Integrand size = 18, antiderivative size = 279 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}+\frac {2 b c (a+b \text {arccosh}(c x)) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b c (a+b \text {arccosh}(c x)) \log \left (1+\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \sqrt {c^2 d^2-e^2}} \]
-(a+b*arccosh(c*x))^2/e/(e*x+d)+2*b*c*(a+b*arccosh(c*x))*ln(1+e*(c*x+(c*x- 1)^(1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)-2 *b*c*(a+b*arccosh(c*x))*ln(1+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2 *d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)+2*b^2*c*polylog(2,-e*(c*x+(c*x-1)^ (1/2)*(c*x+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e/(c^2*d^2-e^2)^(1/2)-2*b^ 2*c*polylog(2,-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2 )))/e/(c^2*d^2-e^2)^(1/2)
Result contains complex when optimal does not.
Time = 3.64 (sec) , antiderivative size = 950, normalized size of antiderivative = 3.41 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=-\frac {\frac {a^2}{d+e x}+2 a b c \left (\frac {\text {arccosh}(c x)}{c d+c e x}+\frac {2 \arctan \left (\frac {\sqrt {c d-e} \sqrt {\frac {-1+c x}{1+c x}}}{\sqrt {-c d-e}}\right )}{\sqrt {-c d-e} \sqrt {c d-e}}\right )+b^2 c \left (\frac {\text {arccosh}(c x)^2}{c d+c e x}+\frac {2 \left (2 \text {arccosh}(c x) \arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )-2 i \arccos \left (-\frac {c d}{e}\right ) \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\left (\arccos \left (-\frac {c d}{e}\right )+2 \left (\arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{-\frac {1}{2} \text {arccosh}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c (d+e x)}}\right )+\left (\arccos \left (-\frac {c d}{e}\right )-2 \left (\arctan \left (\frac {(c d+e) \coth \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{\frac {1}{2} \text {arccosh}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c (d+e x)}}\right )-\left (\arccos \left (-\frac {c d}{e}\right )+2 \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {-c^2 d^2+e^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\left (\arccos \left (-\frac {c d}{e}\right )-2 \arctan \left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {-c^2 d^2+e^2}\right ) \left (1+\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c d-i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c d+i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}\right )\right )\right )}{\sqrt {-c^2 d^2+e^2}}\right )}{e} \]
-((a^2/(d + e*x) + 2*a*b*c*(ArcCosh[c*x]/(c*d + c*e*x) + (2*ArcTan[(Sqrt[c *d - e]*Sqrt[(-1 + c*x)/(1 + c*x)])/Sqrt[-(c*d) - e]])/(Sqrt[-(c*d) - e]*S qrt[c*d - e])) + b^2*c*(ArcCosh[c*x]^2/(c*d + c*e*x) + (2*(2*ArcCosh[c*x]* ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] - (2*I)*Ar cCos[-((c*d)/e)]*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2 ) + e^2]] + (ArcCos[-((c*d)/e)] + 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2 ])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sq rt[-(c^2*d^2) + e^2]]))*Log[Sqrt[-(c^2*d^2) + e^2]/(Sqrt[2]*Sqrt[e]*E^(Arc Cosh[c*x]/2)*Sqrt[c*(d + e*x)])] + (ArcCos[-((c*d)/e)] - 2*(ArcTan[((c*d + e)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]] + ArcTan[((-(c*d) + e)*T anh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2]]))*Log[(Sqrt[-(c^2*d^2) + e^2] *E^(ArcCosh[c*x]/2))/(Sqrt[2]*Sqrt[e]*Sqrt[c*(d + e*x)])] - (ArcCos[-((c*d )/e)] + 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*d^2) + e^2 ]])*Log[((c*d + e)*(c*d - e + I*Sqrt[-(c^2*d^2) + e^2])*(-1 + Tanh[ArcCosh [c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))] - (ArcCos[-((c*d)/e)] - 2*ArcTan[((-(c*d) + e)*Tanh[ArcCosh[c*x]/2])/Sqrt[ -(c^2*d^2) + e^2]])*Log[((c*d + e)*(-(c*d) + e + I*Sqrt[-(c^2*d^2) + e^2]) *(1 + Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sqrt[-(c^2*d^2) + e^2]*Tanh[A rcCosh[c*x]/2]))] + I*(PolyLog[2, ((c*d - I*Sqrt[-(c^2*d^2) + e^2])*(c*d + e - I*Sqrt[-(c^2*d^2) + e^2]*Tanh[ArcCosh[c*x]/2]))/(e*(c*d + e + I*Sq...
