Integrand size = 29, antiderivative size = 568 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 a b x \sqrt {-1+c x} \sqrt {1+c x}}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 b^2 x \sqrt {-1+c x} \sqrt {1+c x} \text {arccosh}(c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b x \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {b x^3 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arccosh}(c x))^2}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}-\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{3 c^6 d^2 \sqrt {d-c^2 d x^2}} \]
1/3*x^4*(a+b*arccosh(c*x))^2/c^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*x^2/c^4/d^ 2/(-c^2*d*x^2+d)^(1/2)-7/3*b^2*(-c*x+1)*(c*x+1)/c^6/d^2/(-c^2*d*x^2+d)^(1/ 2)-4/3*x^2*(a+b*arccosh(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-16/3*a*b*x*(c *x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-16/3*b^2*x*arccosh( c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+11/3*b*x*(a+ b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+1 /3*b*x^3*(a+b*arccosh(c*x))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^3/d^2/(-c^2*x^2+ 1)/(-c^2*d*x^2+d)^(1/2)-22/3*b*(a+b*arccosh(c*x))*arctanh(c*x+(c*x-1)^(1/2 )*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)- 11/3*b^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1) ^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)+11/3*b^2*polylog(2,c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-8/3 *(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/c^6/d^3
Time = 4.39 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.86 \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=-\frac {8 a^2 \left (8-12 c^2 x^2+3 c^4 x^4\right )+2 a b \left (25 \text {arccosh}(c x)-36 \text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))+3 \text {arccosh}(c x) \cosh (4 \text {arccosh}(c x))+33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-33 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )+4 \sinh (2 \text {arccosh}(c x))-11 \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right ) \sinh (3 \text {arccosh}(c x))+11 \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right ) \sinh (3 \text {arccosh}(c x))-3 \sinh (4 \text {arccosh}(c x))\right )+b^2 \left (22+25 \text {arccosh}(c x)^2-4 \left (7+9 \text {arccosh}(c x)^2\right ) \cosh (2 \text {arccosh}(c x))+3 \left (2+\text {arccosh}(c x)^2\right ) \cosh (4 \text {arccosh}(c x))-66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right )+66 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right )+88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,-e^{-\text {arccosh}(c x)}\right )-88 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arccosh}(c x)}\right )+8 \text {arccosh}(c x) \sinh (2 \text {arccosh}(c x))+22 \text {arccosh}(c x) \log \left (1-e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-22 \text {arccosh}(c x) \log \left (1+e^{-\text {arccosh}(c x)}\right ) \sinh (3 \text {arccosh}(c x))-6 \text {arccosh}(c x) \sinh (4 \text {arccosh}(c x))\right )}{24 c^6 d \left (d-c^2 d x^2\right )^{3/2}} \]
-1/24*(8*a^2*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*a*b*(25*ArcCosh[c*x] - 36*Ar cCosh[c*x]*Cosh[2*ArcCosh[c*x]] + 3*ArcCosh[c*x]*Cosh[4*ArcCosh[c*x]] + 33 *Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] - 33*Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] + 4*Sinh[2*ArcCo sh[c*x]] - 11*Log[Cosh[ArcCosh[c*x]/2]]*Sinh[3*ArcCosh[c*x]] + 11*Log[Sinh [ArcCosh[c*x]/2]]*Sinh[3*ArcCosh[c*x]] - 3*Sinh[4*ArcCosh[c*x]]) + b^2*(22 + 25*ArcCosh[c*x]^2 - 4*(7 + 9*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] + 3*( 2 + ArcCosh[c*x]^2)*Cosh[4*ArcCosh[c*x]] - 66*Sqrt[(-1 + c*x)/(1 + c*x)]*( 1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 66*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 88*((-1 + c*x)/ (1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, -E^(-ArcCosh[c*x])] - 88*((-1 + c* x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 8*ArcCosh[ c*x]*Sinh[2*ArcCosh[c*x]] + 22*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sin h[3*ArcCosh[c*x]] - 22*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])]*Sinh[3*ArcC osh[c*x]] - 6*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(c^6*d*(d - c^2*d*x^2)^( 3/2))
Result contains complex when optimal does not.
Time = 3.47 (sec) , antiderivative size = 569, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {6349, 6327, 6349, 25, 109, 27, 83, 6327, 6329, 2009, 6353, 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.
