Integrand size = 24, antiderivative size = 302 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=-\frac {b^2 x}{12 d^3 \left (1-c^2 x^2\right )}+\frac {b (a+b \text {arccosh}(c x))}{6 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b (a+b \text {arccosh}(c x))}{4 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))^2}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x))^2 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c d^3}-\frac {5 b^2 \text {arctanh}(c x)}{6 c d^3}+\frac {3 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{4 c d^3}-\frac {3 b (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{4 c d^3}-\frac {3 b^2 \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )}{4 c d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )}{4 c d^3} \] Output:
-1/12*b^2*x/d^3/(-c^2*x^2+1)+1/6*b*(a+b*arccosh(c*x))/c/d^3/(c*x-1)^(3/2)/ (c*x+1)^(3/2)-3/4*b*(a+b*arccosh(c*x))/c/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1 /4*x*(a+b*arccosh(c*x))^2/d^3/(-c^2*x^2+1)^2+3/8*x*(a+b*arccosh(c*x))^2/d^ 3/(-c^2*x^2+1)+3/4*(a+b*arccosh(c*x))^2*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^ (1/2))/c/d^3-5/6*b^2*arctanh(c*x)/c/d^3+3/4*b*(a+b*arccosh(c*x))*polylog(2 ,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^3-3/4*b*(a+b*arccosh(c*x))*polylog( 2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/c/d^3-3/4*b^2*polylog(3,-c*x-(c*x-1)^(1 /2)*(c*x+1)^(1/2))/c/d^3+3/4*b^2*polylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2) )/c/d^3
Leaf count is larger than twice the leaf count of optimal. \(660\) vs. \(2(302)=604\).
Time = 7.56 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^3,x]
Output:
(a^2*x)/(4*d^3*(-1 + c^2*x^2)^2) - (3*a^2*x)/(8*d^3*(-1 + c^2*x^2)) - (3*a ^2*Log[1 - c*x])/(16*c*d^3) + (3*a^2*Log[1 + c*x])/(16*c*d^3) - (2*a*b*((( -2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*ArcCosh[c*x])/(48*(-1 + c*x)^2) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x])/(48*(1 + c*x) ^2) - (3*(-(Sqrt[1 + c*x]/Sqrt[-1 + c*x]) - ArcCosh[c*x]/(-1 + c*x)))/16 - (3*(Sqrt[-1 + c*x]/Sqrt[1 + c*x] - ArcCosh[c*x]/(1 + c*x)))/16 - (3*(-1/2 *ArcCosh[c*x]^2 + 2*ArcCosh[c*x]*Log[1 + E^ArcCosh[c*x]] + 2*PolyLog[2, -E ^ArcCosh[c*x]]))/16 + (3*(-1/2*ArcCosh[c*x]^2 + 2*ArcCosh[c*x]*Log[1 - E^A rcCosh[c*x]] + 2*PolyLog[2, E^ArcCosh[c*x]]))/16))/(c*d^3) - (b^2*(80*ArcC osh[c*x]*Coth[ArcCosh[c*x]/2] + 2*(-2 + 9*ArcCosh[c*x]^2)*Csch[ArcCosh[c*x ]/2]^2 - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Csch[ArcCosh[ c*x]/2]^4 - 3*ArcCosh[c*x]^2*Csch[ArcCosh[c*x]/2]^4 - 160*Log[Tanh[ArcCosh [c*x]/2]] + 72*(ArcCosh[c*x]^2*Log[1 - E^(-ArcCosh[c*x])] - ArcCosh[c*x]^2 *Log[1 + E^(-ArcCosh[c*x])] + 2*ArcCosh[c*x]*PolyLog[2, -E^(-ArcCosh[c*x]) ] - 2*ArcCosh[c*x]*PolyLog[2, E^(-ArcCosh[c*x])] + 2*PolyLog[3, -E^(-ArcCo sh[c*x])] - 2*PolyLog[3, E^(-ArcCosh[c*x])]) + 2*(-2 + 9*ArcCosh[c*x]^2)*S ech[ArcCosh[c*x]/2]^2 + 3*ArcCosh[c*x]^2*Sech[ArcCosh[c*x]/2]^4 - (32*ArcC osh[c*x]*Sinh[ArcCosh[c*x]/2]^4)/(((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3 ) - 80*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]))/(192*c*d^3)
Result contains complex when optimal does not.
