Integrand size = 29, antiderivative size = 241 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {40 b^2 \sqrt {d-c^2 d x^2}}{27 c^4 d}-\frac {2 b^2 x^2 \sqrt {d-c^2 d x^2}}{27 c^2 d}+\frac {4 b x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{3 c^3 d \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{9 c d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^4 d}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d} \] Output:
-40/27*b^2*(-c^2*d*x^2+d)^(1/2)/c^4/d-2/27*b^2*x^2*(-c^2*d*x^2+d)^(1/2)/c^ 2/d+4/3*b*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c^3/d/(c*x-1)^(1/2)/(c *x+1)^(1/2)+2/9*b*x^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/c/d/(c*x-1)^ (1/2)/(c*x+1)^(1/2)-2/3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^4/d-1/ 3*x^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/c^2/d
Time = 0.41 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.83 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (6 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (6+c^2 x^2\right )-9 a^2 \left (-2+c^2 x^2+c^4 x^4\right )-2 b^2 \left (-20+19 c^2 x^2+c^4 x^4\right )+6 b \left (b c x \sqrt {-1+c x} \sqrt {1+c x} \left (6+c^2 x^2\right )-3 a \left (-2+c^2 x^2+c^4 x^4\right )\right ) \text {arccosh}(c x)-9 b^2 \left (-2+c^2 x^2+c^4 x^4\right ) \text {arccosh}(c x)^2\right )}{27 c^4 d (-1+c x) (1+c x)} \] Input:
Integrate[(x^3*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[d - c^2*d*x^2]*(6*a*b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(6 + c^2*x^2) - 9*a^2*(-2 + c^2*x^2 + c^4*x^4) - 2*b^2*(-20 + 19*c^2*x^2 + c^4*x^4) + 6 *b*(b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(6 + c^2*x^2) - 3*a*(-2 + c^2*x^2 + c^4*x^4))*ArcCosh[c*x] - 9*b^2*(-2 + c^2*x^2 + c^4*x^4)*ArcCosh[c*x]^2))/ (27*c^4*d*(-1 + c*x)*(1 + c*x))
Time = 1.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {6353, 6298, 111, 27, 83, 6329, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 6353 |
| \(\displaystyle -\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int x^2 (a+b \text {arccosh}(c x))dx}{3 c \sqrt {d-c^2 d x^2}}+\frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c x-1} \sqrt {c x+1}}dx\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {\int \frac {2 x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \int \frac {x}{\sqrt {c x-1} \sqrt {c x+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}\) |
| \(\Big \downarrow \) 83 |
| \(\displaystyle \frac {2 \int \frac {x (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6329 |
| \(\displaystyle \frac {2 \left (-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \int (a+b \text {arccosh}(c x))dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}\right )}{3 c^2}-\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{3 c^2 d}+\frac {2 \left (-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (a x+b x \text {arccosh}(c x)-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c}\right )}{c \sqrt {d-c^2 d x^2}}\right )}{3 c^2}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (\frac {1}{3} x^3 (a+b \text {arccosh}(c x))-\frac {1}{3} b c \left (\frac {2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^4}+\frac {x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2}\right )\right )}{3 c \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x^3*(a + b*ArcCosh[c*x])^2)/Sqrt[d - c^2*d*x^2],x]
Output:
-1/3*(x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(c^2*d) - (2*b*Sqrt[ -1 + c*x]*Sqrt[1 + c*x]*(-1/3*(b*c*((2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^ 4) + (x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^2))) + (x^3*(a + b*ArcCosh[c* x]))/3))/(3*c*Sqrt[d - c^2*d*x^2]) + (2*(-((Sqrt[d - c^2*d*x^2]*(a + b*Arc Cosh[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])))/ (3*c^2)
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*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
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_.)*(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/(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]
Leaf count of result is larger than twice the leaf count of optimal. \(532\) vs. \(2(209)=418\).
Time = 0.59 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.21
| method | result | size |
| orering | \(\frac {\left (19 c^{6} x^{6}+100 c^{4} x^{4}-380 c^{2} x^{2}+240\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{27 c^{6} x^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {2 \left (c x -1\right ) \left (c x +1\right ) \left (c^{4} x^{4}+12 c^{2} x^{2}-20\right ) \left (\frac {3 x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {2 x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{9 c^{6} x^{4}}+\frac {\left (c^{2} x^{2}+20\right ) \left (c x +1\right )^{2} \left (c x -1\right )^{2} \left (\frac {6 x \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {12 x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b c}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \sqrt {c x +1}}+\frac {7 x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{2} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {2 x^{3} b^{2} c^{2}}{\left (c x -1\right ) \left (c x +1\right ) \sqrt {-c^{2} d \,x^{2}+d}}+\frac {4 x^{4} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{3} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \sqrt {c x -1}\, \sqrt {c x +1}}-\frac {x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {-c^{2} d \,x^{2}+d}\, \left (c x -1\right )^{\frac {3}{2}} \sqrt {c x +1}}-\frac {x^{3} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) b \,c^{2}}{\sqrt {-c^{2} d \,x^{2}+d}\, \sqrt {c x -1}\, \left (c x +1\right )^{\frac {3}{2}}}+\frac {3 x^{5} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2} c^{4} d^{2}}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}\right )}{27 c^{6} x^{3}}\) | \(533\) |
