Integrand size = 25, antiderivative size = 284 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b c^3}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b c}{6 d^3 x^2 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {29 b c^3}{24 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3}-\frac {3 c^2 (a+b \text {arccosh}(c x))}{d^3 x}+\frac {c^4 x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {11 c^4 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {19 b c^3 \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d^3}+\frac {35 c^3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 d^3}+\frac {35 b c^3 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 d^3}-\frac {35 b c^3 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 d^3} \] Output:
-1/12*b*c^3/d^3/(c*x-1)^(3/2)/(c*x+1)^(3/2)+1/6*b*c/d^3/x^2/(c*x-1)^(3/2)/ (c*x+1)^(3/2)-29/24*b*c^3/d^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*(a+b*arccosh (c*x))/d^3/x^3-3*c^2*(a+b*arccosh(c*x))/d^3/x+1/4*c^4*x*(a+b*arccosh(c*x)) /d^3/(-c^2*x^2+1)^2+11/8*c^4*x*(a+b*arccosh(c*x))/d^3/(-c^2*x^2+1)+19/6*b* c^3*arctan((c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3+35/4*c^3*(a+b*arccosh(c*x))*ar ctanh(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^3+35/8*b*c^3*polylog(2,-c*x-(c*x- 1)^(1/2)*(c*x+1)^(1/2))/d^3-35/8*b*c^3*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1) ^(1/2))/d^3
Time = 1.27 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.66 \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {16 a}{x^3}-\frac {144 a c^2}{x}+\frac {12 a c^4 x}{\left (-1+c^2 x^2\right )^2}-\frac {66 a c^4 x}{-1+c^2 x^2}-\frac {b c^3 \left ((-2+c x) \sqrt {-1+c x} \sqrt {1+c x}-3 \text {arccosh}(c x)\right )}{(-1+c x)^2}+\frac {b c^3 \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{(1+c x)^2}+33 b c^3 \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )+33 b c^3 \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )+144 b c^2 \left (-\frac {\text {arccosh}(c x)}{x}+\frac {c \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right )+\frac {8 b \left (-2 \text {arccosh}(c x)+\frac {c x \left (-1+c^2 x^2+c^2 x^2 \sqrt {-1+c^2 x^2} \arctan \left (\sqrt {-1+c^2 x^2}\right )\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\right )}{x^3}-105 a c^3 \log (1-c x)+105 a c^3 \log (1+c x)-\frac {105}{2} b c^3 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )+\frac {105}{2} b c^3 \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{48 d^3} \] Input:
Integrate[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]
Output:
((-16*a)/x^3 - (144*a*c^2)/x + (12*a*c^4*x)/(-1 + c^2*x^2)^2 - (66*a*c^4*x )/(-1 + c^2*x^2) - (b*c^3*((-2 + c*x)*Sqrt[-1 + c*x]*Sqrt[1 + c*x] - 3*Arc Cosh[c*x]))/(-1 + c*x)^2 + (b*c^3*(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2 + c*x) - 3*ArcCosh[c*x]))/(1 + c*x)^2 + 33*b*c^3*(-(1/Sqrt[(-1 + c*x)/(1 + c*x)]) + ArcCosh[c*x]/(1 - c*x)) + 33*b*c^3*(Sqrt[(-1 + c*x)/(1 + c*x)] - ArcCos h[c*x]/(1 + c*x)) + 144*b*c^2*(-(ArcCosh[c*x]/x) + (c*Sqrt[-1 + c^2*x^2]*A rcTan[Sqrt[-1 + c^2*x^2]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (8*b*(-2*ArcC osh[c*x] + (c*x*(-1 + c^2*x^2 + c^2*x^2*Sqrt[-1 + c^2*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/x^3 - 105*a*c^3*Log[1 - c*x ] + 105*a*c^3*Log[1 + c*x] - (105*b*c^3*(ArcCosh[c*x]*(ArcCosh[c*x] - 4*Lo g[1 + E^ArcCosh[c*x]]) - 4*PolyLog[2, -E^ArcCosh[c*x]]))/2 + (105*b*c^3*(A rcCosh[c*x]*(ArcCosh[c*x] - 4*Log[1 - E^ArcCosh[c*x]]) - 4*PolyLog[2, E^Ar cCosh[c*x]]))/2)/(48*d^3)
Result contains complex when optimal does not.
