Integrand size = 16, antiderivative size = 292 \[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \text {arccosh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^3}-\frac {b^{3/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \]
1/3*x^3*(a+b*arccosh(c*x))^(3/2)-1/288*b^(3/2)*exp(3*a/b)*erf(3^(1/2)*(a+b *arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3+1/288*b^(3/2)*erfi(3^(1 /2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-3/32 *b^(3/2)*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3+3/32* b^(3/2)*erfi((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3/exp(a/b)-1/3*b *(c*x-1)^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1/2)/c^3-1/6*b*x^2*(c*x-1 )^(1/2)*(c*x+1)^(1/2)*(a+b*arccosh(c*x))^(1/2)/c
Time = 1.95 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.85 \[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=\frac {a e^{-\frac {3 a}{b}} \sqrt {a+b \text {arccosh}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arccosh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arccosh}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \text {arccosh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arccosh}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arccosh}(c x))}{b}\right )\right )}{72 c^3 \sqrt {-\frac {(a+b \text {arccosh}(c x))^2}{b^2}}}+\frac {\sqrt {b} \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \text {arccosh}(c x)}+8 \sqrt {b} c x \text {arccosh}(c x) \sqrt {a+b \text {arccosh}(c x)}+(2 a+3 b) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \text {arccosh}(c x)} (2 \text {arccosh}(c x) \cosh (3 \text {arccosh}(c x))-\sinh (3 \text {arccosh}(c x)))\right )}{288 c^3} \]
(a*Sqrt[a + b*ArcCosh[c*x]]*(9*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)] *Gamma[3/2, a/b + ArcCosh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcCosh[c*x]]*Gamma[3 /2, (-3*(a + b*ArcCosh[c*x]))/b] + 9*E^((2*a)/b)*Sqrt[a/b + ArcCosh[c*x]]* Gamma[3/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b* ArcCosh[c*x])/b)]*Gamma[3/2, (3*(a + b*ArcCosh[c*x]))/b]))/(72*c^3*E^((3*a )/b)*Sqrt[-((a + b*ArcCosh[c*x])^2/b^2)]) + (Sqrt[b]*(9*(-12*Sqrt[b]*Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[a + b*ArcCosh[c*x]] + 8*Sqrt[b]*c*x*A rcCosh[c*x]*Sqrt[a + b*ArcCosh[c*x]] + (2*a + 3*b)*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] - Sinh[a/b]) + (2*a - 3*b)*Sqrt[Pi]*Er f[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]*(Cosh[a/b] + Sinh[a/b])) + (2*a + b)*S qrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]]*(Cosh[(3*a)/b] - Sinh[(3*a)/b]) + (2*a - b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c* x]])/Sqrt[b]]*(Cosh[(3*a)/b] + Sinh[(3*a)/b]) + 12*Sqrt[b]*Sqrt[a + b*ArcC osh[c*x]]*(2*ArcCosh[c*x]*Cosh[3*ArcCosh[c*x]] - Sinh[3*ArcCosh[c*x]])))/( 288*c^3)
Result contains complex when optimal does not.
Time = 2.85 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.35, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {6299, 6354, 6302, 25, 5971, 2009, 6330, 6296, 25, 3042, 26, 3789, 2611, 2633, 2634}
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.
| \(\displaystyle \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 6299 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \int \frac {x^3 \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \int \frac {x \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}-\frac {b \int \frac {x^2}{\sqrt {a+b \text {arccosh}(c x)}}dx}{6 c}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6302 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {\int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arccosh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arccosh}(c x)}}\right )d(a+b \text {arccosh}(c x))}{6 c^4}+\frac {2 \int \frac {x \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \int \frac {x \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6330 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {b \int \frac {1}{\sqrt {a+b \text {arccosh}(c x)}}dx}{2 c}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6296 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c^2}+\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}+\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arccosh}(c x))}{b}\right )}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {i \left (\frac {1}{2} i \int \frac {e^{-\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))-\frac {1}{2} i \int \frac {e^{\text {arccosh}(c x)}}{\sqrt {a+b \text {arccosh}(c x)}}d(a+b \text {arccosh}(c x))\right )}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}-i \int e^{\frac {a+b \text {arccosh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arccosh}(c x)}\right )}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arccosh}(c x)}{b}}d\sqrt {a+b \text {arccosh}(c x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arccosh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {2 \left (\frac {\sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{c^2}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arccosh}(c x)}}{\sqrt {b}}\right )\right )}{2 c^2}\right )}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {a+b \text {arccosh}(c x)}}{3 c^2}\right )\) |
(x^3*(a + b*ArcCosh[c*x])^(3/2))/3 - (b*c*((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c* x]*Sqrt[a + b*ArcCosh[c*x]])/(3*c^2) + (2*((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*S qrt[a + b*ArcCosh[c*x]])/c^2 - ((I/2)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[ Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/E^(a/b)))/c^2))/(3*c^2) - (-1/8*(Sqrt[b]*E^(a/b) *Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b)*Sq rt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/8 + (Sqrt[b]*Sqr t[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[ Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(8*E^((3*a)/b)))/( 6*c^4)))/2
3.2.45.3.1 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[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcCosh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x ], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p _)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e1*e2*(p + 1))), x] - Simp[b*(n/(2 *c*(p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^ p] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, p}, x] && EqQ[e1, c*d1] && E qQ[e2, (-c)*d2] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e 1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*((a + b*ArcCosh[c*x])^n/(e1*e2*( m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, x], x] - Simp[b*f *(n/(c*(m + 2*p + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2, f, p}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && GtQ[n, 0] && IGtQ[m, 1] && N eQ[m + 2*p + 1, 0]
\[\int x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=\int x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]
Exception generated. \[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 (a+b \text {arccosh}(c x))^{3/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2} \,d x \]