Integrand size = 35, antiderivative size = 158 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=-\frac {i b d x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
I*d*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-I *b*d*x*(c^2*x^2+1)^(1/2)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+1/2*d*(a+b*ar csinh(c*x))^2*(c^2*x^2+1)^(1/2)/b/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)
Time = 1.76 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\frac {-2 i \sqrt {d+i c d x} \sqrt {f-i c f x} \left (b c x-a \sqrt {1+c^2 x^2}\right )+2 i b \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2+2 a \sqrt {d} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{2 c f \sqrt {1+c^2 x^2}} \]
((-2*I)*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(b*c*x - a*Sqrt[1 + c^2*x^2]) + (2*I)*b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*ArcSinh[c* x] + b*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 + 2*a*Sqrt[d]*Sq rt[f]*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sq rt[f - I*c*f*x]])/(2*c*f*Sqrt[1 + c^2*x^2])
Time = 0.54 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.61, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, 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.
\(\displaystyle \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx\) |
\(\Big \downarrow \) 6211 |
\(\displaystyle \frac {\sqrt {c^2 x^2+1} \int \frac {d (i c x+1) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d \sqrt {c^2 x^2+1} \int \frac {(i c x+1) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
\(\Big \downarrow \) 6253 |
\(\displaystyle \frac {d \sqrt {c^2 x^2+1} \int \left (\frac {i c x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}\right )dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d \sqrt {c^2 x^2+1} \left (\frac {i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c}-i b x\right )}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
(d*Sqrt[1 + c^2*x^2]*((-I)*b*x + (I*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) )/c + (a + b*ArcSinh[c*x])^2/(2*b*c)))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x ])
3.6.54.3.1 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]))
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {i c d x +d}}{\sqrt {-i c f x +f}}d x\]
\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {-i \, c f x + f}} \,d x } \]
integral((I*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*b*log(c*x + sqrt(c^2*x^2 + 1)) + I*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*a)/(c*f*x + I*f), x)
\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int \frac {\sqrt {i d \left (c x - i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {- i f \left (c x + i\right )}}\, dx \]
\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {-i \, c f x + f}} \,d x } \]
a*(d*arcsinh(c*x)/(c*f*sqrt(d/f)) + I*sqrt(c^2*d*f*x^2 + d*f)/(c*f)) + b*i ntegrate(sqrt(I*c*d*x + d)*log(c*x + sqrt(c^2*x^2 + 1))/sqrt(-I*c*f*x + f) , x)
\[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int { \frac {\sqrt {i \, c d x + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {-i \, c f x + f}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {d+i c d x} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d+c\,d\,x\,1{}\mathrm {i}}}{\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \]