Integrand size = 23, antiderivative size = 377 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {45 b^4 e^3 (c+d x)^2}{128 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}-\frac {45 b^3 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{64 d}-\frac {3 b^3 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{32 d}-\frac {45 b^2 e^3 (a+b \text {arccosh}(c+d x))^2}{128 d}+\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arccosh}(c+d x))^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^2}{16 d}-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{8 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))^3}{4 d}-\frac {3 e^3 (a+b \text {arccosh}(c+d x))^4}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arccosh}(c+d x))^4}{4 d} \] Output:
45/128*b^4*e^3*(d*x+c)^2/d+3/128*b^4*e^3*(d*x+c)^4/d-45/64*b^3*e^3*(d*x+c- 1)^(1/2)*(d*x+c)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d-3/32*b^3*e^3*(d*x+ c-1)^(1/2)*(d*x+c)^3*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))/d-45/128*b^2*e^3 *(a+b*arccosh(d*x+c))^2/d+9/16*b^2*e^3*(d*x+c)^2*(a+b*arccosh(d*x+c))^2/d+ 3/16*b^2*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^2/d-3/8*b*e^3*(d*x+c-1)^(1/2)* (d*x+c)*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^3/d-1/4*b*e^3*(d*x+c-1)^(1/2) *(d*x+c)^3*(d*x+c+1)^(1/2)*(a+b*arccosh(d*x+c))^3/d-3/32*e^3*(a+b*arccosh( d*x+c))^4/d+1/4*e^3*(d*x+c)^4*(a+b*arccosh(d*x+c))^4/d
Time = 0.73 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.49 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=\frac {e^3 \left (9 b^2 \left (8 a^2+5 b^2\right ) (c+d x)^2+\left (32 a^4+24 a^2 b^2+3 b^4\right ) (c+d x)^4+2 a b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (-3 \left (8 a^2+15 b^2\right )-2 \left (8 a^2+3 b^2\right ) (c+d x)^2\right )+2 b (c+d x) \left (72 a b^2 (c+d x)+64 a^3 (c+d x)^3+24 a b^2 (c+d x)^3-72 a^2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-45 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x}-48 a^2 b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}-6 b^3 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)+3 b^2 \left (-24 a^2-15 b^2+24 b^2 (c+d x)^2+64 a^2 (c+d x)^4+8 b^2 (c+d x)^4-48 a b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}-32 a b \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^2+16 b^3 \left (-3 a+8 a (c+d x)^4-3 b \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}-2 b \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}\right ) \text {arccosh}(c+d x)^3+4 b^4 \left (-3+8 (c+d x)^4\right ) \text {arccosh}(c+d x)^4-6 a b \left (8 a^2+15 b^2\right ) \log \left (c+d x+\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )\right )}{128 d} \] Input:
Integrate[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^4,x]
Output:
(e^3*(9*b^2*(8*a^2 + 5*b^2)*(c + d*x)^2 + (32*a^4 + 24*a^2*b^2 + 3*b^4)*(c + d*x)^4 + 2*a*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(-3*(8*a^ 2 + 15*b^2) - 2*(8*a^2 + 3*b^2)*(c + d*x)^2) + 2*b*(c + d*x)*(72*a*b^2*(c + d*x) + 64*a^3*(c + d*x)^3 + 24*a*b^2*(c + d*x)^3 - 72*a^2*b*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 45*b^3*Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x] - 4 8*a^2*b*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x] - 6*b^3*Sqrt[-1 + c + d*x]*(c + d*x)^2*Sqrt[1 + c + d*x])*ArcCosh[c + d*x] + 3*b^2*(-24*a^2 - 15*b^2 + 24*b^2*(c + d*x)^2 + 64*a^2*(c + d*x)^4 + 8*b^2*(c + d*x)^4 - 48*a*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x] - 32*a*b*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^2 + 16*b^3*(-3*a + 8*a*(c + d*x)^4 - 3*b*Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x] - 2* b*Sqrt[-1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x])*ArcCosh[c + d*x]^3 + 4 *b^4*(-3 + 8*(c + d*x)^4)*ArcCosh[c + d*x]^4 - 6*a*b*(8*a^2 + 15*b^2)*Log[ c + d*x + Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x]]))/(128*d)
Time = 2.88 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6411, 27, 6298, 6354, 6298, 6354, 15, 6298, 6308, 6354, 15, 6308}
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.
