Integrand size = 26, antiderivative size = 198 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{c d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1-e^{2 \text {arccosh}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )}{c d \sqrt {d-c^2 d x^2}} \]
x*(a+b*arccosh(c*x))^2/d/(-c^2*d*x^2+d)^(1/2)+(a+b*arccosh(c*x))^2*(c*x-1) ^(1/2)*(c*x+1)^(1/2)/c/d/(-c^2*d*x^2+d)^(1/2)-2*b*(a+b*arccosh(c*x))*ln(1- (c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/d/(-c^2 *d*x^2+d)^(1/2)-b^2*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*(c*x-1) ^(1/2)*(c*x+1)^(1/2)/c/d/(-c^2*d*x^2+d)^(1/2)
Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.64 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))^2+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left ((a+b \text {arccosh}(c x)) \left (a+b \text {arccosh}(c x)-2 b \log \left (1-e^{\text {arccosh}(c x)}\right )-2 b \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-2 b^2 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c}}{d \sqrt {d-c^2 d x^2}} \]
(x*(a + b*ArcCosh[c*x])^2 + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[ c*x])*(a + b*ArcCosh[c*x] - 2*b*Log[1 - E^ArcCosh[c*x]] - 2*b*Log[1 + E^Ar cCosh[c*x]]) - 2*b^2*PolyLog[2, -E^ArcCosh[c*x]] - 2*b^2*PolyLog[2, E^ArcC osh[c*x]]))/c)/(d*Sqrt[d - c^2*d*x^2])
Result contains complex when optimal does not.
Time = 0.66 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.71, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {6314, 6328, 3042, 26, 4199, 25, 2620, 2715, 2838}
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6314 |
| \(\displaystyle \frac {2 b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {x (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6328 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {c x (a+b \text {arccosh}(c x))}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -i (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \tan \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (2 i \int -\frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \int \frac {e^{2 \text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{1-e^{2 \text {arccosh}(c x)}}d\text {arccosh}(c x)-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{2} b \int \log \left (1-e^{2 \text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (\frac {1}{4} b \int e^{-2 \text {arccosh}(c x)} \log \left (1-e^{2 \text {arccosh}(c x)}\right )de^{2 \text {arccosh}(c x)}-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {x (a+b \text {arccosh}(c x))^2}{d \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {c x-1} \sqrt {c x+1} \left (-2 i \left (-\frac {1}{2} \log \left (1-e^{2 \text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 \text {arccosh}(c x)}\right )\right )-\frac {i (a+b \text {arccosh}(c x))^2}{2 b}\right )}{c d \sqrt {d-c^2 d x^2}}\) |
(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*ArcCosh[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])
3.3.9.3.1 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_))))^(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_.)*(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]
Leaf count of result is larger than twice the leaf count of optimal. \(577\) vs. \(2(204)=408\).
Time = 0.98 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.92
| method | result | size |
| default | \(\frac {a^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(578\) |
| parts | \(\frac {a^{2} x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )^{2}}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right )^{2} x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}+\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(578\) |
a^2/d*x/(-c^2*d*x^2+d)^(1/2)-b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x +1)^(1/2)/d^2/c/(c^2*x^2-1)*arccosh(c*x)^2-b^2*(-d*(c^2*x^2-1))^(1/2)*arcc osh(c*x)^2/d^2/(c^2*x^2-1)*x+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c *x+1)^(1/2)/d^2/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)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2 *x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*b^2*(-d*(c^2*x^2-1)) ^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/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)*(c*x-1)^(1/2)*( c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))- 2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1) *arccosh(c*x)-2*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/(c^2*x^2-1)*x+ 2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1) *ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a ^2)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-a*b*c*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d + b^2*integrate(log(c*x + sqrt( c*x + 1)*sqrt(c*x - 1))^2/(-c^2*d*x^2 + d)^(3/2), x) + 2*a*b*x*arccosh(c*x )/(sqrt(-c^2*d*x^2 + d)*d) + a^2*x/(sqrt(-c^2*d*x^2 + d)*d)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]