Integrand size = 37, antiderivative size = 436 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {4 i b^2 d^2 \left (1+c^2 x^2\right )}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {b^2 d^2 x \left (1+c^2 x^2\right )}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b^2 d^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {4 i b d^2 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c d^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \] Output:
4*I*b^2*d^2*(c^2*x^2+1)/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/4*b^2*d^2* x*(c^2*x^2+1)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/4*b^2*d^2*(c^2*x^2+1)^ (1/2)*arcsinh(c*x)/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-4*I*b*d^2*x*(c^2* x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/2*b* c*d^2*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/(d+I*c*d*x)^(1/2)/(f-I*c*f* x)^(1/2)+2*I*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(1/2)/(f-I *c*f*x)^(1/2)-1/2*d^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(1/2) /(f-I*c*f*x)^(1/2)+1/2*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^3/b/c/(d+I *c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)
Time = 7.61 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.21 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {-32 i a b c d x \sqrt {d+i c d x} \sqrt {f-i c f x}+16 i a^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+32 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 a^2 c d x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+4 b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3+2 a b d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+2 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-16 i b c x-4 a (-4 i+c x) \sqrt {1+c^2 x^2}+b \cosh (2 \text {arcsinh}(c x))\right )+12 a^2 d^{3/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+2 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (6 a+8 i b \sqrt {1+c^2 x^2}-b \sinh (2 \text {arcsinh}(c x))\right )}{8 c f \sqrt {1+c^2 x^2}} \] Input:
Integrate[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x ]
Output:
((-32*I)*a*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (16*I)*a^2*d*Sqrt [d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (32*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*a^2*c*d*x*Sqrt[d + I*c*d* x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 4*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 + 2*a*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Co sh[2*ArcSinh[c*x]] + 2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x ]*((-16*I)*b*c*x - 4*a*(-4*I + c*x)*Sqrt[1 + c^2*x^2] + b*Cosh[2*ArcSinh[c *x]]) + 12*a^2*d^(3/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqr t[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] - b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f* x]*ArcSinh[c*x]^2*(6*a + (8*I)*b*Sqrt[1 + c^2*x^2] - b*Sinh[2*ArcSinh[c*x] ]))/(8*c*f*Sqrt[1 + c^2*x^2])
Time = 0.86 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.52, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {6211, 27, 6258, 3042, 3798, 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.
| \(\displaystyle \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\sqrt {c^2 x^2+1} \int \frac {d^2 (i c x+1)^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \int \frac {(i c x+1)^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
| \(\Big \downarrow \) 6258 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \int \left (i x c^2+c\right )^2 (a+b \text {arcsinh}(c x))^2d\text {arcsinh}(c x)}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))^2 (\sin (i \text {arcsinh}(c x)) c+c)^2d\text {arcsinh}(c x)}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
| \(\Big \downarrow \) 3798 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \int \left (-x^2 (a+b \text {arcsinh}(c x))^2 c^4+2 i x (a+b \text {arcsinh}(c x))^2 c^3+(a+b \text {arcsinh}(c x))^2 c^2\right )d\text {arcsinh}(c x)}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \left (\frac {1}{2} b c^4 x^2 (a+b \text {arcsinh}(c x))-4 i b c^3 x (a+b \text {arcsinh}(c x))+2 i c^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {c^2 (a+b \text {arcsinh}(c x))^3}{2 b}-\frac {1}{2} c^3 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} b^2 c^2 \text {arcsinh}(c x)+4 i b^2 c^2 \sqrt {c^2 x^2+1}-\frac {1}{4} b^2 c^3 x \sqrt {c^2 x^2+1}\right )}{c^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
Input:
Int[((d + I*c*d*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/Sqrt[f - I*c*f*x],x]
Output:
(d^2*Sqrt[1 + c^2*x^2]*((4*I)*b^2*c^2*Sqrt[1 + c^2*x^2] - (b^2*c^3*x*Sqrt[ 1 + c^2*x^2])/4 + (b^2*c^2*ArcSinh[c*x])/4 - (4*I)*b*c^3*x*(a + b*ArcSinh[ c*x]) + (b*c^4*x^2*(a + b*ArcSinh[c*x]))/2 + (2*I)*c^2*Sqrt[1 + c^2*x^2]*( a + b*ArcSinh[c*x])^2 - (c^3*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (c^2*(a + b*ArcSinh[c*x])^3)/(2*b)))/(c^3*Sqrt[d + I*c*d*x]*Sqrt[f - I* c*f*x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ ) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 + c^2*x ^2)^q) Int[(d + e*x)^(p - q)*(1 + c^2*x^2)^q*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^ 2 + e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/S qrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[1/(c^(m + 1)*Sqrt[d]) Subst[I nt[(a + b*x)^n*(c*f + g*Sinh[x])^m, x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b , c, d, e, f, g, n}, x] && EqQ[e, c^2*d] && IntegerQ[m] && GtQ[d, 0] && (Gt Q[m, 0] || IGtQ[n, 0])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (367 ) = 734\).
