Integrand size = 35, antiderivative size = 158 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\frac {i b f x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i f \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5796, 5838, 5783, 5798, 8} \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\frac {f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i f \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {i b f x \sqrt {c^2 x^2+1}}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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Rule 8
Rule 5783
Rule 5796
Rule 5798
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} \int \frac {(f-i c f x) (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\sqrt {1+c^2 x^2} \int \left (\frac {f (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}-\frac {i c f x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\left (f \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (i c f \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {i f \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 {f \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}}+\frac {\left (i b f \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {i b f x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {i f \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 {f \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}} \\ \end{align*}
Time = 1.52 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\frac {2 i \sqrt {d+i c d x} \sqrt {f-i c f x} \left (b c x-a \sqrt {1+c^2 x^2}\right )-2 i b \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2+2 a \sqrt {d} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{2 c d \sqrt {1+c^2 x^2}} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {-i c f x +f}}{\sqrt {i c d x +d}}d x\]
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\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {i \, c d x + d}} \,d x } \]
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\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int \frac {\sqrt {- i f \left (c x + i\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {i d \left (c x - i\right )}}\, dx \]
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\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {i \, c d x + d}} \,d x } \]
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\[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int { \frac {\sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {i \, c d x + d}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {f-i c f x} (a+b \text {arcsinh}(c x))}{\sqrt {d+i c d x}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}}}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}} \,d x \]
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