Integrand size = 37, antiderivative size = 260 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\frac {f \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^{1+n}}{b c (1+n) \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}+\frac {e^{-\frac {a}{b}} g \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}}-\frac {e^{a/b} g \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \text {arccosh}(c x)}{b}\right )}{2 c^2 \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \] Output:
f*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1+n)/b/c/(1+n)/(c*d1*x+d 1)^(1/2)/(-c*d2*x+d2)^(1/2)+1/2*g*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh (c*x))^n*GAMMA(1+n,-(a+b*arccosh(c*x))/b)/c^2/exp(a/b)/(c*d1*x+d1)^(1/2)/( -c*d2*x+d2)^(1/2)/((-(a+b*arccosh(c*x))/b)^n)-1/2*exp(a/b)*g*(c*x-1)^(1/2) *(c*x+1)^(1/2)*(a+b*arccosh(c*x))^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)/c^2/(c *d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2)/(((a+b*arccosh(c*x))/b)^n)
Time = 3.11 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.84 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\frac {e^{-\frac {a}{b}} \sqrt {\frac {-1+c x}{1+c x}} \sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x} (a+b \text {arccosh}(c x))^n \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^{-n} \left (-2 c e^{a/b} f (a+b \text {arccosh}(c x)) \left (-\frac {(a+b \text {arccosh}(c x))^2}{b^2}\right )^n+b e^{\frac {2 a}{b}} g (1+n) \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\text {arccosh}(c x)\right )-b g (1+n) \left (\frac {a}{b}+\text {arccosh}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{2 b c^2 \text {d1} \text {d2} (1+n) (-1+c x)} \] Input:
Integrate[((f + g*x)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]),x]
Output:
(Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x]*(a + b*Arc Cosh[c*x])^n*(-2*c*E^(a/b)*f*(a + b*ArcCosh[c*x])*(-((a + b*ArcCosh[c*x])^ 2/b^2))^n + b*E^((2*a)/b)*g*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, a/b + ArcCosh[c*x]] - b*g*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)]))/(2*b*c^2*d1*d2*E^(a/b)*(1 + n)*(-1 + c*x)*(- ((a + b*ArcCosh[c*x])^2/b^2))^n)
Time = 1.05 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.68, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6397, 6395, 3042, 3798, 2009}
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.
\(\displaystyle \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}} \, dx\) |
\(\Big \downarrow \) 6397 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
\(\Big \downarrow \) 6395 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int (c f+c g x) (a+b \text {arccosh}(c x))^nd\text {arccosh}(c x)}{c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))^n \left (c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )\right )d\text {arccosh}(c x)}{c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
\(\Big \downarrow \) 3798 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (c f (a+b \text {arccosh}(c x))^n+c g x (a+b \text {arccosh}(c x))^n\right )d\text {arccosh}(c x)}{c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f (a+b \text {arccosh}(c x))^{n+1}}{b (n+1)}+\frac {1}{2} g e^{-\frac {a}{b}} (a+b \text {arccosh}(c x))^n \left (-\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{2} g e^{a/b} (a+b \text {arccosh}(c x))^n \left (\frac {a+b \text {arccosh}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \text {arccosh}(c x)}{b}\right )\right )}{c^2 \sqrt {c \text {d1} x+\text {d1}} \sqrt {\text {d2}-c \text {d2} x}}\) |
Input:
Int[((f + g*x)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x ]),x]
Output:
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((c*f*(a + b*ArcCosh[c*x])^(1 + n))/(b*(1 + n)) + (g*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/( 2*E^(a/b)*(-((a + b*ArcCosh[c*x])/b))^n) - (E^(a/b)*g*(a + b*ArcCosh[c*x]) ^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(2*((a + b*ArcCosh[c*x])/b)^n)))/ (c^2*Sqrt[d1 + c*d1*x]*Sqrt[d2 - c*d2*x])
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/( Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[1/( c^(m + 1)*Sqrt[(-d1)*d2]) Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ [e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[((-d1 )*d2)^IntPart[p]*(d1 + e1*x)^FracPart[p]*((d2 + e2*x)^FracPart[p]/((-1 + c* x)^FracPart[p]*(1 + c*x)^FracPart[p])) Int[(f + g*x)^m*(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && Inte gerQ[p - 1/2] && !(GtQ[d1, 0] && LtQ[d2, 0])
\[\int \frac {\left (g x +f \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{n}}{\sqrt {c \operatorname {d1} x +\operatorname {d1}}\, \sqrt {-c \operatorname {d2} x +\operatorname {d2}}}d x\]
Input:
int((g*x+f)*(a+b*arccosh(c*x))^n/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x)
Output:
int((g*x+f)*(a+b*arccosh(c*x))^n/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}}} \,d x } \] Input:
integrate((g*x+f)*(a+b*arccosh(c*x))^n/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2 ),x, algorithm="fricas")
Output:
integral(-sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)*(g*x + f)*(b*arccosh(c*x) + a)^n/(c^2*d1*d2*x^2 - d1*d2), x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n} \left (f + g x\right )}{\sqrt {d_{1} \left (c x + 1\right )} \sqrt {- d_{2} \left (c x - 1\right )}}\, dx \] Input:
integrate((g*x+f)*(a+b*acosh(c*x))**n/(c*d1*x+d1)**(1/2)/(-c*d2*x+d2)**(1/ 2),x)
Output:
Integral((a + b*acosh(c*x))**n*(f + g*x)/(sqrt(d1*(c*x + 1))*sqrt(-d2*(c*x - 1))), x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}}} \,d x } \] Input:
integrate((g*x+f)*(a+b*arccosh(c*x))^n/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2 ),x, algorithm="maxima")
Output:
integrate((g*x + f)*(b*arccosh(c*x) + a)^n/(sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)), x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {c d_{1} x + d_{1}} \sqrt {-c d_{2} x + d_{2}}} \,d x } \] Input:
integrate((g*x+f)*(a+b*arccosh(c*x))^n/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2 ),x, algorithm="giac")
Output:
integrate((g*x + f)*(b*arccosh(c*x) + a)^n/(sqrt(c*d1*x + d1)*sqrt(-c*d2*x + d2)), x)
Timed out. \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {d_{1}+c\,d_{1}\,x}\,\sqrt {d_{2}-c\,d_{2}\,x}} \,d x \] Input:
int(((f + g*x)*(a + b*acosh(c*x))^n)/((d1 + c*d1*x)^(1/2)*(d2 - c*d2*x)^(1 /2)),x)
Output:
int(((f + g*x)*(a + b*acosh(c*x))^n)/((d1 + c*d1*x)^(1/2)*(d2 - c*d2*x)^(1 /2)), x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))^n}{\sqrt {\text {d1}+c \text {d1} x} \sqrt {\text {d2}-c \text {d2} x}} \, dx=\frac {\left (\int \frac {\left (\mathit {acosh} \left (c x \right ) b +a \right )^{n}}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) f +\left (\int \frac {\left (\mathit {acosh} \left (c x \right ) b +a \right )^{n} x}{\sqrt {c x +1}\, \sqrt {-c x +1}}d x \right ) g}{\sqrt {\mathit {d2}}\, \sqrt {\mathit {d1}}} \] Input:
int((g*x+f)*(a+b*acosh(c*x))^n/(c*d1*x+d1)^(1/2)/(-c*d2*x+d2)^(1/2),x)
Output:
(int((acosh(c*x)*b + a)**n/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*f + int(((a cosh(c*x)*b + a)**n*x)/(sqrt(c*x + 1)*sqrt( - c*x + 1)),x)*g)/(sqrt(d2)*sq rt(d1))