Integrand size = 26, antiderivative size = 348 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^2 \sqrt {-1+c x}}-\frac {9 \sqrt {1-c x} \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^2 \sqrt {-1+c x}}+\frac {5 \sqrt {1-c x} \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^2 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 b^2 c^2 \sqrt {-1+c x}}+\frac {9 \sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^2 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^2 \sqrt {-1+c x}} \]
-1/8*cosh(a/b)*Shi((a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1 /2)+9/16*cosh(3*a/b)*Shi(3*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^2/(c *x-1)^(1/2)-5/16*cosh(5*a/b)*Shi(5*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^ 2/c^2/(c*x-1)^(1/2)+1/8*Chi((a+b*arccosh(c*x))/b)*sinh(a/b)*(-c*x+1)^(1/2) /b^2/c^2/(c*x-1)^(1/2)-9/16*Chi(3*(a+b*arccosh(c*x))/b)*sinh(3*a/b)*(-c*x+ 1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)+5/16*Chi(5*(a+b*arccosh(c*x))/b)*sinh(5*a/b )*(-c*x+1)^(1/2)/b^2/c^2/(c*x-1)^(1/2)-x*(-c^2*x^2+1)^(3/2)*(c*x-1)^(1/2)* (c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))
Time = 0.75 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-16 b c x+32 b c^3 x^3-16 b c^5 x^5-2 (a+b \text {arccosh}(c x)) \text {Chi}\left (\frac {a}{b}+\text {arccosh}(c x)\right ) \sinh \left (\frac {a}{b}\right )+9 (a+b \text {arccosh}(c x)) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )-5 a \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-5 b \text {arccosh}(c x) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )+2 a \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )+2 b \text {arccosh}(c x) \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arccosh}(c x)\right )-9 a \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-9 b \text {arccosh}(c x) \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 a \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+5 b \text {arccosh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b^2 c^2 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \]
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b*c*x + 32*b*c^3*x^3 - 16*b*c^5*x^5 - 2 *(a + b*ArcCosh[c*x])*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b] + 9*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 5*a*C oshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 5*b*ArcCosh[c*x]*CoshI ntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] + 2*a*Cosh[a/b]*SinhIntegral [a/b + ArcCosh[c*x]] + 2*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + ArcCo sh[c*x]] - 9*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - 9*b*Ar cCosh[c*x]*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] + 5*a*Cosh[( 5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 5*b*ArcCosh[c*x]*Cosh[(5*a) /b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))/(16*b^2*c^2*Sqrt[1 - c^2*x^2]*( a + b*ArcCosh[c*x]))
Result contains complex when optimal does not.
Time = 2.05 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {6357, 25, 6304, 6321, 25, 3042, 26, 3793, 2009, 6327, 6367, 25, 5971, 2009}
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.
| \(\displaystyle \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) |
| \(\Big \downarrow \) 6357 |
| \(\displaystyle -\frac {5 c \sqrt {1-c x} \int -\frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}+\frac {\sqrt {1-c x} \int -\frac {(1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {\sqrt {1-c x} \int \frac {(1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6304 |
| \(\displaystyle -\frac {\sqrt {1-c x} \int \frac {1-c^2 x^2}{a+b \text {arccosh}(c x)}dx}{b c \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6321 |
| \(\displaystyle \frac {\sqrt {1-c x} \int -\frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {1-c x} \int \frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {1-c x} \int \frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^3}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \sqrt {1-c x} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )^3}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {i \sqrt {1-c x} \int \left (\frac {3 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 (a+b \text {arccosh}(c x))}-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}+\frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 c \sqrt {1-c x} \int \frac {x^2 (1-c x) (c x+1)}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6327 |
| \(\displaystyle \frac {5 c \sqrt {1-c x} \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \text {arccosh}(c x)}dx}{b \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 6367 |
| \(\displaystyle -\frac {5 \sqrt {1-c x} \int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \sqrt {1-c x} \int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{a+b \text {arccosh}(c x)}d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {5 \sqrt {1-c x} \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{16 (a+b \text {arccosh}(c x))}-\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{8 (a+b \text {arccosh}(c x))}\right )d(a+b \text {arccosh}(c x))}{b^2 c^2 \sqrt {c x-1}}-\frac {i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i \sqrt {1-c x} \left (\frac {3}{4} i \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{4} i \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {3}{4} i \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{4} i \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \left (\frac {1}{8} \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )+\frac {1}{16} \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{16} \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )-\frac {1}{8} \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arccosh}(c x)}{b}\right )-\frac {1}{16} \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+\frac {1}{16} \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arccosh}(c x))}{b}\right )\right )}{b^2 c^2 \sqrt {c x-1}}-\frac {x \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \text {arccosh}(c x))}\) |
