Integrand size = 16, antiderivative size = 164 \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}-\frac {4 b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {6 b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4} \]
1/2*a*x^2-4*b*x^(3/2)*arctanh(exp(c+d*x^(1/2)))/d-6*b*x*polylog(2,-exp(c+d *x^(1/2)))/d^2+6*b*x*polylog(2,exp(c+d*x^(1/2)))/d^2-12*b*polylog(4,-exp(c +d*x^(1/2)))/d^4+12*b*polylog(4,exp(c+d*x^(1/2)))/d^4+12*b*polylog(3,-exp( c+d*x^(1/2)))*x^(1/2)/d^3-12*b*polylog(3,exp(c+d*x^(1/2)))*x^(1/2)/d^3
Time = 0.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.10 \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}+\frac {2 b \left (d^3 x^{3/2} \log \left (1-e^{c+d \sqrt {x}}\right )-d^3 x^{3/2} \log \left (1+e^{c+d \sqrt {x}}\right )-3 d^2 x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )+3 d^2 x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )+6 d \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )-6 d \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )-6 \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )+6 \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )\right )}{d^4} \]
(a*x^2)/2 + (2*b*(d^3*x^(3/2)*Log[1 - E^(c + d*Sqrt[x])] - d^3*x^(3/2)*Log [1 + E^(c + d*Sqrt[x])] - 3*d^2*x*PolyLog[2, -E^(c + d*Sqrt[x])] + 3*d^2*x *PolyLog[2, E^(c + d*Sqrt[x])] + 6*d*Sqrt[x]*PolyLog[3, -E^(c + d*Sqrt[x]) ] - 6*d*Sqrt[x]*PolyLog[3, E^(c + d*Sqrt[x])] - 6*PolyLog[4, -E^(c + d*Sqr t[x])] + 6*PolyLog[4, E^(c + d*Sqrt[x])]))/d^4
Time = 0.38 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2010, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a x+b x \text {csch}\left (c+d \sqrt {x}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a x^2}{2}-\frac {4 b x^{3/2} \text {arctanh}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {12 b \operatorname {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \operatorname {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {6 b x \operatorname {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {6 b x \operatorname {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}\) |
(a*x^2)/2 - (4*b*x^(3/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (6*b*x*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2 + (6*b*x*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (1 2*b*Sqrt[x]*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (12*b*Sqrt[x]*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (12*b*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (12 *b*PolyLog[4, E^(c + d*Sqrt[x])])/d^4
3.1.33.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
\[\int x \left (a +b \,\operatorname {csch}\left (c +d \sqrt {x}\right )\right )d x\]
\[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
\[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int x \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )\, dx \]
Time = 0.40 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.05 \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\frac {1}{2} \, a x^{2} - \frac {2 \, {\left (\log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{3} + 3 \, {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} - 6 \, \log \left (e^{\left (d \sqrt {x}\right )}\right ) {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{4}} + \frac {2 \, {\left (\log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{3} + 3 \, {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) \log \left (e^{\left (d \sqrt {x}\right )}\right )^{2} - 6 \, \log \left (e^{\left (d \sqrt {x}\right )}\right ) {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 6 \, {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )})\right )} b}{d^{4}} \]
1/2*a*x^2 - 2*(log(e^(d*sqrt(x) + c) + 1)*log(e^(d*sqrt(x)))^3 + 3*dilog(- e^(d*sqrt(x) + c))*log(e^(d*sqrt(x)))^2 - 6*log(e^(d*sqrt(x)))*polylog(3, -e^(d*sqrt(x) + c)) + 6*polylog(4, -e^(d*sqrt(x) + c)))*b/d^4 + 2*(log(-e^ (d*sqrt(x) + c) + 1)*log(e^(d*sqrt(x)))^3 + 3*dilog(e^(d*sqrt(x) + c))*log (e^(d*sqrt(x)))^2 - 6*log(e^(d*sqrt(x)))*polylog(3, e^(d*sqrt(x) + c)) + 6 *polylog(4, e^(d*sqrt(x) + c)))*b/d^4
\[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \operatorname {csch}\left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
Timed out. \[ \int x \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right ) \, dx=\int x\,\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right ) \,d x \]