Integrand size = 16, antiderivative size = 194 \[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4} \]
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Time = 0.13 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {14, 5544, 4265, 2611, 6744, 2320, 6724} \[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2} \]
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Rule 14
Rule 2320
Rule 2611
Rule 4265
Rule 5544
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \left (a x+b x \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx \\ & = \frac {a x^2}{2}+b \int x \text {sech}\left (c+d \sqrt {x}\right ) \, dx \\ & = \frac {a x^2}{2}+(2 b) \text {Subst}\left (\int x^3 \text {sech}(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {(6 i b) \text {Subst}\left (\int x^2 \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(6 i b) \text {Subst}\left (\int x^2 \log \left (1+i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(12 i b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(12 i b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = \frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(12 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(12 i b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = \frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(12 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(12 i b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^4} \\ & = \frac {a x^2}{2}+\frac {4 b x^{3/2} \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07 \[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}+\frac {2 i b \left (d^3 x^{3/2} \log \left (1-i e^{c+d \sqrt {x}}\right )-d^3 x^{3/2} \log \left (1+i e^{c+d \sqrt {x}}\right )-3 d^2 x \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )+3 d^2 x \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )+6 d \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )-6 d \sqrt {x} \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )-6 \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )+6 \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )\right )}{d^4} \]
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\[\int x \left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right )d x\]
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\[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
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\[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int x \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )\, dx \]
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\[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
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\[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
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Timed out. \[ \int x \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx=\int x\,\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
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