Integrand size = 20, antiderivative size = 140 \[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2}{3} a x^{3/2}+\frac {4 b x \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3} \]
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Time = 0.09 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {14, 5544, 4265, 2611, 2320, 6724} \[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2}{3} a x^{3/2}+\frac {4 b x \arctan \left (e^{c+d \sqrt {x}}\right )}{d}+\frac {4 i b \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2} \]
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Rule 14
Rule 2320
Rule 2611
Rule 4265
Rule 5544
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (a \sqrt {x}+b \sqrt {x} \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx \\ & = \frac {2}{3} a x^{3/2}+b \int \sqrt {x} \text {sech}\left (c+d \sqrt {x}\right ) \, dx \\ & = \frac {2}{3} a x^{3/2}+(2 b) \text {Subst}\left (\int x^2 \text {sech}(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{3} a x^{3/2}+\frac {4 b x \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {(4 i b) \text {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(4 i b) \text {Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {2}{3} a x^{3/2}+\frac {4 b x \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(4 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = \frac {2}{3} a x^{3/2}+\frac {4 b x \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(4 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3} \\ & = \frac {2}{3} a x^{3/2}+\frac {4 b x \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.23 \[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2 \left (a d^3 x^{3/2}+3 i b d^2 x \log \left (1-i e^{c+d \sqrt {x}}\right )-3 i b d^2 x \log \left (1+i e^{c+d \sqrt {x}}\right )-6 i b d \sqrt {x} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )+6 i b d \sqrt {x} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )+6 i b \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )-6 i b \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )\right )}{3 d^3} \]
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\[\int \left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right ) \sqrt {x}d x\]
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\[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
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\[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int \sqrt {x} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )\, dx \]
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\[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
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\[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
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Timed out. \[ \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int \sqrt {x}\,\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
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