Integrand size = 35, antiderivative size = 112 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=-\frac {d (i-c x) \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b d \left (1+c^2 x^2\right )^{3/2} \log (i+c x)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {5796, 651, 5837, 12, 641, 31} \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=-\frac {d (-c x+i) \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}}-\frac {b d \left (c^2 x^2+1\right )^{3/2} \log (c x+i)}{c (d+i c d x)^{3/2} (f-i c f x)^{3/2}} \]
[In]
[Out]
Rule 12
Rule 31
Rule 641
Rule 651
Rule 5796
Rule 5837
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+c^2 x^2\right )^{3/2} \int \frac {(d+i c d x) (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 {d (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 {\left (b c \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {d (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 {d (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 {\left (b d \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {i-c x}{1+c^2 x^2} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {d (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 {\left (b d \left (1+c^2 x^2\right )^{3/2}\right ) \int \frac {1}{-i-c x} \, dx}{(d+i c d x)^{3/2} (f-i c f x)^{3/2}} \\ & = -\frac {d (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 {b d \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 = 1.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=\frac {\sqrt {f-i c f x} \left (a+i a c x+(b+i b c x) \text {arcsinh}(c x)-i b \sqrt {1+c^2 x^2} \log (d (-1+i c x))\right )}{c f^2 (i+c x) \sqrt {d+i c d x}} \]
[In]
[Out]
\[\int \frac {a +b \,\operatorname {arcsinh}\left (c x \right )}{\left (-i c f x +f \right )^{\frac {3}{2}} \sqrt {i c d x +d}}d x\]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (88) = 176\).
Time = 0.30 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.96 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=\frac {2 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (c^{2} d f^{2} x + i \, c d f^{2}\right )} \sqrt {\frac {b^{2}}{c^{2} d f^{3}}} \log \left (-\frac {{\left (-i \, b c^{6} x^{2} + 2 \, b c^{5} x + 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} - {\left (i \, c^{9} d f^{2} x^{4} - 2 \, c^{8} d f^{2} x^{3} + i \, c^{7} d f^{2} x^{2} - 2 \, c^{6} d f^{2} x\right )} \sqrt {\frac {b^{2}}{c^{2} d f^{3}}}}{8 \, {\left (b c^{3} x^{3} + i \, b c^{2} x^{2} + b c x + i \, b\right )}}\right ) + {\left (c^{2} d f^{2} x + i \, c d f^{2}\right )} \sqrt {\frac {b^{2}}{c^{2} d f^{3}}} \log \left (-\frac {{\left (-i \, b c^{6} x^{2} + 2 \, b c^{5} x + 2 i \, b c^{4}\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} - {\left (-i \, c^{9} d f^{2} x^{4} + 2 \, c^{8} d f^{2} x^{3} - i \, c^{7} d f^{2} x^{2} + 2 \, c^{6} d f^{2} x\right )} \sqrt {\frac {b^{2}}{c^{2} d f^{3}}}}{8 \, {\left (b c^{3} x^{3} + i \, b c^{2} x^{2} + b c x + i \, b\right )}}\right ) + 2 \, \sqrt {i \, c d x + d} \sqrt {-i \, c f x + f} a}{2 \, {\left (c^{2} d f^{2} x + i \, c d f^{2}\right )}} \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\sqrt {i d \left (c x - i\right )} \left (- i f \left (c x + i\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=-\frac {i \, \sqrt {c^{2} d f x^{2} + d f} b \operatorname {arsinh}\left (c x\right )}{-i \, c^{2} d f^{2} x + c d f^{2}} - \frac {i \, \sqrt {c^{2} d f x^{2} + d f} a}{-i \, c^{2} d f^{2} x + c d f^{2}} - \frac {b \log \left (i \, c x - 1\right )}{c \sqrt {d} f^{\frac {3}{2}}} \]
[In]
[Out]
\[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{\sqrt {i \, c d x + d} {\left (-i \, c f x + f\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+i c d x} (f-i c f x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{\sqrt {d+c\,d\,x\,1{}\mathrm {i}}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
[In]
[Out]