Optimal. Leaf size=229 \[ \frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \text {ArcTan}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b^2 \sqrt {x} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {2 b^2 \text {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 i a b \text {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 i a b \text {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d} \]
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Rubi [A]
time = 0.23, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5544, 4275,
4265, 2611, 2320, 6724, 4269, 3799, 2221, 2317, 2438} \begin {gather*} \frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \text {ArcTan}\left (e^{c+d \sqrt {x}}\right )}{d}+\frac {8 i a b \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 i a b \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {2 b^2 \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {4 b^2 \sqrt {x} \log \left (e^{2 \left (c+d \sqrt {x}\right )}+1\right )}{d^2}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {2 b^2 x}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3799
Rule 4265
Rule 4269
Rule 4275
Rule 5544
Rule 6724
Rubi steps
\begin {align*} \int \sqrt {x} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \text {Subst}\left (\int x^2 (a+b \text {sech}(c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text {sech}(c+d x)+b^2 x^2 \text {sech}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} a^2 x^{3/2}+(4 a b) \text {Subst}\left (\int x^2 \text {sech}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^2 \text {sech}^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(8 i a b) \text {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(8 i a b) \text {Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {\left (4 b^2\right ) \text {Subst}\left (\int x \tanh (c+d x) \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(8 i a b) \text {Subst}\left (\int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(8 i a b) \text {Subst}\left (\int \text {Li}_2\left (i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x}{1+e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b^2 \sqrt {x} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(8 i a b) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(8 i a b) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^3}+\frac {\left (4 b^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b^2 \sqrt {x} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 i a b \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}\\ &=\frac {2 b^2 x}{d}+\frac {2}{3} a^2 x^{3/2}+\frac {8 a b x \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {4 b^2 \sqrt {x} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {8 i a b \sqrt {x} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 i a b \sqrt {x} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {2 b^2 \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {8 i a b \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {8 i a b \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2 b^2 x \tanh \left (c+d \sqrt {x}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 3.90, size = 309, normalized size = 1.35 \begin {gather*} \frac {2 \cosh \left (c+d \sqrt {x}\right ) \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \left (a^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )+\frac {3 b \cosh \left (c+d \sqrt {x}\right ) \left (\frac {2 b d^2 e^{2 c} x}{1+e^{2 c}}-b \left (2 d \sqrt {x} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )+\text {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )\right )+2 i a \left (d^2 x \log \left (1-i e^{c+d \sqrt {x}}\right )-d^2 x \log \left (1+i e^{c+d \sqrt {x}}\right )-2 d \sqrt {x} \text {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )+2 d \sqrt {x} \text {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )+2 \text {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )-2 \text {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )\right )\right )}{d^3}+\frac {3 b^2 x \text {sech}(c) \sinh \left (d \sqrt {x}\right )}{d}\right )}{3 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )^{2} \sqrt {x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {x}\,{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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