| \(\displaystyle \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \text {arccosh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \text {arccosh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \int \frac {(c+d x)^4 (a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \int (c+d x)^3 (a+b \text {arccosh}(c+d x))^2d(c+d x)+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \int \frac {(c+d x)^4 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{4} b \int (c+d x)^3d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))\right )\right )+\frac {3}{4} \left (-\frac {3}{2} b \int (c+d x) (a+b \text {arccosh}(c+d x))^2d(c+d x)+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )+\frac {3}{4} \left (-\frac {3}{2} b \int (c+d x) (a+b \text {arccosh}(c+d x))^2d(c+d x)+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )+\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )+\frac {1}{2} \int \frac {(a+b \text {arccosh}(c+d x))^3}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )+\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arccosh}(c+d x))}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)\right )+\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6354 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))\right )+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )+\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))\right )\right )+\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )+\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )+\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)+\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )\right )+\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )+\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6308 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^4-b \left (\frac {1}{4} \sqrt {c+d x-1} (c+d x)^3 \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arccosh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3 (a+b \text {arccosh}(c+d x))+\frac {3}{4} \left (\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))+\frac {(a+b \text {arccosh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )-\frac {1}{16} b (c+d x)^4\right )\right )+\frac {3}{4} \left (\frac {(a+b \text {arccosh}(c+d x))^4}{8 b}+\frac {1}{2} \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))^3-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))^2-b \left (\frac {1}{2} \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x) (a+b \text {arccosh}(c+d x))+\frac {(a+b \text {arccosh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )\right )\right )\right )\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^3*(a + b*ArcCosh[c + d*x])^4,x]
Output:
(e^3*(((c + d*x)^4*(a + b*ArcCosh[c + d*x])^4)/4 - b*((Sqrt[-1 + c + d*x]* (c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/4 - (3*b*(((c + d*x)^4*(a + b*ArcCosh[c + d*x])^2)/4 - (b*(-1/16*(b*(c + d*x)^4) + (Sqrt[- 1 + c + d*x]*(c + d*x)^3*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/4 + ( 3*(-1/4*(b*(c + d*x)^2) + (Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]* (a + b*ArcCosh[c + d*x]))/2 + (a + b*ArcCosh[c + d*x])^2/(4*b)))/4))/2))/4 + (3*((Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x])^3)/2 + (a + b*ArcCosh[c + d*x])^4/(8*b) - (3*b*(((c + d*x)^2*(a + b* ArcCosh[c + d*x])^2)/2 - b*(-1/4*(b*(c + d*x)^2) + (Sqrt[-1 + c + d*x]*(c + d*x)*Sqrt[1 + c + d*x]*(a + b*ArcCosh[c + d*x]))/2 + (a + b*ArcCosh[c + d*x])^2/(4*b))))/2))/4)))/d
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -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_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sq rt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]]*(a + b*ArcCosh[ c*x])^(n + 1), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1 ] && EqQ[e2, (-c)*d2] && NeQ[n, -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[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.42 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.75
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{8}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{32}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{64}-\frac {45 \operatorname {arccosh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}+\frac {45 \left (d x +c \right )^{2}}{128}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{16}-\frac {9 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{128}-\frac {45 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{256}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right )}{256}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )+4 e^{3} a^{3} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\) | \(658\) |
| default | \(\frac {\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{8}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{32}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{64}-\frac {45 \operatorname {arccosh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}+\frac {45 \left (d x +c \right )^{2}}{128}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{16}-\frac {9 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{128}-\frac {45 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{256}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right )}{256}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )+4 e^{3} a^{3} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\) | \(658\) |
