrt[b]*c*(a + b*ArcCosh[c + d*x])*(-2*a + b - 2*b*ArcCosh[
c + d*x] + 2*E^(a/b + ArcCosh[c + d*x])*Sqrt[a/b + ArcCosh[c + d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, a/b +
ArcCosh[c + d*x]]))/E^ArcCosh[c + d*x] - (2*Sqrt[b]*c*(a + b*ArcCosh[c + d*x])*(E^(a/b + ArcCosh[c + d*x])*(2
*a + b + 2*b*ArcCosh[c + d*x]) + 2*b*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcCosh[c + d*x
])/b)]))/E^(a/b) - 4*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*Sinh[
a/b] + 4*c*Sqrt[Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*Sinh[a/b] + 8*Sq
rt[2*Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] - 8*
Sqrt[2*Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]]*Sinh[(2*a)/b] -
16*a^2*Sqrt[b]*Sinh[2*ArcCosh[c + d*x]] - 3*b^(5/2)*Sinh[2*ArcCosh[c + d*x]] - 32*a*b^(3/2)*ArcCosh[c + d*x]*
Sinh[2*ArcCosh[c + d*x]] - 16*b^(5/2)*ArcCosh[c + d*x]^2*Sinh[2*ArcCosh[c + d*x]]))/(15*b^(7/2)*d*(a + b*ArcCo
sh[c + d*x])^(5/2))

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)/(b*arccosh(d*x + c) + a)^(7/2), x)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {d e x +c e}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)/(a+b*arccosh(d*x+c))^(7/2),x)

[Out]

int((d*e*x+c*e)/(a+b*arccosh(d*x+c))^(7/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*e*x + c*e)/(b*arccosh(d*x + c) + a)^(7/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)/(a + b*acosh(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)/(a + b*acosh(c + d*x))^(7/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e \left (\int \frac {c}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)/(a+b*acosh(d*x+c))**(7/2),x)

[Out]

e*(Integral(c/(a**3*sqrt(a + b*acosh(c + d*x)) + 3*a**2*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + 3*a*b**2
*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**2 + b**3*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**3), x) + Integ
ral(d*x/(a**3*sqrt(a + b*acosh(c + d*x)) + 3*a**2*b*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x) + 3*a*b**2*sqrt(
a + b*acosh(c + d*x))*acosh(c + d*x)**2 + b**3*sqrt(a + b*acosh(c + d*x))*acosh(c + d*x)**3), x))

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