Time = 1.70 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.43, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.080, Rules used = {6347, 27, 114, 27, 115, 27, 115, 27, 103, 218, 6347, 115, 27, 115, 27, 103, 218, 6316, 83, 6316, 83, 6318, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {7}{3} c^2 \int \frac {a+b \text {arccosh}(c x)}{d^3 x^2 \left (1-c^2 x^2\right )^3}dx+\frac {b c \int \frac {1}{x^3 (c x-1)^{5/2} (c x+1)^{5/2}}dx}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \int \frac {1}{x^3 (c x-1)^{5/2} (c x+1)^{5/2}}dx}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {1}{2} \int \frac {5 c^2}{x (c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \int \frac {1}{x (c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (-\frac {\int \frac {3 c}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx}{3 c}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (-\int \frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (\frac {\int \frac {c}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{c}+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}+\frac {b c \left (\frac {5}{2} c^2 \left (c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 c^2 \int \frac {a+b \text {arccosh}(c x)}{x^2 \left (1-c^2 x^2\right )^3}dx}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 6347 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx+b c \int \frac {1}{x (c x-1)^{5/2} (c x+1)^{5/2}}dx-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx+b c \left (-\frac {\int \frac {3 c}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx}{3 c}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx+b c \left (-\int \frac {1}{x (c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx+b c \left (\frac {\int \frac {c}{x \sqrt {c x-1} \sqrt {c x+1}}dx}{c}+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx+b c \left (\int \frac {1}{x \sqrt {c x-1} \sqrt {c x+1}}dx+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx+b c \left (c \int \frac {1}{(c x-1) (c x+1) c+c}d\left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^3}dx-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx-\frac {1}{4} b c \int \frac {x}{(c x-1)^{5/2} (c x+1)^{5/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arccosh}(c x)}{\left (1-c^2 x^2\right )^2}dx+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 6316 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {1}{2} b c \int \frac {x}{(c x-1)^{3/2} (c x+1)^{3/2}}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 6318 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {7 c^2 \left (5 c^2 \left (\frac {3}{4} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {x (a+b \text {arccosh}(c x))}{2 \left (1-c^2 x^2\right )}-\frac {b}{2 c \sqrt {c x-1} \sqrt {c x+1}}\right )+\frac {x (a+b \text {arccosh}(c x))}{4 \left (1-c^2 x^2\right )^2}+\frac {b}{12 c (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {a+b \text {arccosh}(c x)}{x \left (1-c^2 x^2\right )^2}+b c \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arccosh}(c x)}{3 d^3 x^3 \left (1-c^2 x^2\right )^2}+\frac {b c \left (\frac {5}{2} c^2 \left (\arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right )+\frac {1}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {1}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )+\frac {1}{2 x^2 (c x-1)^{3/2} (c x+1)^{3/2}}\right )}{3 d^3}\) |
Input:
Int[(a + b*ArcCosh[c*x])/(x^4*(d - c^2*d*x^2)^3),x]
Output:
-1/3*(a + b*ArcCosh[c*x])/(d^3*x^3*(1 - c^2*x^2)^2) + (b*c*(1/(2*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (5*c^2*(-1/3*1/((-1 + c*x)^(3/2)*(1 + c*x)^ (3/2)) + 1/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]))/2))/(3*d^3) + (7*c^2*(-((a + b*ArcCosh[c*x])/(x*(1 - c^2*x^2)^2)) + b*c*(-1/3*1/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + 1/(Sqrt[-1 + c*x]*Sqrt [1 + c*x]) + ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]]) + 5*c^2*(b/(12*c*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (x*(a + b*ArcCosh[c*x]))/(4*(1 - c^2*x^2)^2) + (3*(-1/2*b/(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (x*(a + b*ArcCosh[c*x]))/ (2*(1 - c^2*x^2)) - ((I/2)*((2*I)*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c *x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^ArcCosh[c*x]]))/ c))/4)))/(3*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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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_.) + 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_.)*((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/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1 ))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Simp [b*c*(n/(f*(m + 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, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.53 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(c^{3} \left (-\frac {a \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {105 \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+29 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-175 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-27 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+56 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +8 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\frac {35 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\right )\) | \(314\) |
| default | \(c^{3} \left (-\frac {a \left (\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {11}{16 \left (c x -1\right )}+\frac {35 \ln \left (c x -1\right )}{16}+\frac {1}{16 \left (c x +1\right )^{2}}+\frac {11}{16 \left (c x +1\right )}-\frac {35 \ln \left (c x +1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {105 \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+29 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-175 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-27 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+56 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +8 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\frac {35 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\right )\) | \(314\) |
