Integrand size = 37, antiderivative size = 972 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\frac {8 i a b d^4 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 {8 i b^2 d^4 \left (1+c^2 x^2\right )^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {b^2 d^4 x \left (1+c^2 x^2\right )^2}{4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{4 c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 i b^2 d^4 x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b c d^4 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 i d^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 d^4 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {8 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {4 i d^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {d^4 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2}{2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {5 d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^3}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {32 i b d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {16 b d^4 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {16 b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {16 b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {8 b^2 d^4 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
-8*I*b^2*d^4*(c^2*x^2+1)^2/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-4*I*d^4*( c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+1/ 4*b^2*d^4*x*(c^2*x^2+1)^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-1/4*b^2*d^4* (c^2*x^2+1)^(3/2)*arcsinh(c*x)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+32*I* b*d^4*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c /(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-1/2*b*c*d^4*x^2*(c^2*x^2+1)^(3/2)*(a+ b*arcsinh(c*x))/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*I*b^2*d^4*x*(c^2*x^2 +1)^(3/2)*arcsinh(c*x)/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*d^4*x*(c^2*x^ 2+1)*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*d^4*(c^2*x ^2+1)^(3/2)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+8*I *a*b*d^4*x*(c^2*x^2+1)^(3/2)/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+1/2*d^4*x *(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-5/ 2*d^4*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))^3/b/c/(d+I*c*d*x)^(3/2)/(f-I*c* f*x)^(3/2)-8*I*d^4*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(3/2)/(f -I*c*f*x)^(3/2)-16*b*d^4*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*ln(1+(c*x+(c ^2*x^2+1)^(1/2))^2)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)+16*b^2*d^4*(c^2* x^2+1)^(3/2)*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(3/2)/(f- I*c*f*x)^(3/2)-16*b^2*d^4*(c^2*x^2+1)^(3/2)*polylog(2,I*(c*x+(c^2*x^2+1)^( 1/2)))/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3/2)-8*b^2*d^4*(c^2*x^2+1)^(3/2)*p olylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c/(d+I*c*d*x)^(3/2)/(f-I*c*f*x)^(3...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2143\) vs. \(2(972)=1944\).
Time = 25.04 (sec) , antiderivative size = 2143, normalized size of antiderivative = 2.20 \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Result too large to show} \]
(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*(((-4*I)*a^2*d^2)/f^2 + (a^2* c*d^2*x)/(2*f^2) + (8*a^2*d^2)/(f^2*(I + c*x))))/c - (15*a^2*d^(5/2)*Log[c *d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]])/(2* c*f^(3/2)) - ((4*I)*a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f* x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]*(-(c*x) + 2*ArcSinh[c *x] + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - I*ArcSinh[c*x]^2 + 4*ArcTan[Coth[Ar cSinh[c*x]/2]] - (2*I)*Log[Sqrt[1 + c^2*x^2]]) - ((-I)*c*x - (2*I)*ArcSinh [c*x] + I*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]^2 + (4*I)*ArcTan[C oth[ArcSinh[c*x]/2]] + 2*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2]))/(c *f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSi nh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])) - (a*b*d^2*Sqrt[I*((-I)*d + c*d*x)]* Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(Cosh[ArcSinh[c*x]/2]* (8*ArcTan[Tanh[ArcSinh[c*x]/2]] + I*(ArcSinh[c*x]*(4*I + ArcSinh[c*x]) + 4 *Log[Sqrt[1 + c^2*x^2]])) + (ArcSinh[c*x]*(-4*I + ArcSinh[c*x]) - (8*I)*Ar cTan[Tanh[ArcSinh[c*x]/2]] + 4*Log[Sqrt[1 + c^2*x^2]])*Sinh[ArcSinh[c*x]/2 ]))/(c*f^2*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*Sqrt[1 + c^2*x^2]*(I*Co sh[ArcSinh[c*x]/2] + Sinh[ArcSinh[c*x]/2])) - (b^2*d^2*(-I + c*x)*Sqrt[I*( (-I)*d + c*d*x)]*Sqrt[(-I)*(I*f + c*f*x)]*Sqrt[-(d*f*(1 + c^2*x^2))]*(-18* Pi*ArcSinh[c*x] - (6 - 6*I)*ArcSinh[c*x]^2 + I*ArcSinh[c*x]^3 - 12*(Pi - ( 2*I)*ArcSinh[c*x])*Log[1 + I/E^ArcSinh[c*x]] + 24*Pi*Log[1 + E^ArcSinh[...
