Optimal. Leaf size=363 \[ -\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (c+d \sqrt {x}\right )}{d}+\frac {8 b^2 x^{3/2} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 a b x^{3/2} \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 b^2 x \text {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 a b x \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b^2 \sqrt {x} \text {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 a b \sqrt {x} \text {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \sqrt {x} \text {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {6 b^2 \text {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {96 a b \text {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 a b \text {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5} \]
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Rubi [A]
time = 0.38, antiderivative size = 363, 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 = {5545, 4275,
4267, 2611, 6744, 2320, 6724, 4269, 3797, 2221} \begin {gather*} \frac {2}{5} a^2 x^{5/2}+\frac {96 a b \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 a b \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 a b \sqrt {x} \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \sqrt {x} \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {48 a b x \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {16 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}-\frac {8 a b x^2 \tanh ^{-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 (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^2 \coth \left (c+d \sqrt {x}\right )}{d}-\frac {2 b^2 x^2}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 4267
Rule 4269
Rule 4275
Rule 5545
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^{3/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \text {Subst}\left (\int x^4 (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (a^2 x^4+2 a b x^4 \text {csch}(c+d x)+b^2 x^4 \text {csch}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{5} a^2 x^{5/2}+(4 a b) \text {Subst}\left (\int x^4 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^4 \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {2}{5} a^2 x^{5/2}-\frac {8 a b x^2 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (c+d \sqrt {x}\right )}{d}-\frac {(16 a b) \text {Subst}\left (\int x^3 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(16 a b) \text {Subst}\left (\int x^3 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (8 b^2\right ) \text {Subst}\left (\int x^3 \coth (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 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (c+d \sqrt {x}\right )}{d}-\frac {16 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(48 a b) \text {Subst}\left (\int x^2 \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(48 a b) \text {Subst}\left (\int x^2 \text {Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (16 b^2\right ) \text {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 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (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 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {48 a b x \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(96 a b) \text {Subst}\left (\int x \text {Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(96 a b) \text {Subst}\left (\int x \text {Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (24 b^2\right ) \text {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 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (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 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (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 a b x \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {96 a b \sqrt {x} \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \sqrt {x} \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(96 a b) \text {Subst}\left (\int \text {Li}_4\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(96 a b) \text {Subst}\left (\int \text {Li}_4\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {\left (24 b^2\right ) \text {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 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (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 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (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 a b x \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {Li}_3\left (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 a b \sqrt {x} \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \sqrt {x} \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(96 a b) \text {Subst}\left (\int \frac {\text {Li}_4(-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(96 a b) \text {Subst}\left (\int \frac {\text {Li}_4(x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^5}+\frac {\left (12 b^2\right ) \text {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 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (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 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (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 a b x \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {Li}_3\left (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 a b \sqrt {x} \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \sqrt {x} \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 a b \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}+\frac {\left (6 b^2\right ) \text {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 \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^2 \coth \left (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 a b x^{3/2} \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 a b x^{3/2} \text {Li}_2\left (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 a b x \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 a b x \text {Li}_3\left (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 a b \sqrt {x} \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 a b \sqrt {x} \text {Li}_4\left (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 a b \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 a b \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}\\ \end {align*}
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Mathematica [A]
time = 6.85, size = 439, normalized size = 1.21 \begin {gather*} \frac {2}{5} a^2 x^{5/2}+\frac {2 b \left (-\frac {2 b d^4 e^{2 c} x^2}{-1+e^{2 c}}+2 a d^4 x^2 \log \left (1-e^{c+d \sqrt {x}}\right )-2 a d^4 x^2 \log \left (1+e^{c+d \sqrt {x}}\right )+4 b d^3 x^{3/2} \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )-8 a d^3 x^{3/2} \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )+8 a d^3 x^{3/2} \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )+6 b d^2 x \text {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )+24 a d^2 x \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )-24 a d^2 x \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )-6 b d \sqrt {x} \text {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )-48 a d \sqrt {x} \text {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )+48 a d \sqrt {x} \text {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )+3 b \text {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )+48 a \text {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )-48 a \text {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )\right )}{d^5}+\frac {b^2 x^2 \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{d}-\frac {b^2 x^2 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d \sqrt {x}\right )\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{\frac {3}{2}} \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 422, normalized size = 1.16 \begin {gather*} \frac {2}{5} \, a^{2} x^{\frac {5}{2}} - \frac {4 \, b^{2} x^{2}}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{4} x^{2} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 4 \, d^{3} x^{\frac {3}{2}} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 12 \, d^{2} x {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 24 \, d \sqrt {x} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 24 \, {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{5}} + \frac {4 \, {\left (d^{4} x^{2} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 4 \, d^{3} x^{\frac {3}{2}} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 12 \, d^{2} x {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 24 \, d \sqrt {x} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 24 \, {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{5}} + \frac {8 \, {\left (d^{3} x^{\frac {3}{2}} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 3 \, d^{2} x {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 6 \, d \sqrt {x} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{5}} + \frac {8 \, {\left (d^{3} x^{\frac {3}{2}} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 3 \, d^{2} x {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 6 \, d \sqrt {x} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{5}} - \frac {2 \, {\left (2 \, a b d^{5} x^{\frac {5}{2}} + 5 \, b^{2} d^{4} x^{2}\right )}}{5 \, d^{5}} + \frac {2 \, {\left (2 \, a b d^{5} x^{\frac {5}{2}} - 5 \, b^{2} d^{4} x^{2}\right )}}{5 \, d^{5}} \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 x^{\frac {3}{2}} \left (a + b \operatorname {csch}{\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 x^{3/2}\,{\left (a+\frac {b}{\mathrm {sinh}\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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