Integrand size = 23, antiderivative size = 118 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\frac {(a+b \text {arccosh}(c+d x))^3}{3 b d e}+\frac {(a+b \text {arccosh}(c+d x))^2 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )}{d e}-\frac {b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )}{d e}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(c+d x)}\right )}{2 d e} \]
1/3*(a+b*arccosh(d*x+c))^3/b/d/e+(a+b*arccosh(d*x+c))^2*ln(1+1/(d*x+c+(d*x +c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e-b*(a+b*arccosh(d*x+c))*polylog(2,-1/(d *x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e-1/2*b^2*polylog(3,-1/(d*x+c+( d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)/d/e
Time = 0.47 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\frac {a b \text {arccosh}(c+d x)^2+\frac {1}{3} b^2 \text {arccosh}(c+d x)^3+2 a b \text {arccosh}(c+d x) \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+b^2 \text {arccosh}(c+d x)^2 \log \left (1+e^{-2 \text {arccosh}(c+d x)}\right )+a^2 \log (c+d x)-b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(c+d x)}\right )-\frac {1}{2} b^2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arccosh}(c+d x)}\right )}{d e} \]
(a*b*ArcCosh[c + d*x]^2 + (b^2*ArcCosh[c + d*x]^3)/3 + 2*a*b*ArcCosh[c + d *x]*Log[1 + E^(-2*ArcCosh[c + d*x])] + b^2*ArcCosh[c + d*x]^2*Log[1 + E^(- 2*ArcCosh[c + d*x])] + a^2*Log[c + d*x] - b*(a + b*ArcCosh[c + d*x])*PolyL og[2, -E^(-2*ArcCosh[c + d*x])] - (b^2*PolyLog[3, -E^(-2*ArcCosh[c + d*x]) ])/2)/(d*e)
Result contains complex when optimal does not.
Time = 0.71 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6411, 27, 6297, 25, 3042, 26, 4201, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^2}{e (c+d x)}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arccosh}(c+d x))^2}{c+d x}d(c+d x)}{d e}\) |
| \(\Big \downarrow \) 6297 |
| \(\displaystyle \frac {\int -(a+b \text {arccosh}(c+d x))^2 \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (a+b \text {arccosh}(c+d x))^2 \tanh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c+d x)}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int -i (a+b \text {arccosh}(c+d x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \int (a+b \text {arccosh}(c+d x))^2 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c+d x))}{b}\right )d(a+b \text {arccosh}(c+d x))}{b d e}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {i \left (2 i \int \frac {e^{\frac {2 (a-c-d x)}{b}} (a+b \text {arccosh}(c+d x))^2}{1+e^{\frac {2 (a-c-d x)}{b}}}d(a+b \text {arccosh}(c+d x))-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )}{b d e}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {i \left (2 i \left (b \int (a+b \text {arccosh}(c+d x)) \log \left (1+e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )}{b d e}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )-\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )d(a+b \text {arccosh}(c+d x))\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )}{b d e}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {i \left (2 i \left (b \left (\frac {1}{4} b^2 \int e^{-\frac {2 (a-c-d x)}{b}} \operatorname {PolyLog}(2,-c-d x)de^{\frac {2 (a-c-d x)}{b}}+\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )}{b d e}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {i \left (2 i \left (b \left (\frac {1}{2} b (a+b \text {arccosh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\frac {2 (a-c-d x)}{b}}\right )+\frac {1}{4} b^2 \operatorname {PolyLog}(3,-c-d x)\right )-\frac {1}{2} b (a+b \text {arccosh}(c+d x))^2 \log \left (e^{\frac {2 (a-c-d x)}{b}}+1\right )\right )-\frac {1}{3} i (a+b \text {arccosh}(c+d x))^3\right )}{b d e}\) |
(I*((-1/3*I)*(a + b*ArcCosh[c + d*x])^3 + (2*I)*(-1/2*(b*(a + b*ArcCosh[c + d*x])^2*Log[1 + E^((2*(a - c - d*x))/b)]) + b*((b*(a + b*ArcCosh[c + d*x ])*PolyLog[2, -E^((2*(a - c - d*x))/b)])/2 + (b^2*PolyLog[3, -c - d*x])/4) )))/(b*d*e)
3.2.9.3.1 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[(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_))))^(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[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.67 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.86
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{3}+\operatorname {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}}{d}\) | \(220\) |
| default | \(\frac {\frac {a^{2} \ln \left (d x +c \right )}{e}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{3}+\operatorname {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e}}{d}\) | \(220\) |
| parts | \(\frac {a^{2} \ln \left (d x +c \right )}{e d}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{3}}{3}+\operatorname {arccosh}\left (d x +c \right )^{2} \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\operatorname {arccosh}\left (d x +c \right ) \operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e d}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )^{2}}{2}+\operatorname {arccosh}\left (d x +c \right ) \ln \left (1+\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )^{2}\right )}{2}\right )}{e d}\) | \(225\) |
1/d*(a^2/e*ln(d*x+c)+b^2/e*(-1/3*arccosh(d*x+c)^3+arccosh(d*x+c)^2*ln(1+(d *x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+arccosh(d*x+c)*polylog(2,-(d*x+c+ (d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)-1/2*polylog(3,-(d*x+c+(d*x+c-1)^(1/2)* (d*x+c+1)^(1/2))^2))+2*a*b/e*(-1/2*arccosh(d*x+c)^2+arccosh(d*x+c)*ln(1+(d *x+c+(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2))^2)+1/2*polylog(2,-(d*x+c+(d*x+c-1)^( 1/2)*(d*x+c+1)^(1/2))^2)))
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\frac {\int \frac {a^{2}}{c + d x}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]
(Integral(a**2/(c + d*x), x) + Integral(b**2*acosh(c + d*x)**2/(c + d*x), x) + Integral(2*a*b*acosh(c + d*x)/(c + d*x), x))/e
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
a^2*log(d*e*x + c*e)/(d*e) + integrate(b^2*log(d*x + sqrt(d*x + c + 1)*sqr t(d*x + c - 1) + c)^2/(d*e*x + c*e) + 2*a*b*log(d*x + sqrt(d*x + c + 1)*sq rt(d*x + c - 1) + c)/(d*e*x + c*e), x)
\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{d e x + c e} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2}{c\,e+d\,e\,x} \,d x \]