Integrand size = 18, antiderivative size = 284 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b c \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Time = 0.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6425, 1588, 958, 733, 430, 947, 174, 552, 551} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b c \sqrt {c^2 x^2+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{\left (-c^2\right )^{3/2} x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b d \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}} \]
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Rule 174
Rule 430
Rule 551
Rule 552
Rule 733
Rule 947
Rule 958
Rule 1588
Rule 6425
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {(2 b) \int \frac {\sqrt {d+e x}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2} \, dx}{c e} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{x \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b d \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {\left (2 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-\sqrt {-c^2} x} \sqrt {1+\sqrt {-c^2} x} \sqrt {d+e x}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d-\frac {\sqrt {-c^2} e}{c^2}}} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 \left (d-\frac {\sqrt {-c^2} e}{c^2}\right )}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{\sqrt {-c^2}}-\frac {e x^2}{\sqrt {-c^2}}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2} \sqrt {1+\frac {e \left (-1+\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{\sqrt {-c^2} \left (d+\frac {e}{\sqrt {-c^2}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ & = \frac {2 \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e}+\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c^3 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b d \sqrt {1+c^2 x^2} \sqrt {1-\frac {e \left (1-\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{c e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 8.45 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \left (a e (d+e x)-\frac {b \left (e+\frac {d}{x}\right ) \left (-c e x \text {csch}^{-1}(c x)+\frac {\sqrt {2} \sqrt {1+i c x} \left (-e^2 (i+c x) \sqrt {\frac {c (d+e x)}{c d-i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )+c d (i c d+e) \sqrt {-\frac {e (i+c x)}{c d-i e}} \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {-\frac {e (i+c x)}{c d-i e}} (c d+c e x)}\right )}{c}\right )}{e^2 \sqrt {d+e x}} \]
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Result contains complex when optimal does not.
Time = 5.38 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.39
method | result | size |
derivativedivides | \(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \operatorname {arccsch}\left (c x \right )+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\) | \(395\) |
default | \(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \operatorname {arccsch}\left (c x \right )+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\) | \(395\) |
parts | \(\frac {2 a \sqrt {e x +d}}{e}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arccsch}\left (c x \right )+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )\right )}{c \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right )}{e}\) | \(398\) |
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\sqrt {d + e x}}\, dx \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
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\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{\sqrt {d+e\,x}} \,d x \]
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