| parts | \(-\frac {a \left (\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}+\frac {c^{3}}{16 \left (c x +1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x +1\right )}-\frac {35 c^{3} \ln \left (c x +1\right )}{16}-\frac {c^{3}}{16 \left (c x -1\right )^{2}}+\frac {11 c^{3}}{16 \left (c x -1\right )}+\frac {35 c^{3} \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \,c^{3} \left (\frac {105 \,\operatorname {arccosh}\left (c x \right ) c^{6} x^{6}+29 c^{5} x^{5} \sqrt {c x -1}\, \sqrt {c x +1}-175 \,\operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-27 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+56 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-4 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +8 \,\operatorname {arccosh}\left (c x \right )}{24 \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) c^{3} x^{3}}-\frac {19 \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3}-\frac {35 \operatorname {dilog}\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \operatorname {dilog}\left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {35 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}\) | \(328\) |
Input:
int((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
c^3*(-a/d^3*(1/3/c^3/x^3+3/c/x-1/16/(c*x-1)^2+11/16/(c*x-1)+35/16*ln(c*x-1 )+1/16/(c*x+1)^2+11/16/(c*x+1)-35/16*ln(c*x+1))-b/d^3*(1/24*(105*arccosh(c *x)*c^6*x^6+29*c^5*x^5*(c*x-1)^(1/2)*(c*x+1)^(1/2)-175*arccosh(c*x)*c^4*x^ 4-27*c^3*x^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+56*c^2*x^2*arccosh(c*x)-4*(c*x-1) ^(1/2)*(c*x+1)^(1/2)*c*x+8*arccosh(c*x))/(c^4*x^4-2*c^2*x^2+1)/c^3/x^3-19/ 3*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-35/8*dilog(c*x+(c*x-1)^(1/2)*(c* x+1)^(1/2))-35/8*dilog(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-35/8*arccosh(c*x )*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))))
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b*arccosh(c*x) + a)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3*c^2*d^3*x ^6 - d^3*x^4), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*acosh(c*x))/x**4/(-c**2*d*x**2+d)**3,x)
Output:
Timed out
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
1/6144*(1935360*c^9*integrate(1/96*x^7*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d ^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x) - 1680*c^8*(2*(5*c^2*x^3 - 3*x)/(c^8 *d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3) + 3*log(c*x + 1)/(c^5*d^3) - 3*log(c*x - 1)/(c^5*d^3)) - 645120*c^8*integrate(1/96*x^6*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x) + 630*(c*(2*(5*c^2*x^2 + 3 *c*x - 6)/(c^8*d^3*x^3 - c^7*d^3*x^2 - c^6*d^3*x + c^5*d^3) - 5*log(c*x + 1)/(c^5*d^3) + 5*log(c*x - 1)/(c^5*d^3)) + 16*(2*c^2*x^2 - 1)*log(c*x - 1) /(c^8*d^3*x^4 - 2*c^6*d^3*x^2 + c^4*d^3))*c^7 + 2800*c^6*(2*(c^2*x^3 + x)/ (c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3) - log(c*x + 1)/(c^3*d^3) + log(c*x - 1)/(c^3*d^3)) + 1290240*c^6*integrate(1/96*x^4*log(c*x - 1)/(c^6*d^3*x^ 8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x) + 315*(c*(2*(3*c^2*x^2 - 3*c*x - 2)/(c^6*d^3*x^3 - c^5*d^3*x^2 - c^4*d^3*x + c^3*d^3) - 3*log(c*x + 1)/(c^3*d^3) + 3*log(c*x - 1)/(c^3*d^3)) - 16*log(c*x - 1)/(c^6*d^3*x^4 - 2*c^4*d^3*x^2 + c^2*d^3))*c^5 + 896*c^4*(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)) - 645120*c^4*integrate(1/96*x^2*log(c*x - 1)/(c^6*d^3*x^8 - 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 - d^3*x^2), x) + 128*c^2*(2*(15*c^4*x^4 - 25*c^2*x^2 + 8) /(c^4*d^3*x^5 - 2*c^2*d^3*x^3 + d^3*x) - 15*c*log(c*x + 1)/d^3 + 15*c*log( c*x - 1)/d^3) - 32*(105*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*log(c*x + 1)^2 + 2 10*(c^7*x^7 - 2*c^5*x^5 + c^3*x^3)*log(c*x + 1)*log(c*x - 1) + 4*(210*c...
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{4}} \,d x } \] Input:
integrate((a+b*arccosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
integrate(-(b*arccosh(c*x) + a)/((c^2*d*x^2 - d)^3*x^4), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^3),x)
Output:
int((a + b*acosh(c*x))/(x^4*(d - c^2*d*x^2)^3), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{x^4 \left (d-c^2 d x^2\right )^3} \, dx=\frac {-48 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b \,c^{4} x^{7}+96 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b \,c^{2} x^{5}-48 \left (\int \frac {\mathit {acosh} \left (c x \right )}{c^{6} x^{10}-3 c^{4} x^{8}+3 c^{2} x^{6}-x^{4}}d x \right ) b \,x^{3}-105 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{7} x^{7}+210 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{5} x^{5}-105 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{3} x^{3}+105 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{7} x^{7}-210 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{5} x^{5}+105 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{3} x^{3}-210 a \,c^{6} x^{6}+350 a \,c^{4} x^{4}-112 a \,c^{2} x^{2}-16 a}{48 d^{3} x^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*acosh(c*x))/x^4/(-c^2*d*x^2+d)^3,x)
Output:
( - 48*int(acosh(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x**4),x)*b *c**4*x**7 + 96*int(acosh(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2*x**6 - x **4),x)*b*c**2*x**5 - 48*int(acosh(c*x)/(c**6*x**10 - 3*c**4*x**8 + 3*c**2 *x**6 - x**4),x)*b*x**3 - 105*log(c**2*x - c)*a*c**7*x**7 + 210*log(c**2*x - c)*a*c**5*x**5 - 105*log(c**2*x - c)*a*c**3*x**3 + 105*log(c**2*x + c)* a*c**7*x**7 - 210*log(c**2*x + c)*a*c**5*x**5 + 105*log(c**2*x + c)*a*c**3 *x**3 - 210*a*c**6*x**6 + 350*a*c**4*x**4 - 112*a*c**2*x**2 - 16*a)/(48*d* *3*x**3*(c**4*x**4 - 2*c**2*x**2 + 1))