Integrand size = 35, antiderivative size = 283 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\frac {i b d^3 x \left (1+c^2 x^2\right )^{3/2}}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i d^3 (1+i c x) \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {i d^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {3 d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 b d^3 \left (1+c^2 x^2\right )^{3/2} \log (i+c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
I*b*d^3*x*(c^2*x^2+1)^(3/2)/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-4*I*d^3*(1 +I*c*x)*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/ 2)-I*d^3*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^ (3/2)-3/2*d^3*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^2/b/c/(d+I*c*d*x)^(3/2) /(f-I*c*f*x)^(3/2)-4*b*d^3*(c^2*x^2+1)^(3/2)*ln(c*x+I)/c/(d+I*c*d*x)^(3/2) /(f-I*c*f*x)^(3/2)
Time = 7.29 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.82 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\frac {\frac {2 a d (5-i c x) \sqrt {d+i c d x} \sqrt {f-i c f x}}{f^2 (i+c x)}-\frac {6 a d^{3/2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{f^{3/2}}+\frac {b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+4 \text {arcsinh}(c x) \left (-i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \left (4 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (1+c^2 x^2\right )\right ) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{f^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}+\frac {2 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \left (-\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\left (c x-4 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (1+c^2 x^2\right )\right ) \left (i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+\sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\text {arcsinh}(c x) \left (-i \left (2+\sqrt {1+c^2 x^2}\right ) \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\left (-2+\sqrt {1+c^2 x^2}\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{f^2 \sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}}{2 c} \]
((2*a*d*(5 - I*c*x)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])/(f^2*(I + c*x)) - (6*a*d^(3/2)*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c *f*x]])/f^(3/2) + (b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x] ^2*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + 4*ArcSinh[c*x]*((-I) *Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2]) + 2*(4*ArcTan[Tanh[ArcSinh[c *x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/ 2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2 ])) + (2*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(-(ArcSinh[c*x]^2*(Cosh[A rcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) + (c*x - 4*ArcTan[Coth[ArcSinh[c *x]/2]] + I*Log[1 + c^2*x^2])*(I*Cosh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/ 2]) + ArcSinh[c*x]*((-I)*(2 + Sqrt[1 + c^2*x^2])*Cosh[ArcSinh[c*x]/2] - (- 2 + Sqrt[1 + c^2*x^2])*Sinh[ArcSinh[c*x]/2])))/(f^2*Sqrt[1 + c^2*x^2]*(Cos h[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])))/(2*c)
Time = 0.70 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6211, 27, 6259, 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.
| \(\displaystyle \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\left (c^2 x^2+1\right )^{3/2} \int \frac {d^3 (i c x+1)^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^{3/2} \int \frac {(i c x+1)^3 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
| \(\Big \downarrow \) 6259 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^{3/2} \int \left (-\frac {i c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {4 i (i-c x) (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}\right )dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^3 \left (c^2 x^2+1\right )^{3/2} \left (-\frac {i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {4 i (1+i c x) (a+b \text {arcsinh}(c x))}{c \sqrt {c^2 x^2+1}}-\frac {3 (a+b \text {arcsinh}(c x))^2}{2 b c}-\frac {4 b \log (c x+i)}{c}+i b x\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
(d^3*(1 + c^2*x^2)^(3/2)*(I*b*x - ((4*I)*(1 + I*c*x)*(a + b*ArcSinh[c*x])) /(c*Sqrt[1 + c^2*x^2]) - (I*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c - (3 *(a + b*ArcSinh[c*x])^2)/(2*b*c) - (4*b*Log[I + c*x])/c))/((d + I*c*d*x)^( 3/2)*(f - I*c*f*x)^(3/2))
3.6.59.3.1 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[(a + b*ArcSinh[c* x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; FreeQ[{ a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IntegerQ[m] && ILtQ[p + 1/2, 0 ] && GtQ[d, 0] && IGtQ[n, 0]
\[\int \frac {\left (i c d x +d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{\left (-i c f x +f \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
integral(((-I*b*c*d*x - 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*c*d*x - a*d)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f))/(c^2*f^2*x^2 + 2*I*c*f^2*x - f^2), x)
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\int \frac {\left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (- i f \left (c x + i\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
a*(-I*(c^2*d*f*x^2 + d*f)^(3/2)/(c^3*f^3*x^2 + 2*I*c^2*f^3*x - c*f^3) - 6* I*sqrt(c^2*d*f*x^2 + d*f)*d/(-I*c^2*f^2*x + c*f^2) - 3*d^2*arcsinh(c*x)/(c *f^2*sqrt(d/f))) + b*integrate((I*c*d*x + d)^(3/2)*log(c*x + sqrt(c^2*x^2 + 1))/(-I*c*f*x + f)^(3/2), x)
Exception generated. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to transpose Error: Bad Argu ment Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l ) Error:
Timed out. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{(f-i c f x)^{3/2}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}}{{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]