Optimal. Leaf size=1303 \[ \frac {2 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {6 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {6 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {12 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {12 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {12 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {12 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {2 x^{3/2} b^2}{a^2 \left (a^2+b^2\right ) d}+\frac {6 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {6 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {12 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}+\frac {12 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}-\frac {12 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^4}-\frac {12 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^4}-\frac {2 x^{3/2} \cosh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {4 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}+\frac {4 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}-\frac {12 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {24 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {24 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {24 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {24 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {x^2}{2 a^2} \]
[Out]
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Rubi [A] time = 2.30, antiderivative size = 1303, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5437, 4191, 3324, 3322, 2264, 2190, 2531, 6609, 2282, 6589, 5561} \[ \frac {2 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {2 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {6 x \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {6 x \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {12 \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {12 \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {12 \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {12 \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^3}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {2 x^{3/2} b^2}{a^2 \left (a^2+b^2\right ) d}+\frac {6 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {6 x \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b^2}{a^2 \left (a^2+b^2\right ) d^2}+\frac {12 \sqrt {x} \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}+\frac {12 \sqrt {x} \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^3}-\frac {12 \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^4}-\frac {12 \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b^2}{a^2 \left (a^2+b^2\right ) d^4}-\frac {2 x^{3/2} \cosh \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {4 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b-\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}+\frac {4 x^{3/2} \log \left (\frac {e^{c+d \sqrt {x}} a}{b+\sqrt {a^2+b^2}}+1\right ) b}{a^2 \sqrt {a^2+b^2} d}-\frac {12 x \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 x \text {PolyLog}\left (2,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^2}+\frac {24 \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {24 \sqrt {x} \text {PolyLog}\left (3,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^3}-\frac {24 \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {24 \text {PolyLog}\left (4,-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right ) b}{a^2 \sqrt {a^2+b^2} d^4}+\frac {x^2}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3322
Rule 3324
Rule 4191
Rule 5437
Rule 5561
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{(a+b \text {csch}(c+d x))^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {x^3}{a^2}+\frac {b^2 x^3}{a^2 (b+a \sinh (c+d x))^2}-\frac {2 b x^3}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^2}{2 a^2}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {x^3}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{(b+a \sinh (c+d x))^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=\frac {x^2}{2 a^2}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {(8 b) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^3}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2 \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right ) d}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^3}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )}-\frac {(8 b) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^3}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2}}+\frac {(8 b) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^3}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \sqrt {a^2+b^2}}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right ) d}+\frac {\left (6 b^2\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^2}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right ) d}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (4 b^3\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^3}{2 b-2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right )^{3/2}}-\frac {\left (4 b^3\right ) \operatorname {Subst}\left (\int \frac {e^{c+d x} x^3}{2 b+2 \sqrt {a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt {x}\right )}{a \left (a^2+b^2\right )^{3/2}}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int x \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {(12 b) \operatorname {Subst}\left (\int x^2 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {(12 b) \operatorname {Subst}\left (\int x^2 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {(24 b) \operatorname {Subst}\left (\int x \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {(24 b) \operatorname {Subst}\left (\int x \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {\left (6 b^3\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {(24 b) \operatorname {Subst}\left (\int \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {(24 b) \operatorname {Subst}\left (\int \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {12 b^2 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {12 b^3 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {12 b^2 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {12 b^3 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}-\frac {(24 b) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \sqrt {a^2+b^2} d^4}+\frac {(24 b) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \sqrt {a^2+b^2} d^4}+\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b-2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-\frac {2 a e^{c+d x}}{2 b+2 \sqrt {a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {12 b^2 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {12 b^3 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {12 b^2 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {12 b^3 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {24 b \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}+\frac {24 b \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}+\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {\left (12 b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^4}\\ &=-\frac {2 b^2 x^{3/2}}{a^2 \left (a^2+b^2\right ) d}+\frac {x^2}{2 a^2}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {6 b^2 x \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^3 x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac {4 b x^{3/2} \log \left (1+\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {12 b^2 \sqrt {x} \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {6 b^3 x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac {12 b x \text {Li}_2\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {12 b^2 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}-\frac {12 b^3 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {12 b^2 \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^4}+\frac {12 b^3 \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac {24 b \sqrt {x} \text {Li}_3\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {12 b^3 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^4}-\frac {24 b \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {12 b^3 \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^4}+\frac {24 b \text {Li}_4\left (-\frac {a e^{c+d \sqrt {x}}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {2 b^2 x^{3/2} \cosh \left (c+d \sqrt {x}\right )}{a \left (a^2+b^2\right ) d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )}\\ \end {align*}
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Mathematica [A] time = 16.24, size = 1333, normalized size = 1.02 \[ \frac {\text {csch}^2\left (c+d \sqrt {x}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right ) \left (\frac {4 x^{3/2} \text {csch}(c) \left (b \cosh (c)+a \sinh \left (d \sqrt {x}\right )\right ) b^2}{\left (a^2+b^2\right ) d}-\frac {4 e^c \left (2 b e^c x^{3/2}+\frac {e^{-c} \left (-1+e^{2 c}\right ) \left (2 a^2 e^c x^{3/2} \log \left (\frac {e^{2 c+d \sqrt {x}} a}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^3+b^2 e^c x^{3/2} \log \left (\frac {e^{2 c+d \sqrt {x}} a}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^3-2 a^2 e^c x^{3/2} \log \left (\frac {e^{2 c+d \sqrt {x}} a}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^3-b^2 e^c x^{3/2} \log \left (\frac {e^{2 c+d \sqrt {x}} a}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^3-3 b \sqrt {\left (a^2+b^2\right ) e^{2 c}} x \log \left (\frac {e^{2 c+d \sqrt {x}} a}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2-3 b \sqrt {\left (a^2+b^2\right ) e^{2 c}} x \log \left (\frac {e^{2 c+d \sqrt {x}} a}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}+1\right ) d^2-3 \left (2 d e^c \sqrt {x} a^2+2 b \sqrt {\left (a^2+b^2\right ) e^{2 c}}+b^2 d e^c \sqrt {x}\right ) \sqrt {x} \text {Li}_2\left (-\frac {a e^{2 c+d \sqrt {x}}}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d-12 a^2 e^c \sqrt {x} \text {Li}_3\left (-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d-6 b^2 e^c \sqrt {x} \text {Li}_3\left (-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d+12 a^2 e^c \sqrt {x} \text {Li}_3\left (-\frac {a e^{2 c+d \sqrt {x}}}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d+6 b^2 e^c \sqrt {x} \text {Li}_3\left (-\frac {a e^{2 c+d \sqrt {x}}}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) d+\left (6 a^2 e^c x d^2+3 b^2 e^c x d^2-6 b \sqrt {\left (a^2+b^2\right ) e^{2 c}} \sqrt {x} d\right ) \text {Li}_2\left (-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 b \sqrt {\left (a^2+b^2\right ) e^{2 c}} \text {Li}_3\left (-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 b \sqrt {\left (a^2+b^2\right ) e^{2 c}} \text {Li}_3\left (-\frac {a e^{2 c+d \sqrt {x}}}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+12 a^2 e^c \text {Li}_4\left (-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 b^2 e^c \text {Li}_4\left (-\frac {a e^{2 c+d \sqrt {x}}}{b e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-12 a^2 e^c \text {Li}_4\left (-\frac {a e^{2 c+d \sqrt {x}}}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 b^2 e^c \text {Li}_4\left (-\frac {a e^{2 c+d \sqrt {x}}}{e^c b+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{d^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right ) \left (b+a \sinh \left (c+d \sqrt {x}\right )\right ) b}{\left (a^2+b^2\right ) d \left (-1+e^{2 c}\right )}+x^2 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )\right )}{2 a^2 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{b^{2} \operatorname {csch}\left (d \sqrt {x} + c\right )^{2} + 2 \, a b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {x}{{\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {8 \, a b^{2} x^{\frac {3}{2}} - {\left (a^{3} d e^{\left (2 \, c\right )} + a b^{2} d e^{\left (2 \, c\right )}\right )} x^{2} e^{\left (2 \, d \sqrt {x}\right )} + {\left (a^{3} d + a b^{2} d\right )} x^{2} - 2 \, {\left (4 \, b^{3} x^{\frac {3}{2}} e^{c} + {\left (a^{2} b d e^{c} + b^{3} d e^{c}\right )} x^{2}\right )} e^{\left (d \sqrt {x}\right )}}{2 \, {\left (a^{5} d + a^{3} b^{2} d - {\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d \sqrt {x}\right )} - 2 \, {\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} e^{\left (d \sqrt {x}\right )}\right )}} - \int \frac {2 \, {\left (3 \, a b^{2} \sqrt {x} - {\left (3 \, b^{3} \sqrt {x} e^{c} + {\left (2 \, a^{2} b d e^{c} + b^{3} d e^{c}\right )} x\right )} e^{\left (d \sqrt {x}\right )}\right )}}{a^{5} d + a^{3} b^{2} d - {\left (a^{5} d e^{\left (2 \, c\right )} + a^{3} b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d \sqrt {x}\right )} - 2 \, {\left (a^{4} b d e^{c} + a^{2} b^{3} d e^{c}\right )} e^{\left (d \sqrt {x}\right )}}\,{d x} \]
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 \[ \int \frac {x}{{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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