Integrand size = 23, antiderivative size = 100 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=-\frac {(a+b \text {arcsinh}(c+d x))^2}{d e^2 (c+d x)}-\frac {4 b (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {2 b^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {2 b^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \]
-(a+b*arcsinh(d*x+c))^2/d/e^2/(d*x+c)-4*b*(a+b*arcsinh(d*x+c))*arctanh(d*x +c+(1+(d*x+c)^2)^(1/2))/d/e^2-2*b^2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))/ d/e^2+2*b^2*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2
Time = 0.55 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.64 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=\frac {-\frac {a^2}{c+d x}-2 a b \left (\frac {\text {arcsinh}(c+d x)}{c+d x}+\log \left (\frac {1}{2} (c+d x) \text {csch}\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )+b^2 \left (\text {arcsinh}(c+d x) \left (-\frac {\text {arcsinh}(c+d x)}{c+d x}+2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )\right )}{d e^2} \]
(-(a^2/(c + d*x)) - 2*a*b*(ArcSinh[c + d*x]/(c + d*x) + Log[((c + d*x)*Csc h[ArcSinh[c + d*x]/2])/2] - Log[Sinh[ArcSinh[c + d*x]/2]]) + b^2*(ArcSinh[ c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSinh[c + d*x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSinh[c + d*x]) ] - 2*PolyLog[2, E^(-ArcSinh[c + d*x])]))/(d*e^2)
Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.88, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6274, 27, 6191, 6231, 3042, 26, 4670, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^2}{e^2 (c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x)^2}d(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {2 b \int \frac {a+b \text {arcsinh}(c+d x)}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {2 b \int \frac {a+b \text {arcsinh}(c+d x)}{c+d x}d\text {arcsinh}(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 b \int i (a+b \text {arcsinh}(c+d x)) \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \int (a+b \text {arcsinh}(c+d x)) \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (i b \int \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-i b \int \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )}{d e^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (-i b \int e^{-\text {arcsinh}(c+d x)} \log \left (1+e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}+i b \int e^{-\text {arcsinh}(c+d x)} \log (-c-d x+1)de^{\text {arcsinh}(c+d x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )}{d e^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{c+d x}+2 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )}{d e^2}\) |
(-((a + b*ArcSinh[c + d*x])^2/(c + d*x)) + (2*I)*b*((2*I)*(a + b*ArcSinh[c + d*x])*ArcTanh[E^ArcSinh[c + d*x]] - I*b*PolyLog[2, E^ArcSinh[c + d*x]] + I*b*PolyLog[2, -c - d*x]))/(d*e^2)
3.2.33.3.1 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[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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Int[((a_.) + ArcSinh[(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* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.46 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.85
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(185\) |
| default | \(\frac {-\frac {a^{2}}{e^{2} \left (d x +c \right )}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(185\) |
| parts | \(-\frac {a^{2}}{e^{2} \left (d x +c \right ) d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\) | \(190\) |
1/d*(-a^2/e^2/(d*x+c)+b^2/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^2-2*arcsinh(d*x+c )*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))-2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+ 2*arcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+2*polylog(2,d*x+c+(1+(d*x +c)^2)^(1/2)))+2*a*b/e^2*(-1/(d*x+c)*arcsinh(d*x+c)-arctanh(1/(1+(d*x+c)^2 )^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
integral((b^2*arcsinh(d*x + c)^2 + 2*a*b*arcsinh(d*x + c) + a^2)/(d^2*e^2* x^2 + 2*c*d*e^2*x + c^2*e^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{2}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
(Integral(a**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**2*asinh(c + d*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(2*a*b*asinh(c + d*x)/( c**2 + 2*c*d*x + d**2*x**2), x))/e**2
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]
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>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]