Time = 1.49 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.41, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, 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.
| \(\displaystyle \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\left (c^2 x^2+1\right )^{3/2} \int \frac {d^4 (i c x+1)^4 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{3/2}}dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^4 \left (c^2 x^2+1\right )^{3/2} \int \frac {(i c x+1)^4 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{3/2}}dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
| \(\Big \downarrow \) 6259 |
| \(\displaystyle \frac {d^4 \left (c^2 x^2+1\right )^{3/2} \int \left (\frac {c^2 x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {4 i c x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {7 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {8 i (i-c x) (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{3/2}}\right )dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^4 \left (c^2 x^2+1\right )^{3/2} \left (\frac {32 i b \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c}+\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2-\frac {4 i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c}+\frac {8 x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {8 i (a+b \text {arcsinh}(c x))^2}{c \sqrt {c^2 x^2+1}}-\frac {1}{2} b c x^2 (a+b \text {arcsinh}(c x))-\frac {5 (a+b \text {arcsinh}(c x))^3}{2 b c}+\frac {8 (a+b \text {arcsinh}(c x))^2}{c}-\frac {16 b \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c}+8 i a b x+\frac {16 b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {16 b^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c}-\frac {8 b^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c}+8 i b^2 x \text {arcsinh}(c x)-\frac {b^2 \text {arcsinh}(c x)}{4 c}+\frac {1}{4} b^2 x \sqrt {c^2 x^2+1}-\frac {8 i b^2 \sqrt {c^2 x^2+1}}{c}\right )}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}\) |
(d^4*(1 + c^2*x^2)^(3/2)*((8*I)*a*b*x - ((8*I)*b^2*Sqrt[1 + c^2*x^2])/c + (b^2*x*Sqrt[1 + c^2*x^2])/4 - (b^2*ArcSinh[c*x])/(4*c) + (8*I)*b^2*x*ArcSi nh[c*x] - (b*c*x^2*(a + b*ArcSinh[c*x]))/2 + (8*(a + b*ArcSinh[c*x])^2)/c - ((8*I)*(a + b*ArcSinh[c*x])^2)/(c*Sqrt[1 + c^2*x^2]) + (8*x*(a + b*ArcSi nh[c*x])^2)/Sqrt[1 + c^2*x^2] - ((4*I)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c* x])^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/2 - (5*(a + b*ArcS inh[c*x])^3)/(2*b*c) + ((32*I)*b*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x ]])/c - (16*b*(a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/c + (16*b^ 2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c - (16*b^2*PolyLog[2, I*E^ArcSinh[c*x] ])/c - (8*b^2*PolyLog[2, -E^(2*ArcSinh[c*x])])/c))/((d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))
3.6.94.3.1 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]
\[\int \frac {\left (i c d x +d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{\left (-i c f x +f \right )^{\frac {3}{2}}}d x\]
\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
integral(((b^2*c^2*d^2*x^2 - 2*I*b^2*c*d^2*x - b^2*d^2)*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^2*d^2*x^2 - 2 *I*a*b*c*d^2*x - a*b*d^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + s qrt(c^2*x^2 + 1)) + (a^2*c^2*d^2*x^2 - 2*I*a^2*c*d^2*x - a^2*d^2)*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)
Timed out. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
1/2*(c^2*d^3*x^3/(sqrt(c^2*d*f*x^2 + d*f)*f) - 8*I*c*d^3*x^2/(sqrt(c^2*d*f *x^2 + d*f)*f) + 17*d^3*x/(sqrt(c^2*d*f*x^2 + d*f)*f) - 15*d^3*arcsinh(c*x )/(sqrt(d*f)*c*f) - 24*I*d^3/(sqrt(c^2*d*f*x^2 + d*f)*c*f))*a^2 + integrat e((I*c*d*x + d)^(5/2)*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(-I*c*f*x + f)^(3 /2) + 2*(I*c*d*x + d)^(5/2)*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(-I*c*f*x + f )^(3/2), x)
Exception generated. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(d+i c d x)^{5/2} (a+b \text {arcsinh}(c x))^2}{(f-i c f x)^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}}{{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]