Integrand size = 26, antiderivative size = 319 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b (a+b \text {arccosh}(c x))}{3 c d^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {2 x (a+b \text {arccosh}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \] Output:
-1/3*b^2*x/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*(a+b*arccosh(c*x))/c/d^2/(c*x-1) ^(1/2)/(c*x+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*x*(a+b*arccosh(c*x))^2/d/(-c ^2*d*x^2+d)^(3/2)+2/3*x*(a+b*arccosh(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)+2/3* (c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^2/c/d^2/(-c^2*d*x^2+d)^(1/2 )-4/3*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^( 1/2)*(c*x+1)^(1/2))^2)/c/d^2/(-c^2*d*x^2+d)^(1/2)-2/3*b^2*(c*x-1)^(1/2)*(c *x+1)^(1/2)*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/c/d^2/(-c^2*d*x ^2+d)^(1/2)
Time = 0.93 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {\frac {a^2 c x \left (-3+2 c^2 x^2\right )}{-1+c^2 x^2}+a b \left (2 c x \left (2+\frac {1}{1-c^2 x^2}\right ) \text {arccosh}(c x)+\frac {\sqrt {\frac {-1+c x}{1+c x}} \left (-1+\left (4-4 c^2 x^2\right ) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )}{-1+c x}\right )+b^2 \left (-\frac {\text {arccosh}(c x) \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)+c x \text {arccosh}(c x)\right )}{-1+c^2 x^2}+c x \left (-1+2 \text {arccosh}(c x)^2\right )-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \left (\text {arccosh}(c x)+2 \log \left (1-e^{-2 \text {arccosh}(c x)}\right )\right )+2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arccosh}(c x)}\right )\right )}{3 c d^2 \sqrt {d-c^2 d x^2}} \] Input:
Integrate[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(5/2),x]
Output:
((a^2*c*x*(-3 + 2*c^2*x^2))/(-1 + c^2*x^2) + a*b*(2*c*x*(2 + (1 - c^2*x^2) ^(-1))*ArcCosh[c*x] + (Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + (4 - 4*c^2*x^2)*Lo g[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)]))/(-1 + c*x)) + b^2*(-((ArcCosh[c* x]*(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) + c*x*ArcCosh[c*x]))/(-1 + c^2*x^ 2)) + c*x*(-1 + 2*ArcCosh[c*x]^2) - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) *ArcCosh[c*x]*(ArcCosh[c*x] + 2*Log[1 - E^(-2*ArcCosh[c*x])]) + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-2*ArcCosh[c*x])]))/(3*c*d^2*Sq rt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 2.20 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.87, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6316, 6314, 6327, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838, 6329, 41}
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.
\(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6316 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6314 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6327 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6328 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 4199 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6329 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {b \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 41 |
\(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {a+b \text {arccosh}(c x)}{2 c^2 \left (1-c^2 x^2\right )}-\frac {b x}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 \left (\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\right )}{3 d}+\frac {x (a+b \text {arccosh}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}\) |
Input:
Int[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^(5/2),x]
Output:
(x*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)) + (2*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/2*(b*x)/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a + b*A rcCosh[c*x])/(2*c^2*(1 - c^2*x^2))))/(3*d^2*Sqrt[d - c^2*d*x^2]) + (2*((x* (a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) + ((2*I)*b*Sqrt[-1 + c*x]* Sqrt[1 + c*x]*(((-1/2*I)*(a + b*ArcCosh[c*x])^2)/b - (2*I)*(-1/2*((a + b*A rcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])]) - (b*PolyLog[2, E^(2*ArcCosh[c*x ])])/4)))/(c*d*Sqrt[d - c^2*d*x^2])))/(3*d)
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[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ .)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp [2*I Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcCosh[c*x])^n/(d*Sqrt[d + e*x^2])), x] + Simp [b*c*(n/d)*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])] Int[x*((a + b*ArcCosh[c*x])^(n - 1)/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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, d, e}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[1/e Subst[Int[(a + b*x)^n*Coth[x], x], x, ArcCosh[c*x] ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(2434\) vs. \(2(301)=602\).
Time = 0.59 (sec) , antiderivative size = 2435, normalized size of antiderivative = 7.63
method | result | size |
default | \(\text {Expression too large to display}\) | \(2435\) |
parts | \(\text {Expression too large to display}\) | \(2435\) |
Input:
int((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
14/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^ 3*arccosh(c*x)*x^5+17/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+1 1*c^2*x^2-4)*c^2/d^3*arccosh(c*x)^2*x^3-16/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3 *c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^2/d^3*arccosh(c*x)*x^3-4/3*b^2*(-d*(c^ 2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)/c/d^3*(c*x-1)^(1/2)*(c *x+1)^(1/2)+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^ 2-4)/d^3*(c*x-1)*(c*x+1)*x-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^ 4*x^4+11*c^2*x^2-4)*c^6/d^3*arccosh(c*x)*x^7-2*b^2*(-d*(c^2*x^2-1))^(1/2)/ (3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c^4/d^3*arccosh(c*x)^2*x^5+4/3*b^2*(c* x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c/(c^2*x^2-1)*arccosh( c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c ^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c/d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh( c*x)*x^2-10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2- 4)*c^2/d^3*(c*x-1)*(c*x+1)*arccosh(c*x)*x^3-14/3*b^2*(-d*(c^2*x^2-1))^(1/2 )/(3*c^6*x^6-10*c^4*x^4+11*c^2*x^2-4)*c/d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ar ccosh(c*x)^2*x^2+4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10*c^4*x^4+11*c ^2*x^2-4)*c^4/d^3*(c*x-1)*(c*x+1)*arccosh(c*x)*x^5+4/3*b^2*(c*x+1)^(1/2)*( c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c/(c^2*x^2-1)*arccosh(c*x)*ln(1+c* x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^6*x^6-10* c^4*x^4+11*c^2*x^2-4)*c^3/d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)^...
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)
Output:
Integral((a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")
Output:
1/3*a*b*c*(sqrt(-d)/(c^4*d^3*x^2 - c^2*d^3) + 2*sqrt(-d)*log(c*x + 1)/(c^2 *d^3) + 2*sqrt(-d)*log(c*x - 1)/(c^2*d^3)) + 2/3*a*b*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arccosh(c*x) + 1/3*a^2*(2*x/(sq rt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + b^2*integrate(lo g(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d*x^2 + d)^(5/2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)^2/(-c^2*d*x^2 + d)^(5/2), x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:
int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^(5/2),x)
Output:
int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^(5/2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{2} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b +3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2}+2 a^{2} c^{2} x^{3}-3 a^{2} x}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \left (c^{2} x^{2}-1\right )} \] Input:
int((a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)
Output:
(6*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*a*b*c** 2*x**2 - 6*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*c **4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x) *a*b + 3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)**2/(sqrt( - c**2*x**2 + 1)* c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x )*b**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 1)*int(acosh(c*x)**2/(sqrt( - c** 2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2* x**2 + 1)),x)*b**2 + 2*a**2*c**2*x**3 - 3*a**2*x)/(3*sqrt(d)*sqrt( - c**2* x**2 + 1)*d**2*(c**2*x**2 - 1))