\(\displaystyle \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6349 |
\(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^4 (a+b \text {arccosh}(c x))}{(1-c x)^2 (c x+1)^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6327 |
\(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (1-c^2 x^2\right )^2}dx}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{3/2}}dx}{3 c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6349 |
\(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \frac {x^3}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int -\frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \int \frac {x^3}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{2 c}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \left (-\frac {\int -\frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c^2}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {b \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c^2}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{(1-c x) (c x+1)}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6327 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}-\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{c^2 d}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6329 |
\(\displaystyle -\frac {4 \left (-\frac {2 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}\right )}{c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \int \frac {x^2 (a+b \text {arccosh}(c x))}{1-c^2 x^2}dx}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6353 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {x (a+b \text {arccosh}(c x))}{c^2}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}+\frac {b \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{c}-\frac {x (a+b \text {arccosh}(c x))}{c^2}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (\frac {\int \frac {a+b \text {arccosh}(c x)}{1-c^2 x^2}dx}{c^2}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 6318 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (-\frac {\int i (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (-\frac {i \int (a+b \text {arccosh}(c x)) \csc (i \text {arccosh}(c x))d\text {arccosh}(c x)}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (-\frac {i \left (i b \int \log \left (1-e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)-i b \int \log \left (1+e^{\text {arccosh}(c x)}\right )d\text {arccosh}(c x)+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (-\frac {i \left (i b \int e^{-\text {arccosh}(c x)} \log \left (1-e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}-i b \int e^{-\text {arccosh}(c x)} \log \left (1+e^{\text {arccosh}(c x)}\right )de^{\text {arccosh}(c x)}+2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {4 \left (\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{c d \sqrt {d-c^2 d x^2}}+\frac {x^2 (a+b \text {arccosh}(c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (-\frac {3 \left (-\frac {i \left (2 i \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{c^3}-\frac {x (a+b \text {arccosh}(c x))}{c^2}+\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c^3}\right )}{2 c^2}+\frac {x^3 (a+b \text {arccosh}(c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{c^4}-\frac {x^2}{c^2 \sqrt {c x-1} \sqrt {c x+1}}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 (a+b \text {arccosh}(c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\) |
(x^4*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (2*b*Sqrt[- 1 + c*x]*Sqrt[1 + c*x]*((b*(-(x^2/(c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])) + (2 *Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^4))/(2*c) + (x^3*(a + b*ArcCosh[c*x]))/(2 *c^2*(1 - c^2*x^2)) - (3*((b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c^3 - (x*(a + b *ArcCosh[c*x]))/c^2 - (I*((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^ 3))/(2*c^2)))/(3*c*d^2*Sqrt[d - c^2*d*x^2]) - (4*((x^2*(a + b*ArcCosh[c*x] )^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (2*(-((Sqrt[d - c^2*d*x^2]*(a + b*ArcCo sh[c*x])^2)/(c^2*d)) - (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a*x - (b*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c + b*x*ArcCosh[c*x]))/(c*Sqrt[d - c^2*d*x^2])))/(c ^2*d) + (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((b*Sqrt[-1 + c*x]*Sqrt[1 + c*x] )/c^3 - (x*(a + b*ArcCosh[c*x]))/c^2 - (I*((2*I)*(a + b*ArcCosh[c*x])*ArcT anh[E^ArcCosh[c*x]] + I*b*PolyLog[2, -E^ArcCosh[c*x]] - I*b*PolyLog[2, E^A rcCosh[c*x]]))/c^3))/(c*d*Sqrt[d - c^2*d*x^2])))/(3*c^2*d)
3.3.14.3.1 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[(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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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), 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_.)*((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_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*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, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - S imp[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c *x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2 *p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x) ^p)] Int[(f*x)^(m - 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*Ar cCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2* d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]
