Integrand size = 38, antiderivative size = 133 \[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c}-\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c} \] Output:
1/2*(a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/b/c-(a+b*arcsinh((-c*x+1 )^(1/2)/(c*x+1)^(1/2)))*ln(1-((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1)/(c* x+1))^(1/2))^2)/c-1/2*b*polylog(2,((-c*x+1)^(1/2)/(c*x+1)^(1/2)+(1+(-c*x+1 )/(c*x+1))^(1/2))^2)/c
Time = 0.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\frac {\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )-2 b \log \left (1-e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )\right )-b^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 b c} \] Input:
Integrate[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]
Output:
((a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])*(a + b*ArcSinh[Sqrt[1 - c*x] /Sqrt[1 + c*x]] - 2*b*Log[1 - E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])]) - b^2*PolyLog[2, E^(2*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])])/(2*b*c)
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.237, Rules used = {7232, 6190, 25, 3042, 26, 4201, 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.
\(\displaystyle \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{1-c^2 x^2} \, dx\) |
\(\Big \downarrow \) 7232 |
\(\displaystyle -\frac {\int \frac {\sqrt {c x+1} \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{\sqrt {1-c x}}d\frac {\sqrt {1-c x}}{\sqrt {c x+1}}}{c}\) |
\(\Big \downarrow \) 6190 |
\(\displaystyle -\frac {\int -\left (\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \coth \left (\frac {a}{b}-\frac {\sqrt {1-c x}}{b \sqrt {c x+1}}\right )\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \coth \left (\frac {a}{b}-\frac {\sqrt {1-c x}}{b \sqrt {c x+1}}\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -i \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \tan \left (\frac {i a}{b}-\frac {i \sqrt {1-c x}}{b \sqrt {c x+1}}+\frac {\pi }{2}\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i \int \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \tan \left (\frac {1}{2} \left (\frac {2 i a}{b}+\pi \right )-\frac {i \sqrt {1-c x}}{b \sqrt {c x+1}}\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b c}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi } \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{1+e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }}d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} b \int \log \left (1+\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )d\left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (1+e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {i \left (2 i \left (-\frac {1}{4} b^2 \int \frac {\sqrt {c x+1} \log \left (1+\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )\right )}{\sqrt {1-c x}}d\exp \left (\frac {2 a}{b}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{b}-i \pi \right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (1+e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}\left (2,-a-b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )-\frac {1}{2} b \left (a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right ) \log \left (1+e^{\frac {2 a}{b}-\frac {2 \sqrt {1-c x}}{b \sqrt {c x+1}}-i \pi }\right )\right )-\frac {i (1-c x)}{2 (c x+1)}\right )}{b c}\) |
Input:
Int[(a + b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]])/(1 - c^2*x^2),x]
Output:
((-I)*(((-1/2*I)*(1 - c*x))/(1 + c*x) + (2*I)*(-1/2*(b*(a + b*ArcSinh[Sqrt [1 - c*x]/Sqrt[1 + c*x]])*Log[1 + E^((2*a)/b - I*Pi - (2*Sqrt[1 - c*x])/(b *Sqrt[1 + c*x]))]) + (b^2*PolyLog[2, -a - b*ArcSinh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]])/4)))/(b*c)
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_.) + (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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Int[((a_.) + (b_.)*(F_)[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.) *(x_)]])^(n_.)/((A_.) + (C_.)*(x_)^2), x_Symbol] :> Simp[2*e*(g/(C*(e*f - d *g))) Subst[Int[(a + b*F[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ[{a, b, c, d, e, f, g, A, C, F}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0] && IGtQ[n, 0]
Time = 0.76 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.94
method | result | size |
default | \(-\frac {a \ln \left (x c -1\right )}{2 c}+\frac {a \ln \left (x c +1\right )}{2 c}-b \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2}}{2 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1-\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1+\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}\right )\) | \(258\) |
parts | \(-\frac {a \ln \left (x c -1\right )}{2 c}+\frac {a \ln \left (x c +1\right )}{2 c}-b \left (-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )^{2}}{2 c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1-\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, \frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right ) \ln \left (1+\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}+\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}+\frac {\operatorname {polylog}\left (2, -\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}-\sqrt {1+\frac {-x c +1}{x c +1}}\right )}{c}\right )\) | \(258\) |
Input:
int((a+b*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2)))/(-c^2*x^2+1),x,method=_RET URNVERBOSE)
Output:
-1/2*a/c*ln(c*x-1)+1/2*a/c*ln(x*c+1)-b*(-1/2*arcsinh((-x*c+1)^(1/2)/(x*c+1 )^(1/2))^2/c+1/c*arcsinh((-x*c+1)^(1/2)/(x*c+1)^(1/2))*ln(1-(-x*c+1)^(1/2) /(x*c+1)^(1/2)-(1+(-x*c+1)/(x*c+1))^(1/2))+1/c*polylog(2,(-x*c+1)^(1/2)/(x *c+1)^(1/2)+(1+(-x*c+1)/(x*c+1))^(1/2))+1/c*arcsinh((-x*c+1)^(1/2)/(x*c+1) ^(1/2))*ln(1+(-x*c+1)^(1/2)/(x*c+1)^(1/2)+(1+(-x*c+1)/(x*c+1))^(1/2))+1/c* polylog(2,-(-x*c+1)^(1/2)/(x*c+1)^(1/2)-(1+(-x*c+1)/(x*c+1))^(1/2)))
\[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x, algo rithm="fricas")
Output:
integral(-(b*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)/(c^2*x^2 - 1), x)
\[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=- \int \frac {a}{c^{2} x^{2} - 1}\, dx - \int \frac {b \operatorname {asinh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \] Input:
integrate((a+b*asinh((-c*x+1)**(1/2)/(c*x+1)**(1/2)))/(-c**2*x**2+1),x)
Output:
-Integral(a/(c**2*x**2 - 1), x) - Integral(b*asinh(sqrt(-c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1), x)
\[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x, algo rithm="maxima")
Output:
-1/8*b*((2*(log(c*x + 1) - log(-c*x + 1))*log(c*x + 1) - log(c*x + 1)^2 + 2*log(c*x + 1)*log(-c*x + 1) - log(-c*x + 1)^2 - 4*(log(c*x + 1) - log(-c* x + 1))*log(sqrt(2) + sqrt(-c*x + 1)))/c + 8*integrate(-1/4*(sqrt(2)*log(c *x + 1) - sqrt(2)*log(-c*x + 1))/(sqrt(2)*c*x + (c*x - 1)*sqrt(-c*x + 1) - sqrt(2)), x)) + 1/2*a*(log(c*x + 1)/c - log(c*x - 1)/c)
\[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int { -\frac {b \operatorname {arsinh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((a+b*arcsinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x, algo rithm="giac")
Output:
integrate(-(b*arcsinh(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)/(c^2*x^2 - 1), x)
Timed out. \[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\int -\frac {a+b\,\mathrm {asinh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \] Input:
int(-(a + b*asinh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))/(c^2*x^2 - 1),x)
Output:
int(-(a + b*asinh((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))/(c^2*x^2 - 1), x)
\[ \int \frac {a+b \text {arcsinh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx=\frac {-2 \left (\int \frac {\mathit {asinh} \left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )}{c^{2} x^{2}-1}d x \right ) b c -\mathrm {log}\left (c^{2} x -c \right ) a +\mathrm {log}\left (c^{2} x +c \right ) a}{2 c} \] Input:
int((a+b*asinh((-c*x+1)^(1/2)/(c*x+1)^(1/2)))/(-c^2*x^2+1),x)
Output:
( - 2*int(asinh(sqrt( - c*x + 1)/sqrt(c*x + 1))/(c**2*x**2 - 1),x)*b*c - l og(c**2*x - c)*a + log(c**2*x + c)*a)/(2*c)