| parts | \(\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{4}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{8}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{32}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{64}-\frac {45 \operatorname {arccosh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}+\frac {45 \left (d x +c \right )^{2}}{128}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )^{2}}{16}\right )}{d}+\frac {4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{16}-\frac {9 \operatorname {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{32}-\frac {3 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{128}-\frac {45 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{256}-\frac {45 \,\operatorname {arccosh}\left (d x +c \right )}{256}+\frac {9 \left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{32}\right )}{d}+\frac {6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{8}-\frac {3 \,\operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arccosh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{2}}{32}\right )}{d}+\frac {4 e^{3} a^{3} b \left (\frac {\left (d x +c \right )^{4} \operatorname {arccosh}\left (d x +c \right )}{4}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{3} \sqrt {\left (d x +c \right )^{2}-1}+3 \left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+3 \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\) | \(669\) |
| orering | \(\text {Expression too large to display}\) | \(2518\) |
Input:
int((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(1/4*e^3*a^4*(d*x+c)^4+e^3*b^4*(1/4*(d*x+c)^4*arccosh(d*x+c)^4-1/4*(d* x+c)^3*arccosh(d*x+c)^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-3/8*arccosh(d*x+c) ^3*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-3/32*arccosh(d*x+c)^4+3/16*(d*x +c)^4*arccosh(d*x+c)^2-3/32*(d*x+c)^3*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+ c+1)^(1/2)-45/64*arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-45 /128*arccosh(d*x+c)^2+3/128*(d*x+c)^4+45/128*(d*x+c)^2+9/16*(d*x+c)^2*arcc osh(d*x+c)^2)+4*e^3*a*b^3*(1/4*(d*x+c)^4*arccosh(d*x+c)^3-3/16*(d*x+c)^3*a rccosh(d*x+c)^2*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-9/32*arccosh(d*x+c)^2*(d*x +c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-3/32*arccosh(d*x+c)^3+3/32*(d*x+c)^4*a rccosh(d*x+c)-3/128*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)^3-45/256*(d*x+ c-1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-45/256*arccosh(d*x+c)+9/32*(d*x+c)^2*ar ccosh(d*x+c))+6*e^3*a^2*b^2*(1/4*(d*x+c)^4*arccosh(d*x+c)^2-1/8*(d*x+c)^3* arccosh(d*x+c)*(d*x+c-1)^(1/2)*(d*x+c+1)^(1/2)-3/16*arccosh(d*x+c)*(d*x+c- 1)^(1/2)*(d*x+c+1)^(1/2)*(d*x+c)-3/32*arccosh(d*x+c)^2+1/32*(d*x+c)^4+3/32 *(d*x+c)^2)+4*e^3*a^3*b*(1/4*(d*x+c)^4*arccosh(d*x+c)-1/32*(d*x+c-1)^(1/2) *(d*x+c+1)^(1/2)*(2*(d*x+c)^3*((d*x+c)^2-1)^(1/2)+3*(d*x+c)*((d*x+c)^2-1)^ (1/2)+3*ln(d*x+c+((d*x+c)^2-1)^(1/2)))/((d*x+c)^2-1)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1236 vs. \(2 (339) = 678\).
Time = 0.13 (sec) , antiderivative size = 1236, normalized size of antiderivative = 3.28 \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=\text {Too large to display} \] Input:
integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x, algorithm="fricas")
Output:
1/128*((32*a^4 + 24*a^2*b^2 + 3*b^4)*d^4*e^3*x^4 + 4*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c*d^3*e^3*x^3 + 3*(24*a^2*b^2 + 15*b^4 + 2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^2)*d^2*e^3*x^2 + 2*(2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^3 + 9*(8*a ^2*b^2 + 5*b^4)*c)*d*e^3*x + 4*(8*b^4*d^4*e^3*x^4 + 32*b^4*c*d^3*e^3*x^3 + 48*b^4*c^2*d^2*e^3*x^2 + 32*b^4*c^3*d*e^3*x + (8*b^4*c^4 - 3*b^4)*e^3)*lo g(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^4 + 16*(8*a*b^3*d^4*e^3*x^4 + 32*a*b^3*c*d^3*e^3*x^3 + 48*a*b^3*c^2*d^2*e^3*x^2 + 32*a*b^3*c^3*d*e^3* x + (8*a*b^3*c^4 - 3*a*b^3)*e^3 - (2*b^4*d^3*e^3*x^3 + 6*b^4*c*d^2*e^3*x^2 + 3*(2*b^4*c^2 + b^4)*d*e^3*x + (2*b^4*c^3 + 3*b^4*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^3 + 3*(8*(8*a^2*b^2 + b^4)*d^4*e^3*x^4 + 32*(8*a^2*b^2 + b^4)*c*d^3*e^3*x^3 + 24*(b^4 + 2*(8*a^2*b^2 + b^4)*c^2)*d^2*e^3*x^2 + 16*(3*b^4*c + 2*(8*a^2*b^ 2 + b^4)*c^3)*d*e^3*x + (24*b^4*c^2 + 8*(8*a^2*b^2 + b^4)*c^4 - 24*a^2*b^2 - 15*b^4)*e^3 - 16*(2*a*b^3*d^3*e^3*x^3 + 6*a*b^3*c*d^2*e^3*x^2 + 3*(2*a* b^3*c^2 + a*b^3)*d*e^3*x + (2*a*b^3*c^3 + 3*a*b^3*c)*e^3)*sqrt(d^2*x^2 + 2 *c*d*x + c^2 - 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1))^2 + 2* (8*(8*a^3*b + 3*a*b^3)*d^4*e^3*x^4 + 32*(8*a^3*b + 3*a*b^3)*c*d^3*e^3*x^3 + 24*(3*a*b^3 + 2*(8*a^3*b + 3*a*b^3)*c^2)*d^2*e^3*x^2 + 16*(9*a*b^3*c + 2 *(8*a^3*b + 3*a*b^3)*c^3)*d*e^3*x + (72*a*b^3*c^2 + 8*(8*a^3*b + 3*a*b^3)* c^4 - 24*a^3*b - 45*a*b^3)*e^3 - 3*(2*(8*a^2*b^2 + b^4)*d^3*e^3*x^3 + 6...