| default | \(a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}-6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}+6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 (1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(752\) |
| parts | \(a^{2} \left (-\frac {x^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {-c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}-6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (9 \operatorname {arccosh}\left (c x \right )^{2}+6 \,\operatorname {arccosh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +1\right ) \left (-1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 (-1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {3 \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 (1+\operatorname {arccosh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 c^{3} x^{3} \sqrt {c x -1}\, \sqrt {c x +1}+4 c^{4} x^{4}+3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x -5 c^{2} x^{2}+1\right ) \left (1+3 \,\operatorname {arccosh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}-1\right )}\right )\) | \(752\) |
Input:
int(x^3*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/27*(19*c^6*x^6+100*c^4*x^4-380*c^2*x^2+240)/c^6/x^2*(a+b*arccosh(c*x))^2 /(-c^2*d*x^2+d)^(1/2)-2/9*(c*x-1)*(c*x+1)*(c^4*x^4+12*c^2*x^2-20)/c^6/x^4* (3*x^2*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2)+2*x^3*(a+b*arccosh(c*x))/ (-c^2*d*x^2+d)^(1/2)*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+x^4*(a+b*arccosh(c*x) )^2/(-c^2*d*x^2+d)^(3/2)*c^2*d)+1/27*(c^2*x^2+20)/c^6*(c*x+1)^2*(c*x-1)^2/ x^3*(6*x*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2)+12*x^2*(a+b*arccosh(c*x ))/(-c^2*d*x^2+d)^(1/2)*b*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)+7*x^3*(a+b*arccosh (c*x))^2/(-c^2*d*x^2+d)^(3/2)*c^2*d+2*x^3*b^2*c^2/(c*x-1)/(c*x+1)/(-c^2*d* x^2+d)^(1/2)+4*x^4*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2)*b*c^3/(c*x-1)^( 1/2)/(c*x+1)^(1/2)*d-x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)*b*c^2/(c* x-1)^(3/2)/(c*x+1)^(1/2)-x^3*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)*b*c^2 /(c*x-1)^(1/2)/(c*x+1)^(3/2)+3*x^5*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(5/ 2)*c^4*d^2)
Time = 0.10 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.17 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {9 \, {\left (b^{2} c^{4} x^{4} + b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 6 \, {\left (a b c^{3} x^{3} + 6 \, a b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 6 \, {\left ({\left (b^{2} c^{3} x^{3} + 6 \, b^{2} c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 3 \, {\left (a b c^{4} x^{4} + a b c^{2} x^{2} - 2 \, a b\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} + {\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} - c^{4} d\right )}} \] Input:
integrate(x^3*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fric as")
Output:
-1/27*(9*(b^2*c^4*x^4 + b^2*c^2*x^2 - 2*b^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1))^2 - 6*(a*b*c^3*x^3 + 6*a*b*c*x)*sqrt(-c^2*d*x^2 + d)* sqrt(c^2*x^2 - 1) - 6*((b^2*c^3*x^3 + 6*b^2*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt (c^2*x^2 - 1) - 3*(a*b*c^4*x^4 + a*b*c^2*x^2 - 2*a*b)*sqrt(-c^2*d*x^2 + d) )*log(c*x + sqrt(c^2*x^2 - 1)) + ((9*a^2 + 2*b^2)*c^4*x^4 + (9*a^2 + 38*b^ 2)*c^2*x^2 - 18*a^2 - 40*b^2)*sqrt(-c^2*d*x^2 + d))/(c^6*d*x^2 - c^4*d)
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate(x**3*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral(x**3*(a + b*acosh(c*x))**2/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
Time = 0.14 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {1}{3} \, b^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{3} \, a b {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {-c^{2} d x^{2} + d} x^{2}}{c^{2} d} + \frac {2 \, \sqrt {-c^{2} d x^{2} + d}}{c^{4} d}\right )} - \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-d} x^{2} + \frac {20 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d}}{c^{2}}}{c^{2} d} - \frac {3 \, {\left (c^{2} \sqrt {-d} x^{3} + 6 \, \sqrt {-d} x\right )} \operatorname {arcosh}\left (c x\right )}{c^{3} d}\right )} + \frac {2 \, {\left (c^{2} \sqrt {-d} x^{3} + 6 \, \sqrt {-d} x\right )} a b}{9 \, c^{3} d} \] Input:
integrate(x^3*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxi ma")
Output:
-1/3*b^2*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4*d ))*arccosh(c*x)^2 - 2/3*a*b*(sqrt(-c^2*d*x^2 + d)*x^2/(c^2*d) + 2*sqrt(-c^ 2*d*x^2 + d)/(c^4*d))*arccosh(c*x) - 1/3*a^2*(sqrt(-c^2*d*x^2 + d)*x^2/(c^ 2*d) + 2*sqrt(-c^2*d*x^2 + d)/(c^4*d)) - 2/27*b^2*((sqrt(c^2*x^2 - 1)*sqrt (-d)*x^2 + 20*sqrt(c^2*x^2 - 1)*sqrt(-d)/c^2)/(c^2*d) - 3*(c^2*sqrt(-d)*x^ 3 + 6*sqrt(-d)*x)*arccosh(c*x)/(c^3*d)) + 2/9*(c^2*sqrt(-d)*x^3 + 6*sqrt(- d)*x)*a*b/(c^3*d)
Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a+b*arccosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac ")
Output:
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. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((x^3*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(1/2),x)
Output:
int((x^3*(a + b*acosh(c*x))^2)/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {x^3 (a+b \text {arccosh}(c x))^2}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {acosh} \left (c x \right ) x^{3}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{4}+3 \left (\int \frac {\mathit {acosh} \left (c x \right )^{2} x^{3}}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{4}}{3 \sqrt {d}\, c^{4}} \] Input:
int(x^3*(a+b*acosh(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)
Output:
( - sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a**2 + 6*int((acosh(c*x)*x**3)/sqrt( - c**2*x**2 + 1),x)*a*b*c**4 + 3*int((acos h(c*x)**2*x**3)/sqrt( - c**2*x**2 + 1),x)*b**2*c**4)/(3*sqrt(d)*c**4)