Integrand size = 14, antiderivative size = 230 \[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=-\frac {105 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \text {arccosh}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) (a+b \text {arccosh}(c+d x))^{3/2}}{4 d}-\frac {7 b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^{5/2}}{2 d}+\frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}}{d}-\frac {105 b^{7/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {105 b^{7/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{32 d} \]
35/4*b^2*(d*x+c)*(a+b*arccosh(d*x+c))^(3/2)/d+(d*x+c)*(a+b*arccosh(d*x+c)) ^(7/2)/d-105/32*b^(7/2)*exp(a/b)*erf((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*P i^(1/2)/d+105/32*b^(7/2)*erfi((a+b*arccosh(d*x+c))^(1/2)/b^(1/2))*Pi^(1/2) /d/exp(a/b)-7/2*b*(a+b*arccosh(d*x+c))^(5/2)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/ 2)/d-105/8*b^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^(1/2)/ d
Leaf count is larger than twice the leaf count of optimal. \(748\) vs. \(2(230)=460\).
Time = 5.06 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.25 \[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=\frac {-4 b \sqrt {a+b \text {arccosh}(c+d x)} \left (-2 b (c+d x) \left (-10 a+35 b \text {arccosh}(c+d x)+4 b \text {arccosh}(c+d x)^3\right )+\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \left (4 a^2-4 a b \text {arccosh}(c+d x)+7 b^2 \left (15+4 \text {arccosh}(c+d x)^2\right )\right )\right )+16 a^3 e^{-\frac {a}{b}} \sqrt {a+b \text {arccosh}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )}{\sqrt {\frac {a}{b}+\text {arccosh}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \text {arccosh}(c+d x)}{b}}}\right )+\sqrt {b} \left (8 a^3+36 a^2 b+90 a b^2+105 b^3\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\sqrt {b} \left (-8 a^3+36 a^2 b-90 a b^2+105 b^3\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+12 a^2 b \left (-12 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+8 (c+d x) \text {arccosh}(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}+\frac {(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}+\frac {(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )}{\sqrt {b}}\right )+6 a \left (4 b \sqrt {a+b \text {arccosh}(c+d x)} \left (2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) (a-5 b \text {arccosh}(c+d x))+b (c+d x) \left (15+4 \text {arccosh}(c+d x)^2\right )\right )-\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )-\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{32 d} \]
(-4*b*Sqrt[a + b*ArcCosh[c + d*x]]*(-2*b*(c + d*x)*(-10*a + 35*b*ArcCosh[c + d*x] + 4*b*ArcCosh[c + d*x]^3) + Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(4*a^2 - 4*a*b*ArcCosh[c + d*x] + 7*b^2*(15 + 4*ArcCosh[c + d*x ]^2))) + (16*a^3*Sqrt[a + b*ArcCosh[c + d*x]]*((E^((2*a)/b)*Gamma[3/2, a/b + ArcCosh[c + d*x]])/Sqrt[a/b + ArcCosh[c + d*x]] + Gamma[3/2, -((a + b*A rcCosh[c + d*x])/b)]/Sqrt[-((a + b*ArcCosh[c + d*x])/b)]))/E^(a/b) + Sqrt[ b]*(8*a^3 + 36*a^2*b + 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCos h[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) - Sqrt[b]*(-8*a^3 + 36*a^2*b - 90*a*b^2 + 105*b^3)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*( Cosh[a/b] + Sinh[a/b]) + 12*a^2*b*(-12*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]* (1 + c + d*x)*Sqrt[a + b*ArcCosh[c + d*x]] + 8*(c + d*x)*ArcCosh[c + d*x]* Sqrt[a + b*ArcCosh[c + d*x]] + ((2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCo sh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]))/Sqrt[b] + ((2*a - 3*b)*Sqrt [Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b]))/Sq rt[b]) + 6*a*(4*b*Sqrt[a + b*ArcCosh[c + d*x]]*(2*Sqrt[(-1 + c + d*x)/(1 + c + d*x)]*(1 + c + d*x)*(a - 5*b*ArcCosh[c + d*x]) + b*(c + d*x)*(15 + 4* ArcCosh[c + d*x]^2)) - Sqrt[b]*(4*a^2 + 12*a*b + 15*b^2)*Sqrt[Pi]*Erfi[Sqr t[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) - Sqrt[b]*(4*a^ 2 - 12*a*b + 15*b^2)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c + d*x]]/Sqrt[b]]*(C osh[a/b] + Sinh[a/b])))/(32*d)
Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6410, 6294, 6330, 6294, 6330, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634}
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.
| \(\displaystyle \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx\) |
| \(\Big \downarrow \) 6410 |
| \(\displaystyle \frac {\int (a+b \text {arccosh}(c+d x))^{7/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \int \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{5/2}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \int (a+b \text {arccosh}(c+d x))^{3/2}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6294 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \int \frac {(c+d x) \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )\right )}{d}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} b \int \frac {1}{\sqrt {a+b \text {arccosh}(c+d x)}}d(c+d x)\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6296 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))+\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}+\frac {1}{2} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-i \int e^{\frac {a+b \text {arccosh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {(c+d x) (a+b \text {arccosh}(c+d x))^{7/2}-\frac {7}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}-\frac {5}{2} b \left ((c+d x) (a+b \text {arccosh}(c+d x))^{3/2}-\frac {3}{2} b \left (\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )\right )}{d}\) |
((c + d*x)*(a + b*ArcCosh[c + d*x])^(7/2) - (7*b*(Sqrt[-1 + c + d*x]*Sqrt[ 1 + c + d*x]*(a + b*ArcCosh[c + d*x])^(5/2) - (5*b*((c + d*x)*(a + b*ArcCo sh[c + d*x])^(3/2) - (3*b*(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]*Sqrt[a + b *ArcCosh[c + d*x]] - (I/2)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b* ArcCosh[c + d*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCo sh[c + d*x]]/Sqrt[b]])/E^(a/b))))/2))/2))/2)/d
3.2.72.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
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] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcCosh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt [1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d , n}, x]
\[\int \left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}d x\]
Exception generated. \[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
\[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (a+b \text {arccosh}(c+d x))^{7/2} \, dx=\int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]