Integrand size = 23, antiderivative size = 266 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b d (a+b \text {arccosh}(c+d x))^{5/2}}+\frac {4 e}{15 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {8 e (c+d x)^2}{15 b^2 d (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {32 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {8 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {8 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d} \]
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*arccos h(d*x+c))^(3/2)+8/15*e*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+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)
Leaf count is larger than twice the leaf count of optimal. \(916\) vs. \(2(266)=532\).
Time = 3.06 (sec) , antiderivative size = 916, normalized size of antiderivative = 3.44 \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\frac {e \left (4 a b^{3/2} c (c+d x)+8 a^2 \sqrt {b} c \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)+4 b^{5/2} c (c+d x) \text {arccosh}(c+d x)+16 a b^{3/2} c \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)+8 b^{5/2} c \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)^2-4 a b^{3/2} \cosh (2 \text {arccosh}(c+d x))-4 b^{5/2} \text {arccosh}(c+d x) \cosh (2 \text {arccosh}(c+d x))-4 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{5/2} \cosh \left (\frac {a}{b}\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{5/2} \cosh \left (\frac {2 a}{b}\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-4 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{5/2} \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{5/2} \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-2 \sqrt {b} c e^{-\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \left (-2 a+b-2 b \text {arccosh}(c+d x)+2 e^{\frac {a}{b}+\text {arccosh}(c+d x)} \sqrt {\frac {a}{b}+\text {arccosh}(c+d x)} (a+b \text {arccosh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arccosh}(c+d x)\right )\right )-2 \sqrt {b} c e^{-\frac {a}{b}} (a+b \text {arccosh}(c+d x)) \left (e^{\frac {a}{b}+\text {arccosh}(c+d x)} (2 a+b+2 b \text {arccosh}(c+d x))+2 b \left (-\frac {a+b \text {arccosh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arccosh}(c+d x)}{b}\right )\right )-4 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{5/2} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+4 c \sqrt {\pi } (a+b \text {arccosh}(c+d x))^{5/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+8 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{5/2} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-8 \sqrt {2 \pi } (a+b \text {arccosh}(c+d x))^{5/2} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-16 a^2 \sqrt {b} \sinh (2 \text {arccosh}(c+d x))-3 b^{5/2} \sinh (2 \text {arccosh}(c+d x))-32 a b^{3/2} \text {arccosh}(c+d x) \sinh (2 \text {arccosh}(c+d x))-16 b^{5/2} \text {arccosh}(c+d x)^2 \sinh (2 \text {arccosh}(c+d x))\right )}{15 b^{7/2} d (a+b \text {arccosh}(c+d x))^{5/2}} \]
(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]*C osh[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*ArcC osh[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[S qrt[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[(Sqrt[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*ArcCos h[c + d*x])*Gamma[1/2, a/b + ArcCosh[c + d*x]]))/E^ArcCosh[c + d*x] - (2*S qrt[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*S qrt[Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Erfi[Sqrt[a + b*ArcCosh[c + d*x]]/S qrt[b]]*Sinh[a/b] + 8*Sqrt[2*Pi]*(a + b*ArcCosh[c + d*x])^(5/2)*Erf[(Sq...
Time = 1.83 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.02, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6411, 27, 6301, 6308, 6366, 6300, 25, 3042, 3788, 26, 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.
\(\displaystyle \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {e (c+d x)}{(a+b \text {arccosh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \int \frac {c+d x}{(a+b \text {arccosh}(c+d x))^{7/2}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6301 |
\(\displaystyle \frac {e \left (-\frac {2 \int \frac {1}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{5 b}+\frac {4 \int \frac {(c+d x)^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{5 b}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6308 |
\(\displaystyle \frac {e \left (\frac {4 \int \frac {(c+d x)^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^{5/2}}d(c+d x)}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6366 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \int \frac {c+d x}{(a+b \text {arccosh}(c+d x))^{3/2}}d(c+d x)}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 6300 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (-\frac {2 \int -\frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (\frac {2 \int \frac {\cosh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}-\frac {2 \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {e \left (\frac {4 \left (-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 \left (-\frac {2 \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1}}{b \sqrt {a+b \text {arccosh}(c+d x)}}+\frac {2 \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arccosh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))}{b^2}\right )}{3 b}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 3788 |
\(\displaystyle \frac {e \left (\frac {4 \left (-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}+\frac {4 \left (-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}-\frac {2 \left (\frac {1}{2} i \int \frac {i e^{-\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} i \int -\frac {i e^{\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b^2}\right )}{3 b}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (-\frac {2 \left (-\frac {1}{2} \int \frac {e^{-\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))-\frac {1}{2} \int \frac {e^{\frac {2 (a-c-d x)}{b}}}{\sqrt {a+b \text {arccosh}(c+d x)}}d(a+b \text {arccosh}(c+d x))\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2611 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (-\frac {2 \left (-\int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\int e^{\frac {2 (a+b \text {arccosh}(c+d x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (-\frac {2 \left (-\int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arccosh}(c+d x))}{b}}d\sqrt {a+b \text {arccosh}(c+d x)}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {e \left (\frac {4 \left (\frac {4 \left (-\frac {2 \left (-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arccosh}(c+d x)}}{\sqrt {b}}\right )\right )}{b^2}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b \sqrt {a+b \text {arccosh}(c+d x)}}\right )}{3 b}-\frac {2 (c+d x)^2}{3 b (a+b \text {arccosh}(c+d x))^{3/2}}\right )}{5 b}+\frac {4}{15 b^2 (a+b \text {arccosh}(c+d x))^{3/2}}-\frac {2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b (a+b \text {arccosh}(c+d x))^{5/2}}\right )}{d}\) |
(e*((-2*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x])/(5*b*(a + b*ArcCos h[c + d*x])^(5/2)) + 4/(15*b^2*(a + b*ArcCosh[c + d*x])^(3/2)) + (4*((-2*( c + d*x)^2)/(3*b*(a + b*ArcCosh[c + d*x])^(3/2)) + (4*((-2*Sqrt[-1 + c + d *x]*(c + d*x)*Sqrt[1 + c + d*x])/(b*Sqrt[a + b*ArcCosh[c + d*x]]) - (2*(-1 /2*(Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d*x ]])/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcCosh[c + d *x]])/Sqrt[b]])/(2*E^((2*a)/b))))/b^2))/(3*b)))/(5*b)))/d
3.2.95.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [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]
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] + Simp[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] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
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] + (-Simp[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] + Simp[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]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -1]
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] - Simp[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, e 1, d2, e2, f, m}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && LtQ[n, -1]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[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]
\[\int \frac {d e x +c e}{\left (a +b \,\operatorname {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {c e+d e x}{(a+b \text {arccosh}(c+d x))^{7/2}} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]