Time = 1.17 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6378, 6395, 3042, 3801, 2694, 27, 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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 6378 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)}dx}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 6395 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \text {arccosh}(c x)}{c d+c e x}d\text {arccosh}(c x)}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}+\frac {2 b c \int \frac {a+b \text {arccosh}(c x)}{c d+e \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{e}\) |
| \(\Big \downarrow \) 3801 |
| \(\displaystyle \frac {4 b c \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 c e^{\text {arccosh}(c x)} d+e e^{2 \text {arccosh}(c x)}+e}d\text {arccosh}(c x)}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {4 b c \left (\frac {e \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 \left (c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 d^2-e^2}}-\frac {e \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 \left (c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4 b c \left (\frac {e \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{c d+e e^{\text {arccosh}(c x)}-\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{c d+e e^{\text {arccosh}(c x)}+\sqrt {c^2 d^2-e^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {4 b c \left (\frac {e \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}-\frac {b \int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {b \int \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )d\text {arccosh}(c x)}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {4 b c \left (\frac {e \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}-\frac {b \int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d-\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}-\frac {b \int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} e}{c d+\sqrt {c^2 d^2-e^2}}+1\right )de^{\text {arccosh}(c x)}}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {4 b c \left (\frac {e \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}-\frac {e \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e e^{\text {arccosh}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arccosh}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e}\right )}{2 \sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {(a+b \text {arccosh}(c x))^2}{e (d+e x)}\) |
-((a + b*ArcCosh[c*x])^2/(e*(d + e*x))) + (4*b*c*((e*(((a + b*ArcCosh[c*x] )*Log[1 + (e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2])])/e + (b*PolyLog[ 2, -((e*E^ArcCosh[c*x])/(c*d - Sqrt[c^2*d^2 - e^2]))])/e))/(2*Sqrt[c^2*d^2 - e^2]) - (e*(((a + b*ArcCosh[c*x])*Log[1 + (e*E^ArcCosh[c*x])/(c*d + Sqr t[c^2*d^2 - e^2])])/e + (b*PolyLog[2, -((e*E^ArcCosh[c*x])/(c*d + Sqrt[c^2 *d^2 - e^2]))])/e))/(2*Sqrt[c^2*d^2 - e^2])))/e
3.1.25.3.1 Defintions of rubi rules used
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[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2 Int[((c + d*x)^m*(E^((-I)*e + f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) *e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c , d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
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])
Time = 0.96 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.91
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} c^{2}}{\left (e c x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(532\) |
| default | \(\frac {-\frac {a^{2} c^{2}}{\left (e c x +c d \right ) e}+b^{2} c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2}}{e \left (e c x +c d \right )}+\frac {2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 \,\operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}\right )+2 a b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(532\) |
| parts | \(-\frac {a^{2}}{\left (e x +d \right ) e}+\frac {b^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )^{2} c^{2}}{e \left (e c x +c d \right )}+\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {arccosh}\left (c x \right ) \ln \left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}+\frac {2 c^{2} \operatorname {dilog}\left (\frac {-c d -e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{-c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}-\frac {2 c^{2} \operatorname {dilog}\left (\frac {c d +e \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\sqrt {c^{2} d^{2}-e^{2}}}{c d +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{e \sqrt {c^{2} d^{2}-e^{2}}}\right )}{c}-\frac {2 a b c \,\operatorname {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {2 a b c \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (-\frac {2 \left (d \,c^{2} x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}\) | \(535\) |
1/c*(-a^2*c^2/(c*e*x+c*d)/e+b^2*c^2*(-arccosh(c*x)^2/e/(c*e*x+c*d)+2/e*arc cosh(c*x)/(c^2*d^2-e^2)^(1/2)*ln((-c*d-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) +(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2)))-2/e*arccosh(c*x)/(c^2*d^ 2-e^2)^(1/2)*ln((c*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/ 2))/(c*d+(c^2*d^2-e^2)^(1/2)))+2/e/(c^2*d^2-e^2)^(1/2)*dilog((-c*d-e*(c*x+ (c*x-1)^(1/2)*(c*x+1)^(1/2))+(c^2*d^2-e^2)^(1/2))/(-c*d+(c^2*d^2-e^2)^(1/2 )))-2/e/(c^2*d^2-e^2)^(1/2)*dilog((c*d+e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)) +(c^2*d^2-e^2)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2))))+2*a*b*c^2*(-1/(c*e*x+c*d )/e*arccosh(c*x)-1/e^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(-2*(d*c^2*x-(c^2*x^2 -1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(c*e*x+c*d))/((c^2*d^2-e^2)/e^2)^ (1/2)/(c^2*x^2-1)^(1/2)))
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{(d+e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^2} \,d x \]