3.2.93 \(\int \frac {(c e+d e x)^3}{(a+b \cosh ^{-1}(c+d x))^{7/2}} \, dx\) [193]

Optimal. Leaf size=441 \[ -\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^3 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {16 e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^3 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d} \]

[Out]

4/5*e^3*(d*x+c)^2/b^2/d/(a+b*arccosh(d*x+c))^(3/2)-16/15*e^3*(d*x+c)^4/b^2/d/(a+b*arccosh(d*x+c))^(3/2)+16/15*
e^3*exp(4*a/b)*erf(2*(a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d+16/15*e^3*erfi(2*(a+b*arccosh(d*x+
c))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/d/exp(4*a/b)+4/15*e^3*exp(2*a/b)*erf(2^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/b^
(1/2))*2^(1/2)*Pi^(1/2)/b^(7/2)/d+4/15*e^3*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^3*(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)/b/d/(a+b*arccosh(d*x+c))^(5/2)+16/5*e^3*
(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)-128/15*e^3*(d*x+c)^3*(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 = 0.92, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5996, 12, 5886, 5951, 5885, 3388, 2211, 2236, 2235} \begin {gather*} \frac {16 \sqrt {\pi } e^3 e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 \sqrt {2 \pi } e^3 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 {16 \sqrt {\pi } e^3 e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 \sqrt {2 \pi } e^3 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 {128 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(-2*e^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(5*b*d*(a + b*ArcCosh[c + d*x])^(5/2)) + (4*e^3*(c +
d*x)^2)/(5*b^2*d*(a + b*ArcCosh[c + d*x])^(3/2)) - (16*e^3*(c + d*x)^4)/(15*b^2*d*(a + b*ArcCosh[c + d*x])^(3
/2)) + (16*e^3*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(5*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) - (128*e
^3*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])/(15*b^3*d*Sqrt[a + b*ArcCosh[c + d*x]]) + (16*e^3*E^((4*a
)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (4*e^3*E^((2*a)/b)*Sqrt[2*Pi]*Er
f[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d) + (16*e^3*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcCos
h[c + d*x]])/Sqrt[b]])/(15*b^(7/2)*d*E^((4*a)/b)) + (4*e^3*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 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3388

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 5885

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^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 5886

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 5951

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*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 +
e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], x] - Dist[f*(m/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]
]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], 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] && EqQ[e2, (-c)*d2] && LtQ[n, -1]

Rule 5996

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)^3}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac {\left (6 e^3\right ) \text {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 {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\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^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^3\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (128 e^3\right ) \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {a+b x}}-\frac {\cosh (4 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^3\right ) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}-\frac {\left (16 e^3\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (32 e^3\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^3 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 )}{5 b^{7/2} d}-\frac {4 e^3 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 )}{5 b^{7/2} d}+\frac {\left (64 e^3\right ) \text {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (64 e^3\right ) \text {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 {\left (64 e^3\right ) \text {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 {\left (64 e^3\right ) \text {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^3 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 {16 e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^3 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 [A]
time = 2.45, size = 445, normalized size = 1.01 \begin {gather*} \frac {e^3 \left (-4 e^{-4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (16 b e^{4 \cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 a}{b}} \left (b+8 a \left (-1+e^{8 \cosh ^{-1}(c+d x)}\right )-8 b \cosh ^{-1}(c+d x)+b e^{8 \cosh ^{-1}(c+d x)} \left (1+8 \cosh ^{-1}(c+d x)\right )+16 e^{4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )\right )-2 \left (\left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 e^{-2 \cosh ^{-1}(c+d x)} \left (b+4 a \left (-1+e^{4 \cosh ^{-1}(c+d x)}\right )-4 b \cosh ^{-1}(c+d x)+b e^{4 \cosh ^{-1}(c+d x)} \left (1+4 \cosh ^{-1}(c+d x)\right )\right )+8 \sqrt {2} b e^{-\frac {2 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+8 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )+3 b^2 \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )-3 b^2 \sinh \left (4 \cosh ^{-1}(c+d x)\right )\right )}{60 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3/(a + b*ArcCosh[c + d*x])^(7/2),x]

[Out]

(e^3*((-4*(a + b*ArcCosh[c + d*x])*(16*b*E^(4*ArcCosh[c + d*x])*(-((a + b*ArcCosh[c + d*x])/b))^(3/2)*Gamma[1/
2, (-4*(a + b*ArcCosh[c + d*x]))/b] + E^((4*a)/b)*(b + 8*a*(-1 + E^(8*ArcCosh[c + d*x])) - 8*b*ArcCosh[c + d*x
] + b*E^(8*ArcCosh[c + d*x])*(1 + 8*ArcCosh[c + d*x]) + 16*E^(4*(a/b + ArcCosh[c + d*x]))*Sqrt[a/b + ArcCosh[c
+ d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (4*(a + b*ArcCosh[c + d*x]))/b])))/E^(4*(a/b + ArcCosh[c + d*x]))
- 2*((a + b*ArcCosh[c + d*x])*((2*(b + 4*a*(-1 + E^(4*ArcCosh[c + d*x])) - 4*b*ArcCosh[c + d*x] + b*E^(4*ArcC
osh[c + d*x])*(1 + 4*ArcCosh[c + d*x])))/E^(2*ArcCosh[c + d*x]) + (8*Sqrt[2]*b*(-((a + b*ArcCosh[c + d*x])/b))
^(3/2)*Gamma[1/2, (-2*(a + b*ArcCosh[c + d*x]))/b])/E^((2*a)/b) + 8*Sqrt[2]*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c +
d*x]]*(a + b*ArcCosh[c + d*x])*Gamma[1/2, (2*(a + b*ArcCosh[c + d*x]))/b]) + 3*b^2*Sinh[2*ArcCosh[c + d*x]])
- 3*b^2*Sinh[4*ArcCosh[c + d*x]]))/(60*b^3*d*(a + b*ArcCosh[c + d*x])^(5/2))

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (d e x +c e \right )^{3}}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^(7/2),x)

[Out]

int((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^(7/2),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)^3/(b*arccosh(d*x + c) + a)^(7/2), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{3} \left (\int \frac {c^{3}}{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^{3} x^{3}}{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 {3 c d^{2} x^{2}}{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 {3 c^{2} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3/(a+b*acosh(d*x+c))**(7/2),x)

[Out]

e**3*(Integral(c**3/(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) +
Integral(d**3*x**3/(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) +
Integral(3*c*d**2*x**2/(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) + Integral(3*c**2*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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3/(a+b*arccosh(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3/(b*arccosh(d*x + c) + a)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^3/(a + b*acosh(c + d*x))^(7/2),x)

[Out]

int((c*e + d*e*x)^3/(a + b*acosh(c + d*x))^(7/2), x)

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