\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=e^{3} \left (\int a^{4} c^{3}\, dx + \int a^{4} d^{3} x^{3}\, dx + \int b^{4} c^{3} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 4 a b^{3} c^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a^{2} b^{2} c^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a^{3} b c^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 3 a^{4} c d^{2} x^{2}\, dx + \int 3 a^{4} c^{2} d x\, dx + \int b^{4} d^{3} x^{3} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 4 a b^{3} d^{3} x^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 6 a^{2} b^{2} d^{3} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 4 a^{3} b d^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 3 b^{4} c d^{2} x^{2} \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 3 b^{4} c^{2} d x \operatorname {acosh}^{4}{\left (c + d x \right )}\, dx + \int 12 a b^{3} c d^{2} x^{2} \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 12 a b^{3} c^{2} d x \operatorname {acosh}^{3}{\left (c + d x \right )}\, dx + \int 18 a^{2} b^{2} c d^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 18 a^{2} b^{2} c^{2} d x \operatorname {acosh}^{2}{\left (c + d x \right )}\, dx + \int 12 a^{3} b c d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 12 a^{3} b c^{2} d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((d*e*x+c*e)**3*(a+b*acosh(d*x+c))**4,x)
Output:
e**3*(Integral(a**4*c**3, x) + Integral(a**4*d**3*x**3, x) + Integral(b**4 *c**3*acosh(c + d*x)**4, x) + Integral(4*a*b**3*c**3*acosh(c + d*x)**3, x) + Integral(6*a**2*b**2*c**3*acosh(c + d*x)**2, x) + Integral(4*a**3*b*c** 3*acosh(c + d*x), x) + Integral(3*a**4*c*d**2*x**2, x) + Integral(3*a**4*c **2*d*x, x) + Integral(b**4*d**3*x**3*acosh(c + d*x)**4, x) + Integral(4*a *b**3*d**3*x**3*acosh(c + d*x)**3, x) + Integral(6*a**2*b**2*d**3*x**3*aco sh(c + d*x)**2, x) + Integral(4*a**3*b*d**3*x**3*acosh(c + d*x), x) + Inte gral(3*b**4*c*d**2*x**2*acosh(c + d*x)**4, x) + Integral(3*b**4*c**2*d*x*a cosh(c + d*x)**4, x) + Integral(12*a*b**3*c*d**2*x**2*acosh(c + d*x)**3, x ) + Integral(12*a*b**3*c**2*d*x*acosh(c + d*x)**3, x) + Integral(18*a**2*b **2*c*d**2*x**2*acosh(c + d*x)**2, x) + Integral(18*a**2*b**2*c**2*d*x*aco sh(c + d*x)**2, x) + Integral(12*a**3*b*c*d**2*x**2*acosh(c + d*x), x) + I ntegral(12*a**3*b*c**2*d*x*acosh(c + d*x), x))
\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4} \,d x } \] Input:
integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x, algorithm="maxima")
Output:
1/4*a^4*d^3*e^3*x^4 + a^4*c*d^2*e^3*x^3 + 3/2*a^4*c^2*d*e^3*x^2 + 3*(2*x^2 *arccosh(d*x + c) - d*(3*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d* x + c^2 - 1)*d)/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x/d^2 - (c^2 - 1)* log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^3 - 3*sqrt( d^2*x^2 + 2*c*d*x + c^2 - 1)*c/d^3))*a^3*b*c^2*d*e^3 + 2/3*(6*x^3*arccosh( d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^2/d^2 - 15*c^3*log(2*d ^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x/d^3 + 9*(c^2 - 1)*c*log(2*d^2*x + 2*c*d + 2*sqrt (d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 )*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)/d^4))*a^3*b*c*d^ 2*e^3 + 1/24*(24*x^4*arccosh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c*x^2/d^3 + 105*c^4*log( 2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 + 35*sqrt(d^2 *x^2 + 2*c*d*x + c^2 - 1)*c^2*x/d^4 - 90*(c^2 - 1)*c^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*d)/d^5 - 105*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*x/d^4 + 9*(c^2 - 1)^2*log(2*d^2*x + 2*c*d + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)* d)/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1)*(c^2 - 1)*c/d^5)*d)*a^3*b*d^ 3*e^3 + a^4*c^3*e^3*x + 4*((d*x + c)*arccosh(d*x + c) - sqrt((d*x + c)^2 - 1))*a^3*b*c^3*e^3/d + 1/4*(b^4*d^3*e^3*x^4 + 4*b^4*c*d^2*e^3*x^3 + 6*b...
\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4} \,d x } \] Input:
integrate((d*e*x+c*e)^3*(a+b*arccosh(d*x+c))^4,x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^3*(b*arccosh(d*x + c) + a)^4, x)
Timed out. \[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \] Input:
int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^4,x)
Output:
int((c*e + d*e*x)^3*(a + b*acosh(c + d*x))^4, x)
\[ \int (c e+d e x)^3 (a+b \text {arccosh}(c+d x))^4 \, dx =\text {Too large to display} \] Input:
int((d*e*x+c*e)^3*(a+b*acosh(d*x+c))^4,x)
Output:
(e**3*(32*acosh(c + d*x)*a**3*b*c**4 + 32*acosh(c + d*x)*a**3*b*c**3*d*x + 48*acosh(c + d*x)*a**3*b*c**2*d**2*x**2 + 32*acosh(c + d*x)*a**3*b*c*d**3 *x**3 + 8*acosh(c + d*x)*a**3*b*d**4*x**4 + 30*sqrt(c**2 + 2*c*d*x + d**2* x**2 - 1)*a**3*b*c**3 - 6*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b*c**2 *d*x - 6*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b*c*d**2*x**2 - 3*sqrt( c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b*c - 2*sqrt(c**2 + 2*c*d*x + d**2*x* *2 - 1)*a**3*b*d**3*x**3 - 3*sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1)*a**3*b*d *x - 32*sqrt(c + d*x + 1)*sqrt(c + d*x - 1)*a**3*b*c**3 + 8*int(acosh(c + d*x)**4,x)*b**4*c**3*d + 32*int(acosh(c + d*x)**3,x)*a*b**3*c**3*d + 48*in t(acosh(c + d*x)**2,x)*a**2*b**2*c**3*d + 8*int(acosh(c + d*x)**4*x**3,x)* b**4*d**4 + 24*int(acosh(c + d*x)**4*x**2,x)*b**4*c*d**3 + 24*int(acosh(c + d*x)**4*x,x)*b**4*c**2*d**2 + 32*int(acosh(c + d*x)**3*x**3,x)*a*b**3*d* *4 + 96*int(acosh(c + d*x)**3*x**2,x)*a*b**3*c*d**3 + 96*int(acosh(c + d*x )**3*x,x)*a*b**3*c**2*d**2 + 48*int(acosh(c + d*x)**2*x**3,x)*a**2*b**2*d* *4 + 144*int(acosh(c + d*x)**2*x**2,x)*a**2*b**2*c*d**3 + 144*int(acosh(c + d*x)**2*x,x)*a**2*b**2*c**2*d**2 - 24*log(sqrt(c**2 + 2*c*d*x + d**2*x** 2 - 1) + c + d*x)*a**3*b*c**4 - 3*log(sqrt(c**2 + 2*c*d*x + d**2*x**2 - 1) + c + d*x)*a**3*b + 8*a**4*c**3*d*x + 12*a**4*c**2*d**2*x**2 + 8*a**4*c*d **3*x**3 + 2*a**4*d**4*x**4))/(8*d)