Time = 4.37 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.97, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6316, 27, 6316, 6318, 3042, 26, 4670, 3011, 2720, 6330, 25, 39, 215, 219, 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.
| \(\displaystyle \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \int \frac {(a+b \text {arccosh}(c x))^2}{d^2 \left (1-c^2 x^2\right )^2}dx}{4 d}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {(a+b \text {arccosh}(c x))^2}{\left (1-c^2 x^2\right )^2}dx}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {3 \left (\frac {1}{2} \int \frac {(a+b \text {arccosh}(c x))^2}{1-c^2 x^2}dx+b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {3 \left (b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {\int \frac {(a+b \text {arccosh}(c x))^2}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3 \left (b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {\int i (a+b \text {arccosh}(c x))^2 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {3 \left (b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {i \int (a+b \text {arccosh}(c x))^2 \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (2 i b \int (a+b \text {arccosh}(c x)) \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-2 i b \int (a+b \text {arccosh}(c x)) \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \int \frac {x (a+b \text {arccosh}(c x))}{(c x-1)^{5/2} (c x+1)^{5/2}}dx}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \left (\frac {b \int -\frac {1}{(1-c x) (c x+1)}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {b \int \frac {1}{(1-c x)^2 (c x+1)^2}dx}{3 c}-\frac {a+b \text {arccosh}(c x)}{3 c^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \left (-\frac {b \int \frac {1}{(1-c x) (c x+1)}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {b \int \frac {1}{(1-c x)^2 (c x+1)^2}dx}{3 c}-\frac {a+b \text {arccosh}(c x)}{3 c^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 39 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \left (-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {b \int \frac {1}{\left (1-c^2 x^2\right )^2}dx}{3 c}-\frac {a+b \text {arccosh}(c x)}{3 c^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \left (-\frac {b \int \frac {1}{1-c^2 x^2}dx}{c}-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {b \left (\frac {1}{2} \int \frac {1}{1-c^2 x^2}dx+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}-\frac {a+b \text {arccosh}(c x)}{3 c^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3 \left (-\frac {i \left (-2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \int e^{-\text {arccosh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2\right )}{2 c}+b c \left (-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \text {arctanh}(c x)}{c^2}\right )+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}-\frac {a+b \text {arccosh}(c x)}{3 c^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {3 \left (b c \left (-\frac {a+b \text {arccosh}(c x)}{c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \text {arctanh}(c x)}{c^2}\right )-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2-2 i b \left (b \operatorname {PolyLog}\left (3,-e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )+2 i b \left (b \operatorname {PolyLog}\left (3,e^{\text {arccosh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))^2}{2 \left (1-c^2 x^2\right )}\right )}{4 d^3}-\frac {b c \left (\frac {b \left (\frac {\text {arctanh}(c x)}{2 c}+\frac {x}{2 \left (1-c^2 x^2\right )}\right )}{3 c}-\frac {a+b \text {arccosh}(c x)}{3 c^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{2 d^3}+\frac {x (a+b \text {arccosh}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
Input:
Int[(a + b*ArcCosh[c*x])^2/(d - c^2*d*x^2)^3,x]
Output:
(x*(a + b*ArcCosh[c*x])^2)/(4*d^3*(1 - c^2*x^2)^2) - (b*c*(-1/3*(a + b*Arc Cosh[c*x])/(c^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (b*(x/(2*(1 - c^2*x^2) ) + ArcTanh[c*x]/(2*c)))/(3*c)))/(2*d^3) + (3*((x*(a + b*ArcCosh[c*x])^2)/ (2*(1 - c^2*x^2)) + b*c*(-((a + b*ArcCosh[c*x])/(c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) - (b*ArcTanh[c*x])/c^2) - ((I/2)*((2*I)*(a + b*ArcCosh[c*x])^2*A rcTanh[E^ArcCosh[c*x]] - (2*I)*b*(-((a + b*ArcCosh[c*x])*PolyLog[2, -E^Arc Cosh[c*x]]) + b*PolyLog[3, -E^ArcCosh[c*x]]) + (2*I)*b*(-((a + b*ArcCosh[c *x])*PolyLog[2, E^ArcCosh[c*x]]) + b*PolyLog[3, E^ArcCosh[c*x]])))/c))/(4* d^3)
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[((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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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[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[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_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcCosh[c*x])^n, x], x] - Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[x*(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}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[-(c*d)^(-1) Subst[Int[(a + b*x)^n*Csch[x], x], x, ArcCosh[c*x ]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(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, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
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 = 0.26 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.85
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}+18 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-15 \operatorname {arccosh}\left (c x \right )^{2} c x -22 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-2 c^{3} x^{3}+2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {5 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}-\frac {3 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c}\) | \(558\) |
| default | \(\frac {-\frac {a^{2} \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}+18 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-15 \operatorname {arccosh}\left (c x \right )^{2} c x -22 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-2 c^{3} x^{3}+2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {5 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}-\frac {3 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}\right )}{d^{3}}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c}\) | \(558\) |
