Integrand size = 18, antiderivative size = 105 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}} \]
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Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6423, 946, 174, 552, 551} \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}} \]
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Rule 174
Rule 551
Rule 552
Rule 946
Rule 6423
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx}{e} \\ & = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}-\frac {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{e} \\ & = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {\left (4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{e} \\ & = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {\left (4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{e \sqrt {d+e x}} \\ & = -\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 18.02 (sec) , antiderivative size = 1675, normalized size of antiderivative = 15.95 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 a}{e \sqrt {d+e x}}-\frac {2 b \text {sech}^{-1}(c x)}{e \sqrt {d+e x}}+\frac {4 i b \left (2 \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}+c d \sqrt {\frac {1-c x}{1+c x}}-e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (-i c d+\sqrt {-c d-e} \sqrt {c d-e}+i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}-c d \sqrt {\frac {1-c x}{1+c x}}+e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (i c d+\sqrt {-c d-e} \sqrt {c d-e}-i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \left (1+\frac {1-c x}{1+c x}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}}\right ),\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )+\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \sqrt {1+\frac {1-c x}{1+c x}} \sqrt {\frac {e-\frac {e (1-c x)}{1+c x}+c d \left (1+\frac {1-c x}{1+c x}\right )}{c d+e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1-c x}{1+c x}}\right ),\frac {c d-e}{c d+e}\right )+2 i \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}+c d \sqrt {\frac {1-c x}{1+c x}}-e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (-i c d+\sqrt {-c d-e} \sqrt {c d-e}+i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}-c d \sqrt {\frac {1-c x}{1+c x}}+e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (i c d+\sqrt {-c d-e} \sqrt {c d-e}-i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \left (1+\frac {1-c x}{1+c x}\right ) \left (\operatorname {EllipticPi}\left (\frac {i \sqrt {-c d-e}-\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}},\arcsin \left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}}\right ),\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )-\operatorname {EllipticPi}\left (\frac {-i \sqrt {-c d-e}+\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}},\arcsin \left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}}\right ),\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )\right )\right )}{e \sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \left (1+\frac {1-c x}{1+c x}\right ) \sqrt {\frac {c d+e+\frac {c d (1-c x)}{1+c x}-\frac {e (1-c x)}{1+c x}}{c+\frac {c (1-c x)}{1+c x}}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(96)=192\).
Time = 11.35 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.39
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(251\) |
default | \(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(251\) |
parts | \(-\frac {2 a}{\sqrt {e x +d}\, e}+\frac {2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c e x}}\, x \sqrt {\frac {c \left (e x +d \right )-c d +e}{c e x}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(255\) |
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\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]
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