Time = 6.21 (sec) , antiderivative size = 978, normalized size of antiderivative = 2.24
| method | result | size |
| default | \(\frac {i a^{2} \left (i c d x +d \right )^{\frac {3}{2}} \sqrt {-i c f x +f}}{2 c f}+\frac {3 i a^{2} d \sqrt {i c d x +d}\, \sqrt {-i c f x +f}}{2 c f}+\frac {3 a^{2} d^{2} \sqrt {\left (-i c f x +f \right ) \left (i c d x +d \right )}\, \ln \left (\frac {c^{2} d f x}{\sqrt {c^{2} d f}}+\sqrt {c^{2} d f \,x^{2}+d f}\right )}{2 \sqrt {i c d x +d}\, \sqrt {-i c f x +f}\, \sqrt {c^{2} d f}}+b^{2} \left (\frac {d \operatorname {arcsinh}\left (x c \right )^{3} \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{2 \sqrt {c^{2} x^{2}+1}\, f c}-\frac {d \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{16 f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right ) \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right ) \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}-\frac {d \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{16 f c \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 d \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{4 \sqrt {c^{2} x^{2}+1}\, f c}-\frac {d \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{16 f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )-1\right ) \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )+1\right ) \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}-\frac {d \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{16 f c \left (c^{2} x^{2}+1\right )}\right )\) | \(978\) |
| parts | \(\frac {i a^{2} \left (i c d x +d \right )^{\frac {3}{2}} \sqrt {-i c f x +f}}{2 c f}+\frac {3 i a^{2} d \sqrt {i c d x +d}\, \sqrt {-i c f x +f}}{2 c f}+\frac {3 a^{2} d^{2} \sqrt {\left (-i c f x +f \right ) \left (i c d x +d \right )}\, \ln \left (\frac {c^{2} d f x}{\sqrt {c^{2} d f}}+\sqrt {c^{2} d f \,x^{2}+d f}\right )}{2 \sqrt {i c d x +d}\, \sqrt {-i c f x +f}\, \sqrt {c^{2} d f}}+b^{2} \left (\frac {d \operatorname {arcsinh}\left (x c \right )^{3} \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{2 \sqrt {c^{2} x^{2}+1}\, f c}-\frac {d \left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{16 f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+2\right ) \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+2\right ) \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}-\frac {d \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{16 f c \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 d \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {-i \left (x c +i\right ) f}\, \sqrt {i \left (x c -i\right ) d}}{4 \sqrt {c^{2} x^{2}+1}\, f c}-\frac {d \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{16 f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )-1\right ) \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}+\frac {i d \left (\operatorname {arcsinh}\left (x c \right )+1\right ) \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{f c \left (c^{2} x^{2}+1\right )}-\frac {d \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}}{16 f c \left (c^{2} x^{2}+1\right )}\right )\) | \(978\) |
Input:
int((d+I*c*d*x)^(3/2)*(a+b*arcsinh(x*c))^2/(f-I*c*f*x)^(1/2),x,method=_RET URNVERBOSE)
Output:
1/2*I*a^2/c/f*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(1/2)+3/2*I*a^2*d/c/f*(d+I*c*d *x)^(1/2)*(f-I*c*f*x)^(1/2)+3/2*a^2*d^2*((f-I*c*f*x)*(d+I*c*d*x))^(1/2)/(d +I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)*ln(c^2*d*f*x/(c^2*d*f)^(1/2)+(c^2*d*f*x^ 2+d*f)^(1/2))/(c^2*d*f)^(1/2)+b^2*(1/2*d*arcsinh(x*c)^3*(I*(x*c-I)*d)^(1/2 )*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/f/c-1/16*d*(2*arcsinh(x*c)^2-2*ar csinh(x*c)+1)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/ 2))*(-I*(I+x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)/f/c/(c^2*x^2+1)+I*d*(arcsinh( x*c)^2-2*arcsinh(x*c)+2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(I*(x*c-I)*d)^( 1/2)*(-I*(I+x*c)*f)^(1/2)/f/c/(c^2*x^2+1)+I*d*(arcsinh(x*c)^2+2*arcsinh(x* c)+2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f) ^(1/2)/f/c/(c^2*x^2+1)-1/16*d*(2*arcsinh(x*c)^2+2*arcsinh(x*c)+1)*(2*x^3*c ^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(-I*(I+x*c)*f)^(1/ 2)*(I*(x*c-I)*d)^(1/2)/f/c/(c^2*x^2+1))+2*a*b*(3/4*d*arcsinh(x*c)^2*(-I*(I +x*c)*f)^(1/2)*(I*(x*c-I)*d)^(1/2)/(c^2*x^2+1)^(1/2)/f/c-1/16*d*(-1+2*arcs inh(x*c))*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))* (I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/f/c/(c^2*x^2+1)+I*d*(arcsinh(x*c) -1)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^( 1/2)/f/c/(c^2*x^2+1)+I*d*(arcsinh(x*c)+1)*(c^2*x^2-(c^2*x^2+1)^(1/2)*x*c+1 )*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/f/c/(c^2*x^2+1)-1/16*d*(1+2*arc sinh(x*c))*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/...