Time = 1.46 (sec) , antiderivative size = 922, normalized size of antiderivative = 1.62
method | result | size |
default | \(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-5 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{3 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(922\) |
parts | \(a^{2} \left (-\frac {x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {\frac {4 x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}}{c^{2}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}-2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (\operatorname {arccosh}\left (c x \right )^{2}+2 \,\operatorname {arccosh}\left (c x \right )+2\right )}{2 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (6 \operatorname {arccosh}\left (c x \right )^{2} x^{2} c^{2}+\sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c x +c^{2} x^{2}-5 \operatorname {arccosh}\left (c x \right )^{2}-1\right )}{3 \left (c^{2} x^{2}-1\right )^{2} d^{3} c^{6}}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}+\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}-\frac {11 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d^{3} c^{6} \left (c^{2} x^{2}-1\right )}\right )-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (6 \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) c^{4} x^{4}-6 c^{5} x^{5}-11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{4} x^{4}+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{4} x^{4}-24 \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, c^{2} x^{2}+11 c^{3} x^{3}+22 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) x^{2} c^{2}-22 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) x^{2} c^{2}+16 \,\operatorname {arccosh}\left (c x \right ) \sqrt {c x -1}\, \sqrt {c x +1}-5 c x -11 \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )+11 \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right )\right )}{3 c^{6} \left (c^{6} x^{6}-3 c^{4} x^{4}+3 c^{2} x^{2}-1\right ) d^{3}}\) | \(922\) |
a^2*(-x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4/c^2*(x^2/c^2/d/(-c^2*d*x^2+d)^(3/2) -2/3/d/c^4/(-c^2*d*x^2+d)^(3/2)))+b^2*(-1/2*(-d*(c^2*x^2-1))^(1/2)*((c*x-1 )^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(arccosh(c*x)^2-2*arccosh(c*x)+2)/d^3 /c^6/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)* c*x+c^2*x^2-1)*(arccosh(c*x)^2+2*arccosh(c*x)+2)/d^3/c^6/(c^2*x^2-1)+1/3*( -d*(c^2*x^2-1))^(1/2)*(6*arccosh(c*x)^2*x^2*c^2+(c*x+1)^(1/2)*arccosh(c*x) *(c*x-1)^(1/2)*c*x+c^2*x^2-5*arccosh(c*x)^2-1)/(c^2*x^2-1)^2/d^3/c^6+11/3* (-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*arc cosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+11/3*(-d*(c^2*x^2-1))^(1/2 )*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^( 1/2)*(c*x+1)^(1/2))-11/3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2 )/d^3/c^6/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-1 1/3*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/c^6/(c^2*x^2-1) *polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-1/3*a*b*(-d*(c^2*x^2-1))^(1/2 )*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*arccosh(c*x)* c^4*x^4-6*c^5*x^5-11*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^4*x^4+11*ln(( c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*c^4*x^4-24*(c*x+1)^(1/2)*arccosh(c*x)*(c *x-1)^(1/2)*c^2*x^2+11*c^3*x^3+22*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*x^ 2*c^2-22*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*x^2*c^2+16*arccosh(c*x)*(c* x-1)^(1/2)*(c*x+1)^(1/2)-5*c*x-11*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))...
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
integral(-(b^2*x^5*arccosh(c*x)^2 + 2*a*b*x^5*arccosh(c*x) + a^2*x^5)*sqrt (-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*a^2*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^ (3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/2)*c^6*d)) - 1/3*(3*b^2*c^4*sqrt(d)* x^4 - 12*b^2*c^2*sqrt(d)*x^2 + 8*b^2*sqrt(d))*sqrt(c*x + 1)*sqrt(-c*x + 1) *log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3) - integrate(2/3*((12*b^2*c^3*x^3 + 3*(a*b*c^5 - b^2*c^5)*x^5 - 8* b^2*c*x)*(c*x + 1)*sqrt(c*x - 1) + (15*b^2*c^4*x^4 + 3*(a*b*c^6 - b^2*c^6) *x^6 - 20*b^2*c^2*x^2 + 8*b^2)*sqrt(c*x + 1))*sqrt(-c*x + 1)*log(c*x + sqr t(c*x + 1)*sqrt(c*x - 1))/(c^12*d^(5/2)*x^7 - 3*c^10*d^(5/2)*x^5 + 3*c^8*d ^(5/2)*x^3 - c^6*d^(5/2)*x + (c^11*d^(5/2)*x^6 - 3*c^9*d^(5/2)*x^4 + 3*c^7 *d^(5/2)*x^2 - c^5*d^(5/2))*sqrt(c*x + 1)*sqrt(c*x - 1)), x)
Exception generated. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^5 (a+b \text {arccosh}(c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]