| parts | \(-\frac {a^{2} \left (\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}\right )}{d^{3}}-\frac {b^{2} \left (\frac {9 \operatorname {arccosh}\left (c x \right )^{2} c^{3} x^{3}+18 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x +1}\, \sqrt {c x -1}\, c^{2} x^{2}-15 \operatorname {arccosh}\left (c x \right )^{2} c x -22 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-2 c^{3} x^{3}+2 c x}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {5 \,\operatorname {arctanh}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}+\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}-\frac {3 \operatorname {polylog}\left (3, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}-\frac {3 \operatorname {arccosh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}+\frac {3 \operatorname {polylog}\left (3, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{4}\right )}{d^{3} c}-\frac {2 a b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3} c}\) | \(578\) |
Input:
int((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
1/c*(-a^2/d^3*(-1/16/(c*x-1)^2+3/16/(c*x-1)+3/16*ln(c*x-1)+1/16/(c*x+1)^2+ 3/16/(c*x+1)-3/16*ln(c*x+1))-b^2/d^3*(1/24*(9*arccosh(c*x)^2*c^3*x^3+18*ar ccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*c^2*x^2-15*arccosh(c*x)^2*c*x-22*ar ccosh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c^3*x^3+2*c*x)/(c^4*x^4-2*c^2*x^2 +1)+5/3*arctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/8*arccosh(c*x)^2*ln(1-c *x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/4*arccosh(c*x)*polylog(2,c*x+(c*x-1)^(1/ 2)*(c*x+1)^(1/2))-3/4*polylog(3,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/8*arcco sh(c*x)^2*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/4*arccosh(c*x)*polylog(2 ,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/4*polylog(3,-c*x-(c*x-1)^(1/2)*(c*x+1 )^(1/2)))-2*a*b/d^3*(1/24*(9*c^3*x^3*arccosh(c*x)+9*(c*x-1)^(1/2)*(c*x+1)^ (1/2)*c^2*x^2-15*c*x*arccosh(c*x)-11*(c*x-1)^(1/2)*(c*x+1)^(1/2))/(c^4*x^4 -2*c^2*x^2+1)+3/8*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+3/8*p olylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-3/8*arccosh(c*x)*ln(1+c*x+(c*x-1 )^(1/2)*(c*x+1)^(1/2))-3/8*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))))
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^6*d^3*x^6 - 3 *c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a^{2}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \] Input:
integrate((a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**3,x)
Output:
-(Integral(a**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Integral (b**2*acosh(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x) + Inte gral(2*a*b*acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1), x))/d** 3
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/16*a^2*(2*(3*c^2*x^3 - 5*x)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) - 3*log (c*x + 1)/(c*d^3) + 3*log(c*x - 1)/(c*d^3)) - 1/16*(6*b^2*c^3*x^3 - 10*b^2 *c*x - 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(c*x + 1) + 3*(b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*log(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1 ))^2/(c^5*d^3*x^4 - 2*c^3*d^3*x^2 + c*d^3) - integrate(-1/8*(6*b^2*c^5*x^5 - 16*b^2*c^3*x^3 + (6*b^2*c^4*x^4 - 10*b^2*c^2*x^2 - 16*a*b - 3*(b^2*c^5* x^5 - 2*b^2*c^3*x^3 + b^2*c*x)*log(c*x + 1) + 3*(b^2*c^5*x^5 - 2*b^2*c^3*x ^3 + b^2*c*x)*log(c*x - 1))*sqrt(c*x + 1)*sqrt(c*x - 1) - 2*(8*a*b*c - 5*b ^2*c)*x - 3*(b^2*c^6*x^6 - 3*b^2*c^4*x^4 + 3*b^2*c^2*x^2 - b^2)*log(c*x + 1) + 3*(b^2*c^6*x^6 - 3*b^2*c^4*x^4 + 3*b^2*c^2*x^2 - b^2)*log(c*x - 1))*l og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(c^7*d^3*x^7 - 3*c^5*d^3*x^5 + 3*c^3 *d^3*x^3 - c*d^3*x + (c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3)*s qrt(c*x + 1)*sqrt(c*x - 1)), x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
integrate(-(b*arccosh(c*x) + a)^2/(c^2*d*x^2 - d)^3, x)
Timed out. \[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^3,x)
Output:
int((a + b*acosh(c*x))^2/(d - c^2*d*x^2)^3, x)
\[ \int \frac {(a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {-32 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{5} x^{4}+64 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b \,c^{3} x^{2}-32 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) a b c -16 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{5} x^{4}+32 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c^{3} x^{2}-16 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2}}{c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1}d x \right ) b^{2} c -3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{4} x^{4}+6 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2} c^{2} x^{2}-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{4} x^{4}-6 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}-6 a^{2} c^{3} x^{3}+10 a^{2} c x}{16 c \,d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^3,x)
Output:
( - 32*int(acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b*c **5*x**4 + 64*int(acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x )*a*b*c**3*x**2 - 32*int(acosh(c*x)/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*a*b*c - 16*int(acosh(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x* *2 - 1),x)*b**2*c**5*x**4 + 32*int(acosh(c*x)**2/(c**6*x**6 - 3*c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2*c**3*x**2 - 16*int(acosh(c*x)**2/(c**6*x**6 - 3 *c**4*x**4 + 3*c**2*x**2 - 1),x)*b**2*c - 3*log(c**2*x - c)*a**2*c**4*x**4 + 6*log(c**2*x - c)*a**2*c**2*x**2 - 3*log(c**2*x - c)*a**2 + 3*log(c**2* x + c)*a**2*c**4*x**4 - 6*log(c**2*x + c)*a**2*c**2*x**2 + 3*log(c**2*x + c)*a**2 - 6*a**2*c**3*x**3 + 10*a**2*c*x)/(16*c*d**3*(c**4*x**4 - 2*c**2*x **2 + 1))