Optimal. Leaf size=407 \[ \frac {2}{5} a^2 x^{5/2}+\frac {96 i a b \text {Li}_5\left (-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \text {Li}_5\left (i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \sqrt {x} \text {Li}_4\left (-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \text {Li}_4\left (i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {48 i a b x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b^2 \text {Li}_4\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {12 b^2 \sqrt {x} \text {Li}_3\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {12 b^2 x \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {8 b^2 x^{3/2} \log \left (e^{2 \left (c+d \sqrt {x}\right )}+1\right )}{d^2}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {2 b^2 x^2}{d} \]
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Rubi [A] time = 0.53, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5436, 4190, 4180, 2531, 6609, 2282, 6589, 4184, 3718, 2190} \[ -\frac {16 i a b x^{3/2} \text {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {48 i a b x \text {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {96 i a b \sqrt {x} \text {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \text {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \text {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \text {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {12 b^2 x \text {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \text {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {6 b^2 \text {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (e^{2 \left (c+d \sqrt {x}\right )}+1\right )}{d^2}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {2 b^2 x^2}{d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 4180
Rule 4184
Rule 4190
Rule 5436
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \operatorname {Subst}\left (\int x^4 (a+b \text {sech}(c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (a^2 x^4+2 a b x^4 \text {sech}(c+d x)+b^2 x^4 \text {sech}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{5} a^2 x^{5/2}+(4 a b) \operatorname {Subst}\left (\int x^4 \text {sech}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \operatorname {Subst}\left (\int x^4 \text {sech}^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(16 i a b) \operatorname {Subst}\left (\int x^3 \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(16 i a b) \operatorname {Subst}\left (\int x^3 \log \left (1+i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {\left (8 b^2\right ) \operatorname {Subst}\left (\int x^3 \tanh (c+d x) \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(48 i a b) \operatorname {Subst}\left (\int x^2 \text {Li}_2\left (-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(48 i a b) \operatorname {Subst}\left (\int x^2 \text {Li}_2\left (i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (16 b^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 (c+d x)} x^3}{1+e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {48 i a b x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(96 i a b) \operatorname {Subst}\left (\int x \text {Li}_3\left (-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(96 i a b) \operatorname {Subst}\left (\int x \text {Li}_3\left (i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {\left (24 b^2\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {96 i a b \sqrt {x} \text {Li}_4\left (-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \text {Li}_4\left (i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(96 i a b) \operatorname {Subst}\left (\int \text {Li}_4\left (-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(96 i a b) \operatorname {Subst}\left (\int \text {Li}_4\left (i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (24 b^2\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}\\ &=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \text {Li}_3\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \text {Li}_4\left (-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \text {Li}_4\left (i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(96 i a b) \operatorname {Subst}\left (\int \frac {\text {Li}_4(-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(96 i a b) \operatorname {Subst}\left (\int \frac {\text {Li}_4(i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}\\ &=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \text {Li}_3\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \text {Li}_4\left (-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \text {Li}_4\left (i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \text {Li}_5\left (-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \text {Li}_5\left (i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}\\ &=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \tan ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \text {Li}_3\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \text {Li}_4\left (-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \text {Li}_4\left (i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {6 b^2 \text {Li}_4\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {96 i a b \text {Li}_5\left (-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \text {Li}_5\left (i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 6.34, size = 487, normalized size = 1.20 \[ \frac {2 \cosh \left (c+d \sqrt {x}\right ) \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \left (a^2 x^{5/2} \cosh \left (c+d \sqrt {x}\right )+\frac {5 b \cosh \left (c+d \sqrt {x}\right ) \left (\frac {2 b e^{2 c} d^4 x^2}{e^{2 c}+1}+i \left (2 a d^4 x^2 \log \left (1-i e^{c+d \sqrt {x}}\right )-2 a d^4 x^2 \log \left (1+i e^{c+d \sqrt {x}}\right )-8 a d^3 x^{3/2} \text {Li}_2\left (-i e^{c+d \sqrt {x}}\right )+8 a d^3 x^{3/2} \text {Li}_2\left (i e^{c+d \sqrt {x}}\right )+24 a d^2 x \text {Li}_3\left (-i e^{c+d \sqrt {x}}\right )-24 a d^2 x \text {Li}_3\left (i e^{c+d \sqrt {x}}\right )-48 a d \sqrt {x} \text {Li}_4\left (-i e^{c+d \sqrt {x}}\right )+48 a d \sqrt {x} \text {Li}_4\left (i e^{c+d \sqrt {x}}\right )+48 a \text {Li}_5\left (-i e^{c+d \sqrt {x}}\right )-48 a \text {Li}_5\left (i e^{c+d \sqrt {x}}\right )+4 i b d^3 x^{3/2} \log \left (e^{2 \left (c+d \sqrt {x}\right )}+1\right )+6 i b d^2 x \text {Li}_2\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )-6 i b d \sqrt {x} \text {Li}_3\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )+3 i b \text {Li}_4\left (-e^{2 \left (c+d \sqrt {x}\right )}\right )\right )\right )}{d^5}+\frac {5 b^2 x^2 \text {sech}(c) \sinh \left (d \sqrt {x}\right )}{d}\right )}{5 \left (a \cosh \left (c+d \sqrt {x}\right )+b\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{\frac {3}{2}} \operatorname {sech}\left (d \sqrt {x} + c\right )^{2} + 2 \, a b x^{\frac {3}{2}} \operatorname {sech}\left (d \sqrt {x} + c\right ) + a^{2} x^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int x^{\frac {3}{2}} \left (a +b \,\mathrm {sech}\left (c +d \sqrt {x}\right )\right )^{2}\, dx \]
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 \[ \frac {2 \, {\left (a^{2} d x^{\frac {5}{2}} e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + a^{2} d x^{\frac {5}{2}} - 10 \, b^{2} x^{2}\right )}}{5 \, {\left (d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + d\right )}} + \int \frac {4 \, {\left (a b d x^{\frac {5}{2}} e^{\left (d \sqrt {x} + c\right )} + 2 \, b^{2} x^{2}\right )}}{d x e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} + d x}\,{d x} \]
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 \[ \int x^{3/2}\,{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
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 \[ \int x^{\frac {3}{2}} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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