Integrand size = 37, antiderivative size = 518 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=-\frac {f^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {4 i b^2 f^3 \left (1+c^2 x^2\right )^{5/2} \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}-\frac {i f^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {2 b f^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {i f^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 b f^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}}+\frac {4 b^2 f^3 \left (1+c^2 x^2\right )^{5/2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c (d+i c d x)^{5/2} (f-i c f x)^{5/2}} \]
-1/3*f^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))^2/c/(d+I*c*d*x)^(5/2)/(f-I*c *f*x)^(5/2)-4/3*I*b^2*f^3*(c^2*x^2+1)^(5/2)*cot(1/4*Pi+1/2*I*arcsinh(c*x)) /c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)-1/3*I*f^3*(c^2*x^2+1)^(5/2)*(a+b*ar csinh(c*x))^2*cot(1/4*Pi+1/2*I*arcsinh(c*x))/c/(d+I*c*d*x)^(5/2)/(f-I*c*f* x)^(5/2)+2/3*b*f^3*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))*csc(1/4*Pi+1/2*I*a rcsinh(c*x))^2/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+1/3*I*f^3*(c^2*x^2+1) ^(5/2)*(a+b*arcsinh(c*x))^2*cot(1/4*Pi+1/2*I*arcsinh(c*x))*csc(1/4*Pi+1/2* I*arcsinh(c*x))^2/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)+4/3*b*f^3*(c^2*x^2 +1)^(5/2)*(a+b*arcsinh(c*x))*ln(1+I*(c*x+(c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x) ^(5/2)/(f-I*c*f*x)^(5/2)+4/3*b^2*f^3*(c^2*x^2+1)^(5/2)*polylog(2,-I*(c*x+( c^2*x^2+1)^(1/2)))/c/(d+I*c*d*x)^(5/2)/(f-I*c*f*x)^(5/2)
Time = 9.85 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\frac {\sqrt {i d (-i+c x)} \sqrt {-i f (i+c x)} \left (-\frac {2 i a^2}{3 d^3 (-i+c x)^2}-\frac {a^2}{3 d^3 (-i+c x)}\right )}{c}+\frac {i a b \sqrt {i (-i d+c d x)} \sqrt {-i (i f+c f x)} \sqrt {-d f \left (1+c^2 x^2\right )} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right ) \left (-i \cosh \left (\frac {3}{2} \text {arcsinh}(c x)\right ) \left (\text {arcsinh}(c x)-2 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-i \log \left (\sqrt {1+c^2 x^2}\right )\right )+\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right ) \left (4+3 i \text {arcsinh}(c x)-6 i \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+3 \log \left (\sqrt {1+c^2 x^2}\right )\right )+2 \left (\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)+2 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (\sqrt {1+c^2 x^2}\right )\right )+2 \left (i+\text {arcsinh}(c x)+2 \arctan \left (\coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+i \log \left (\sqrt {1+c^2 x^2}\right )\right )\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{3 c d^3 (i+c x) \sqrt {-((-i d+c d x) (i f+c f x))} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^4}+\frac {i b^2 (i+c x) \sqrt {i (-i d+c d x)} \sqrt {-i (i f+c f x)} \sqrt {-d f \left (1+c^2 x^2\right )} \left ((-1+i) \text {arcsinh}(c x)^2-\frac {2 \text {arcsinh}(c x) (-2 i+\text {arcsinh}(c x))}{-i+c x}+2 i (\pi +2 i \text {arcsinh}(c x)) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-i \pi \left (\text {arcsinh}(c x)-4 \log \left (1+e^{\text {arcsinh}(c x)}\right )+4 \log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \log \left (\sin \left (\frac {1}{4} (\pi +2 i \text {arcsinh}(c x))\right )\right )\right )+4 \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-\frac {4 \text {arcsinh}(c x)^2 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )^3}+\frac {2 \left (4+\text {arcsinh}(c x)^2\right ) \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}{\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )}\right )}{3 c d^3 \sqrt {-((-i d+c d x) (i f+c f x))} \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} \]