-((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh [c*x]))) - (I*Sqrt[1 - c*x]*(((3*I)/4)*CoshIntegral[(a + b*ArcCosh[c*x])/b ]*Sinh[a/b] - (I/4)*CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b] - ((3*I)/4)*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x])/b] + (I/4)*Cosh[( 3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b]))/(b^2*c^2*Sqrt[-1 + c*x] ) - (5*Sqrt[1 - c*x]*((CoshIntegral[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/8 + (CoshIntegral[(3*(a + b*ArcCosh[c*x]))/b]*Sinh[(3*a)/b])/16 - (CoshIntegr al[(5*(a + b*ArcCosh[c*x]))/b]*Sinh[(5*a)/b])/16 - (Cosh[a/b]*SinhIntegral [(a + b*ArcCosh[c*x])/b])/8 - (Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCos h[c*x]))/b])/16 + (Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b]) /16))/(b^2*c^2*Sqrt[-1 + c*x])
3.4.29.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*( (d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*A rcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(1/(b*c))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Subst[Int[x^n*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + ( e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1 *d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2 , e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c* x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (Simp[ f*(m/(b*c*(n + 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] - Simp[c*((m + 2*p + 1)/(b*f*(n + 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, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x )^p*(-1 + c*x)^p)] Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && Eq Q[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.96 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.68
| method | result | size |
| default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{5} x^{5}-32 b \,c^{6} x^{6}+64 \sqrt {c x -1}\, \sqrt {c x +1}\, b \,c^{3} x^{3}+64 b \,c^{4} x^{4}-32 \sqrt {c x -1}\, \sqrt {c x +1}\, b c x -32 b \,c^{2} x^{2}+9 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-2 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-5 \,\operatorname {arccosh}\left (c x \right ) b \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+5 \,\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}} b \,\operatorname {arccosh}\left (c x \right )-9 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} b \,\operatorname {arccosh}\left (c x \right )+2 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} b \,\operatorname {arccosh}\left (c x \right )+9 a \,\operatorname {Ei}_{1}\left (-3 \,\operatorname {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}}-2 a \,\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+a}{b}}-5 a \,\operatorname {Ei}_{1}\left (-5 \,\operatorname {arccosh}\left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}}+5 \,\operatorname {Ei}_{1}\left (5 \,\operatorname {arccosh}\left (c x \right )+\frac {5 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+5 a}{b}} a -9 \,\operatorname {Ei}_{1}\left (3 \,\operatorname {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\operatorname {arccosh}\left (c x \right )+3 a}{b}} a +2 \,\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{b}} a \right )}{32 \left (c x +1\right ) c^{2} \left (c x -1\right ) b^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}\) | \(583\) |
1/32*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(-32* (c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^5*x^5-32*b*c^6*x^6+64*(c*x-1)^(1/2)*(c*x+1 )^(1/2)*b*c^3*x^3+64*b*c^4*x^4-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c*x-32*b*c ^2*x^2+9*arccosh(c*x)*b*Ei(1,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arccosh(c*x)+ 3*a)/b)-2*arccosh(c*x)*b*Ei(1,-arccosh(c*x)-a/b)*exp(-(-b*arccosh(c*x)+a)/ b)-5*arccosh(c*x)*b*Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-(-b*arccosh(c*x)+5*a) /b)+5*Ei(1,5*arccosh(c*x)+5*a/b)*exp((b*arccosh(c*x)+5*a)/b)*b*arccosh(c*x )-9*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)*b*arccosh(c*x)+ 2*Ei(1,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)*b*arccosh(c*x)+9*a*Ei(1 ,-3*arccosh(c*x)-3*a/b)*exp(-(-b*arccosh(c*x)+3*a)/b)-2*a*Ei(1,-arccosh(c* x)-a/b)*exp(-(-b*arccosh(c*x)+a)/b)-5*a*Ei(1,-5*arccosh(c*x)-5*a/b)*exp(-( -b*arccosh(c*x)+5*a)/b)+5*Ei(1,5*arccosh(c*x)+5*a/b)*exp((b*arccosh(c*x)+5 *a)/b)*a-9*Ei(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)*a+2*Ei(1 ,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)*a)/(c*x+1)/c^2/(c*x-1)/b^2/(a +b*arccosh(c*x))
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
integral(-(c^2*x^3 - x)*sqrt(-c^2*x^2 + 1)/(b^2*arccosh(c*x)^2 + 2*a*b*arc cosh(c*x) + a^2), x)
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
((c^4*x^5 - 2*c^2*x^3 + x)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^6 - 2*c^3*x^4 + c*x^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c *x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*b^2 *c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate((5*(c^ 5*x^5 - c^3*x^3)*(c*x + 1)^(3/2)*(c*x - 1) + (10*c^6*x^6 - 17*c^4*x^4 + 8* c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (5*c^7*x^7 - 12*c^5*x^5 + 9*c^3*x^3 - 2*c*x)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^5*x^4 + (c*x + 1)*(c*x - 1) *a*b*c^3*x^2 - 2*a*b*c^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b*c^2*x)*sqrt(c* x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^2 - 2* b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)
\[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]