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-i \, c f x + f}} \,d x } \] Input:
integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x, algo rithm="fricas")
Output:
integral(-((b^2*c*d*x - I*b^2*d)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log( c*x + sqrt(c^2*x^2 + 1))^2 + 2*(a*b*c*d*x - I*a*b*d)*sqrt(I*c*d*x + d)*sqr t(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (a^2*c*d*x - I*a^2*d)*sqrt( I*c*d*x + d)*sqrt(-I*c*f*x + f))/(c*f*x + I*f), x)
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int \frac {\left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {- i f \left (c x + i\right )}}\, dx \] Input:
integrate((d+I*c*d*x)**(3/2)*(a+b*asinh(c*x))**2/(f-I*c*f*x)**(1/2),x)
Output:
Integral((I*d*(c*x - I))**(3/2)*(a + b*asinh(c*x))**2/sqrt(-I*f*(c*x + I)) , x)
Exception generated. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x, algo rithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{\sqrt {-i \, c f x + f}} \,d x } \] Input:
integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))^2/(f-I*c*f*x)^(1/2),x, algo rithm="giac")
Output:
integrate((I*c*d*x + d)^(3/2)*(b*arcsinh(c*x) + a)^2/sqrt(-I*c*f*x + f), x )
Timed out. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \] Input:
int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(3/2))/(f - c*f*x*1i)^(1/2),x)
Output:
int(((a + b*asinh(c*x))^2*(d + c*d*x*1i)^(3/2))/(f - c*f*x*1i)^(1/2), x)
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))^2}{\sqrt {f-i c f x}} \, dx=\frac {\sqrt {d}\, d \left (6 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a^{2} i -\sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} c x +4 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a^{2} i +4 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right ) x}{\sqrt {-c i x +1}}d x \right ) a b \,c^{2} i +4 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )}{\sqrt {-c i x +1}}d x \right ) a b c +2 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2} x}{\sqrt {-c i x +1}}d x \right ) b^{2} c^{2} i +2 \left (\int \frac {\sqrt {c i x +1}\, \mathit {asinh} \left (c x \right )^{2}}{\sqrt {-c i x +1}}d x \right ) b^{2} c \right )}{2 \sqrt {f}\, c} \] Input:
int((d+I*c*d*x)^(3/2)*(a+b*asinh(c*x))^2/(f-I*c*f*x)^(1/2),x)
Output:
(sqrt(d)*d*(6*asin(sqrt( - c*i*x + 1)/sqrt(2))*a**2*i - sqrt(c*i*x + 1)*sq rt( - c*i*x + 1)*a**2*c*x + 4*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a**2*i + 4*int((sqrt(c*i*x + 1)*asinh(c*x)*x)/sqrt( - c*i*x + 1),x)*a*b*c**2*i + 4* int((sqrt(c*i*x + 1)*asinh(c*x))/sqrt( - c*i*x + 1),x)*a*b*c + 2*int((sqrt (c*i*x + 1)*asinh(c*x)**2*x)/sqrt( - c*i*x + 1),x)*b**2*c**2*i + 2*int((sq rt(c*i*x + 1)*asinh(c*x)**2)/sqrt( - c*i*x + 1),x)*b**2*c))/(2*sqrt(f)*c)