Optimal. Leaf size=140 \[ \frac {2}{3} a x^{3/2}+\frac {4 i b \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {14, 5436, 4180, 2531, 2282, 6589} \[ -\frac {4 i b \sqrt {x} \text {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \text {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \text {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \text {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2}{3} a x^{3/2}+\frac {4 b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2282
Rule 2531
Rule 4180
Rule 5436
Rule 6589
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right ) \, dx &=\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) \operatorname {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 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {(4 i b) \operatorname {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(4 i b) \operatorname {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 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 i b) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(4 i b) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {2}{3} a x^{3/2}+\frac {4 b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(4 i b) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(4 i b) \operatorname {Subst}\left (\int \frac {\text {Li}_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 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 i b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {4 i b \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {4 i b \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 7.66, size = 197, normalized size = 1.41 \[ \frac {2 \left (a d^3 x^{3/2}-6 b d^2 x \tan ^{-1}\left (\cosh \left (c+d \sqrt {x}\right )-\sinh \left (c+d \sqrt {x}\right )\right )-6 i b d \sqrt {x} \text {Li}_2\left (-i \left (\cosh \left (c+d \sqrt {x}\right )-\sinh \left (c+d \sqrt {x}\right )\right )\right )+6 i b d \sqrt {x} \text {Li}_2\left (i \left (\cosh \left (c+d \sqrt {x}\right )-\sinh \left (c+d \sqrt {x}\right )\right )\right )-6 i b \text {Li}_3\left (-i \left (\cosh \left (c+d \sqrt {x}\right )-\sinh \left (c+d \sqrt {x}\right )\right )\right )+6 i b \text {Li}_3\left (i \left (\cosh \left (c+d \sqrt {x}\right )-\sinh \left (c+d \sqrt {x}\right )\right )\right )\right )}{3 d^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \sqrt {x} \operatorname {sech}\left (d \sqrt {x} + c\right ) + a \sqrt {x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right ) \sqrt {x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2}{3} \, a x^{\frac {3}{2}} + 2 \, b \int \frac {\sqrt {x} e^{\left (d \sqrt {x} + c\right )}}{e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {x}\,\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________