Integrand size = 37, antiderivative size = 508 \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {4 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 c}+\frac {1}{4} b^2 d x \sqrt {d+i c d x} \sqrt {f-i c f x}+\frac {2 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right )}{27 c}-\frac {b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)}{4 c \sqrt {1+c^2 x^2}}-\frac {2 i b d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {b c d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}}-\frac {2 i b c^2 d x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {1}{2} d x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2+\frac {i d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^3}{6 b c \sqrt {1+c^2 x^2}} \]
4/9*I*b^2*d*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c+1/4*b^2*d*x*(d+I*c*d*x)^ (1/2)*(f-I*c*f*x)^(1/2)+2/27*I*b^2*d*(c^2*x^2+1)*(d+I*c*d*x)^(1/2)*(f-I*c* f*x)^(1/2)/c+1/2*d*x*(a+b*arcsinh(c*x))^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1 /2)+1/3*I*d*(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)/c-1/4*b^2*d*arcsinh(c*x)*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/c/(c^2 *x^2+1)^(1/2)-2/3*I*b*d*x*(a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2)*(f-I*c*f*x) ^(1/2)/(c^2*x^2+1)^(1/2)-1/2*b*c*d*x^2*(a+b*arcsinh(c*x))*(d+I*c*d*x)^(1/2 )*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)-2/9*I*b*c^2*d*x^3*(a+b*arcsinh(c*x)) *(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/6*d*(a+b*arcsinh( c*x))^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/b/c/(c^2*x^2+1)^(1/2)
Time = 3.23 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.39 \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {-108 i a b c d x \sqrt {d+i c d x} \sqrt {f-i c f x}+72 i a^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+108 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+108 a^2 c d x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+72 i a^2 c^2 d x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+36 b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^3-54 a b d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+4 i b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (3 \text {arcsinh}(c x))+108 a^2 d^{3/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+27 b^2 d \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (2 \text {arcsinh}(c x))+18 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2 \left (6 a+3 i b \sqrt {1+c^2 x^2}+i b \cosh (3 \text {arcsinh}(c x))+3 b \sinh (2 \text {arcsinh}(c x))\right )-12 i a b d \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+6 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (-9 b \cosh (2 \text {arcsinh}(c x))+2 \left (-9 i b c x+9 i a \sqrt {1+c^2 x^2}+3 i a \cosh (3 \text {arcsinh}(c x))+9 a \sinh (2 \text {arcsinh}(c x))-i b \sinh (3 \text {arcsinh}(c x))\right )\right )}{216 c \sqrt {1+c^2 x^2}} \]
((-108*I)*a*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (72*I)*a^2*d*Sqr t[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (108*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 108*a^2*c*d*x*Sqrt[d + I* c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (72*I)*a^2*c^2*d*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36*b^2*d*Sqrt[d + I*c*d*x] *Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 54*a*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I* c*f*x]*Cosh[2*ArcSinh[c*x]] + (4*I)*b^2*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f *x]*Cosh[3*ArcSinh[c*x]] + 108*a^2*d^(3/2)*Sqrt[f]*Sqrt[1 + c^2*x^2]*Log[c *d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 27*b^2*d*S qrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 18*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(6*a + (3*I)*b*Sqrt[1 + c^2*x^2] + I*b*Cosh[3*ArcSinh[c*x]] + 3*b*Sinh[2*ArcSinh[c*x]]) - (12*I)*a*b*d*Sqr t[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 6*b*d*Sqrt[d + I*c *d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(-9*b*Cosh[2*ArcSinh[c*x]] + 2*((-9*I )*b*c*x + (9*I)*a*Sqrt[1 + c^2*x^2] + (3*I)*a*Cosh[3*ArcSinh[c*x]] + 9*a*S inh[2*ArcSinh[c*x]] - I*b*Sinh[3*ArcSinh[c*x]])))/(216*c*Sqrt[1 + c^2*x^2] )
Time = 0.88 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.51, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {6211, 27, 6253, 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.
| \(\displaystyle \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \int d (i c x+1) \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} \int (i c x+1) \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} \int \left (i c x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\frac {2}{9} i b c^2 x^3 (a+b \text {arcsinh}(c x))+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2+\frac {i \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c}-\frac {1}{2} b c x^2 (a+b \text {arcsinh}(c x))-\frac {2}{3} i b x (a+b \text {arcsinh}(c x))+\frac {(a+b \text {arcsinh}(c x))^3}{6 b c}-\frac {b^2 \text {arcsinh}(c x)}{4 c}+\frac {1}{4} b^2 x \sqrt {c^2 x^2+1}+\frac {2 i b^2 \left (c^2 x^2+1\right )^{3/2}}{27 c}+\frac {4 i b^2 \sqrt {c^2 x^2+1}}{9 c}\right )}{\sqrt {c^2 x^2+1}}\) |
(d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*((((4*I)/9)*b^2*Sqrt[1 + c^2*x^2])/ c + (b^2*x*Sqrt[1 + c^2*x^2])/4 + (((2*I)/27)*b^2*(1 + c^2*x^2)^(3/2))/c - (b^2*ArcSinh[c*x])/(4*c) - ((2*I)/3)*b*x*(a + b*ArcSinh[c*x]) - (b*c*x^2* (a + b*ArcSinh[c*x]))/2 - ((2*I)/9)*b*c^2*x^3*(a + b*ArcSinh[c*x]) + (x*Sq rt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 + ((I/3)*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/c + (a + b*ArcSinh[c*x])^3/(6*b*c)))/Sqrt[1 + c^2*x^2 ]
3.6.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
\[\int \left (i c d x +d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {-i c f x +f}d x\]
\[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {3}{2}} \sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \,d x } \]
integral((I*b^2*c*d*x + 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*(-I*a*b*c*d*x - a*b*d)*sqrt(I*c*d*x + d)*sqrt (-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (I*a^2*c*d*x + a^2*d)*sqrt(I *c*d*x + d)*sqrt(-I*c*f*x + f), x)
\[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int \left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
Exception generated. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int (d+i c d x)^{3/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}} \,d x \]