(Sqrt[I*d*(-I + c*x)]*Sqrt[(-I)*f*(I + c*x)]*((((-2*I)/3)*a^2)/(d^3*(-I + c*x)^2) - a^2/(3*d^3*(-I + c*x))))/c + ((I/3)*a*b*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] - I*Sinh[ArcSinh[c*x]/2])*((-I)*Cosh[(3*ArcSinh[c*x])/2]*(ArcSinh[c*x] - 2*ArcTan[Coth[ArcSinh[c*x]/2]] - I*Log[Sqrt[1 + c^2*x^2]]) + Cosh[ArcSinh[ c*x]/2]*(4 + (3*I)*ArcSinh[c*x] - (6*I)*ArcTan[Coth[ArcSinh[c*x]/2]] + 3*L og[Sqrt[1 + c^2*x^2]]) + 2*(Sqrt[1 + c^2*x^2]*(ArcSinh[c*x] + 2*ArcTan[Cot h[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]) + 2*(I + ArcSinh[c*x] + 2*A rcTan[Coth[ArcSinh[c*x]/2]] + I*Log[Sqrt[1 + c^2*x^2]]))*Sinh[ArcSinh[c*x] /2]))/(c*d^3*(I + c*x)*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x))]*(Cosh[ArcSi nh[c*x]/2] + I*Sinh[ArcSinh[c*x]/2])^4) + ((I/3)*b^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))]*((-1 + I )*ArcSinh[c*x]^2 - (2*ArcSinh[c*x]*(-2*I + ArcSinh[c*x]))/(-I + c*x) + (2* I)*(Pi + (2*I)*ArcSinh[c*x])*Log[1 - I/E^ArcSinh[c*x]] - I*Pi*(ArcSinh[c*x ] - 4*Log[1 + E^ArcSinh[c*x]] + 4*Log[Cosh[ArcSinh[c*x]/2]] + 2*Log[Sin[(P i + (2*I)*ArcSinh[c*x])/4]]) + 4*PolyLog[2, I/E^ArcSinh[c*x]] - (4*ArcSinh [c*x]^2*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2] + I*Sinh[ArcSinh[c*x]/ 2])^3 + (2*(4 + ArcSinh[c*x]^2)*Sinh[ArcSinh[c*x]/2])/(Cosh[ArcSinh[c*x]/2 ] + I*Sinh[ArcSinh[c*x]/2])))/(c*d^3*Sqrt[-(((-I)*d + c*d*x)*(I*f + c*f*x) )]*Sqrt[1 + c^2*x^2]*(Cosh[ArcSinh[c*x]/2] - I*Sinh[ArcSinh[c*x]/2])^2)
Time = 1.38 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, 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.
| \(\displaystyle \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\left (c^2 x^2+1\right )^{5/2} \int \frac {f^3 (1-i c x)^3 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {f^3 \left (c^2 x^2+1\right )^{5/2} \int \frac {(1-i c x)^3 (a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^{5/2}}dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 6259 |
| \(\displaystyle \frac {f^3 \left (c^2 x^2+1\right )^{5/2} \int \left (\frac {i (a+b \text {arcsinh}(c x))^2}{(c x-i) \sqrt {c^2 x^2+1}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{(c x-i)^2 \sqrt {c^2 x^2+1}}\right )dx}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f^3 \left (c^2 x^2+1\right )^{5/2} \left (-\frac {(a+b \text {arcsinh}(c x))^2}{3 c}+\frac {4 b \log \left (1+i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c}-\frac {i \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {2 b \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {i \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) \csc ^2\left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right ) (a+b \text {arcsinh}(c x))^2}{3 c}+\frac {4 b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c}-\frac {4 i b^2 \cot \left (\frac {\pi }{4}+\frac {1}{2} i \text {arcsinh}(c x)\right )}{3 c}\right )}{(d+i c d x)^{5/2} (f-i c f x)^{5/2}}\) |
(f^3*(1 + c^2*x^2)^(5/2)*(-1/3*(a + b*ArcSinh[c*x])^2/c - (((4*I)/3)*b^2*C ot[Pi/4 + (I/2)*ArcSinh[c*x]])/c - ((I/3)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSinh[c*x]])/c + (2*b*(a + b*ArcSinh[c*x])*Csc[Pi/4 + (I/2)*ArcS inh[c*x]]^2)/(3*c) + ((I/3)*(a + b*ArcSinh[c*x])^2*Cot[Pi/4 + (I/2)*ArcSin h[c*x]]*Csc[Pi/4 + (I/2)*ArcSinh[c*x]]^2)/c + (4*b*(a + b*ArcSinh[c*x])*Lo g[1 + I*E^ArcSinh[c*x]])/(3*c) + (4*b^2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/( 3*c)))/((d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))
3.6.75.3.1 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]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {-i c f x +f}}{\left (i c d x +d \right )^{\frac {5}{2}}}d x\]
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*((b^2*c*x + I*b^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqr t(c^2*x^2 + 1))^2 - 3*(c^3*d^3*x^2 - 2*I*c^2*d^3*x - c*d^3)*integral(1/3*( 3*I*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a^2 + 2*(sqrt(c^2*x^2 + 1)*sqrt(I *c*d*x + d)*sqrt(-I*c*f*x + f)*b^2 + 3*I*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a*b)*log(c*x + sqrt(c^2*x^2 + 1)))/(c^3*d^3*x^3 - 3*I*c^2*d^3*x^2 - 3* c*d^3*x + I*d^3), x))/(c^3*d^3*x^2 - 2*I*c^2*d^3*x - c*d^3)
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\int \frac {\sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (i d \left (c x - i\right )\right )^{\frac {5}{2}}}\, dx \]
Timed out. \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (i \, c d x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{(d+i c d x)^{5/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]