Integrand size = 35, antiderivative size = 181 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\frac {2 i f^2 (1-i c x) \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {f^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f^2 \left (1+c^2 x^2\right )^{3/2} \log (i-c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
[Out]
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {5796, 5844, 651, 5837, 12, 641, 31, 5783} \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=-\frac {f^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{2 b c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}+\frac {2 i f^2 (1-i c x) \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {2 b f^2 \left (c^2 x^2+1\right )^{3/2} \log (-c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
[In]
[Out]
Rule 12
Rule 31
Rule 641
Rule 651
Rule 5783
Rule 5796
Rule 5837
Rule 5844
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(f-i c f x)^2 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {\left (1+c^2 x^2\right )^{3/2} \int \left (-\frac {2 i \left (i f^2+c f^2 x\right ) (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}}-\frac {f^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {\left (2 i \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {\left (i f^2+c f^2 x\right ) (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {\left (f^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {2 i f^2 (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 {f^2 \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 {\left (2 i b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {f^2 (1-i c x)}{c \left (1+c^2 x^2\right )} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {2 i f^2 (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 {f^2 \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 {\left (2 i b f^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {1-i c x}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {2 i f^2 (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 {f^2 \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 {\left (2 i b f^2 \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {1}{1+i c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = \frac {2 i f^2 (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 {f^2 \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 {2 b f^2 \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}} \\ \end{align*}
Time = 2.55 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=-\frac {-\frac {4 a \sqrt {d+i c d x} \sqrt {f-i c f x}}{-i+c x}+2 a \sqrt {d} \sqrt {f} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+\frac {b \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\text {arcsinh}(c x) \left (-4 i \cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-4 \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\text {arcsinh}(c x)^2 \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 \left (4 i \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+\log \left (1+c^2 x^2\right )\right ) \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )}{\sqrt {1+c^2 x^2} \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c x)\right )+i \sinh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}}{2 c d^2} \]
[In]
[Out]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {-i c f x +f}}{\left (i c d x +d \right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int \frac {\sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (i d \left (c x - i\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{{\left (i \, c d x + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{(d+i c d x)^{3/2}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
[In]
[Out]