Integrand size = 27, antiderivative size = 229 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(1/2)+2*(a+b*arccosh(c*x))*arctan(c*x+ (c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^ (1/2)+b*arctanh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-I* b*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1 /2)/d/(-c^2*d*x^2+d)^(1/2)+I*b*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2 )))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)
Time = 2.14 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.47 \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {\frac {a \sqrt {d-c^2 d x^2}}{-1+c^2 x^2}-a \sqrt {d} \log (x)+a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {i b d \left (i \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}}{d^2} \]
-(((a*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2) - a*Sqrt[d]*Log[x] + a*Sqrt[d]*L og[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (I*b*d*(I*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - Sqrt[(- 1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*S qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] - I*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + Sqrt[(-1 + c*x)/( 1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]]))/Sqrt[d - c^2*d*x^2])/d^2)
Time = 0.73 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.69, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6351, 25, 39, 219, 6361, 3042, 4668, 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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6351 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int -\frac {1}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{(1-c x) (c x+1)}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \int \frac {1}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 6361 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{c x}d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x)) \csc \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )d\text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (-i b \int \log \left (1-i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+i b \int \log \left (1+i e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (-i b \int e^{-\text {arccosh}(c x)} \log \left (1-i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+i b \int e^{-\text {arccosh}(c x)} \log \left (1+i e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (2 \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \text {arccosh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
(a + b*ArcCosh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTanh[c*x])/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x] *(2*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]] - I*b*PolyLog[2, (-I)*E^Ar cCosh[c*x]] + I*b*PolyLog[2, I*E^ArcCosh[c*x]]))/(d*Sqrt[d - c^2*d*x^2])
3.2.20.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Simp[ b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[ (f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x]) ^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] & & GtQ[n, 0] && LtQ[p, -1] && !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x ]/Sqrt[d + e*x^2])] Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]] , x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && Int egerQ[m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (238 ) = 476\).
Time = 1.22 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \left (c^{2} x^{2}-1\right )}\) | \(483\) |
parts | \(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}-i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) x^{2} c^{2}+\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}-i \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+i \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )+\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}+i \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-i \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )-\ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+\ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{\sqrt {c x -1}\, \sqrt {c x +1}\, d^{2} \left (c^{2} x^{2}-1\right )}\) | \(483\) |
a/d/(-c^2*d*x^2+d)^(1/2)-a/d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2)) /x)-b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(I*arccosh(c*x)*l n(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*x^2*c^2-I*arccosh(c*x)*ln(1+I*(c* x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*x^2*c^2-I*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c* x+1)^(1/2)))*x^2*c^2+I*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*x^2*c^ 2+ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^2*c^2-ln((c*x-1)^(1/2)*(c*x+1)^( 1/2)+c*x-1)*x^2*c^2-I*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) ))+I*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+arccosh(c*x)*( c*x-1)^(1/2)*(c*x+1)^(1/2)+I*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))- I*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-ln(1+c*x+(c*x-1)^(1/2)*(c*x +1)^(1/2))+ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1))/d^2/(c^2*x^2-1)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*x^5 - 2*c^2*d^ 2*x^3 + d^2*x), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
-a*(log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(3/2) - 1/(s qrt(-c^2*d*x^2 + d)*d)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1 ))/((-c^2*d*x^2 + d)^(3/2)*x), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]