Integrand size = 20, antiderivative size = 144 \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2}{3} a x^{3/2}-\frac {4 b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 b \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3} \]
2/3*a*x^(3/2)-4*b*x*arctanh(exp(I*(c+d*x^(1/2))))/d-4*b*polylog(3,-exp(I*( c+d*x^(1/2))))/d^3+4*b*polylog(3,exp(I*(c+d*x^(1/2))))/d^3+4*I*b*polylog(2 ,-exp(I*(c+d*x^(1/2))))*x^(1/2)/d^2-4*I*b*polylog(2,exp(I*(c+d*x^(1/2))))* x^(1/2)/d^2
Time = 4.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.33 \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2 \left (a d^3 x^{3/2}-6 b d^2 x \text {arctanh}\left (\cos \left (c+d \sqrt {x}\right )+i \sin \left (c+d \sqrt {x}\right )\right )+6 i b d \sqrt {x} \operatorname {PolyLog}\left (2,-\cos \left (c+d \sqrt {x}\right )-i \sin \left (c+d \sqrt {x}\right )\right )-6 i b d \sqrt {x} \operatorname {PolyLog}\left (2,\cos \left (c+d \sqrt {x}\right )+i \sin \left (c+d \sqrt {x}\right )\right )-6 b \operatorname {PolyLog}\left (3,-\cos \left (c+d \sqrt {x}\right )-i \sin \left (c+d \sqrt {x}\right )\right )+6 b \operatorname {PolyLog}\left (3,\cos \left (c+d \sqrt {x}\right )+i \sin \left (c+d \sqrt {x}\right )\right )\right )}{3 d^3} \]
(2*(a*d^3*x^(3/2) - 6*b*d^2*x*ArcTanh[Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqr t[x]]] + (6*I)*b*d*Sqrt[x]*PolyLog[2, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sq rt[x]]] - (6*I)*b*d*Sqrt[x]*PolyLog[2, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sq rt[x]]] - 6*b*PolyLog[3, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sqrt[x]]] + 6*b *PolyLog[3, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]]))/(3*d^3)
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
\(\displaystyle \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a \sqrt {x}+b \sqrt {x} \csc \left (c+d \sqrt {x}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{3} a x^{3/2}-\frac {4 b x \text {arctanh}\left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {4 b \operatorname {PolyLog}\left (3,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 b \operatorname {PolyLog}\left (3,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,-e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {4 i b \sqrt {x} \operatorname {PolyLog}\left (2,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}\) |
(2*a*x^(3/2))/3 - (4*b*x*ArcTanh[E^(I*(c + d*Sqrt[x]))])/d + ((4*I)*b*Sqrt [x]*PolyLog[2, -E^(I*(c + d*Sqrt[x]))])/d^2 - ((4*I)*b*Sqrt[x]*PolyLog[2, E^(I*(c + d*Sqrt[x]))])/d^2 - (4*b*PolyLog[3, -E^(I*(c + d*Sqrt[x]))])/d^3 + (4*b*PolyLog[3, E^(I*(c + d*Sqrt[x]))])/d^3
3.1.52.3.1 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]]
\[\int \left (a +b \csc \left (c +d \sqrt {x}\right )\right ) \sqrt {x}d x\]
\[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
\[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int \sqrt {x} \left (a + b \csc {\left (c + d \sqrt {x} \right )}\right )\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (110) = 220\).
Time = 0.27 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.57 \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {2 \, {\left (d \sqrt {x} + c\right )}^{3} a - 6 \, {\left (d \sqrt {x} + c\right )}^{2} a c + 6 \, {\left (d \sqrt {x} + c\right )} a c^{2} - 6 \, b c^{2} \log \left (\cot \left (d \sqrt {x} + c\right ) + \csc \left (d \sqrt {x} + c\right )\right ) + 6 \, {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} b + 2 i \, {\left (d \sqrt {x} + c\right )} b c\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), \cos \left (d \sqrt {x} + c\right ) + 1\right ) + 6 \, {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} b + 2 i \, {\left (d \sqrt {x} + c\right )} b c\right )} \arctan \left (\sin \left (d \sqrt {x} + c\right ), -\cos \left (d \sqrt {x} + c\right ) + 1\right ) + 12 \, {\left (i \, {\left (d \sqrt {x} + c\right )} b - i \, b c\right )} {\rm Li}_2\left (-e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) + 12 \, {\left (-i \, {\left (d \sqrt {x} + c\right )} b + i \, b c\right )} {\rm Li}_2\left (e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) - 3 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} b - 2 \, {\left (d \sqrt {x} + c\right )} b c\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} + 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right ) + 3 \, {\left ({\left (d \sqrt {x} + c\right )}^{2} b - 2 \, {\left (d \sqrt {x} + c\right )} b c\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} - 2 \, \cos \left (d \sqrt {x} + c\right ) + 1\right ) - 12 \, b {\rm Li}_{3}(-e^{\left (i \, d \sqrt {x} + i \, c\right )}) + 12 \, b {\rm Li}_{3}(e^{\left (i \, d \sqrt {x} + i \, c\right )})}{3 \, d^{3}} \]
1/3*(2*(d*sqrt(x) + c)^3*a - 6*(d*sqrt(x) + c)^2*a*c + 6*(d*sqrt(x) + c)*a *c^2 - 6*b*c^2*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c)) + 6*(-I*(d*sqr t(x) + c)^2*b + 2*I*(d*sqrt(x) + c)*b*c)*arctan2(sin(d*sqrt(x) + c), cos(d *sqrt(x) + c) + 1) + 6*(-I*(d*sqrt(x) + c)^2*b + 2*I*(d*sqrt(x) + c)*b*c)* arctan2(sin(d*sqrt(x) + c), -cos(d*sqrt(x) + c) + 1) + 12*(I*(d*sqrt(x) + c)*b - I*b*c)*dilog(-e^(I*d*sqrt(x) + I*c)) + 12*(-I*(d*sqrt(x) + c)*b + I *b*c)*dilog(e^(I*d*sqrt(x) + I*c)) - 3*((d*sqrt(x) + c)^2*b - 2*(d*sqrt(x) + c)*b*c)*log(cos(d*sqrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 + 2*cos(d*sqrt( x) + c) + 1) + 3*((d*sqrt(x) + c)^2*b - 2*(d*sqrt(x) + c)*b*c)*log(cos(d*s qrt(x) + c)^2 + sin(d*sqrt(x) + c)^2 - 2*cos(d*sqrt(x) + c) + 1) - 12*b*po lylog(3, -e^(I*d*sqrt(x) + I*c)) + 12*b*polylog(3, e^(I*d*sqrt(x) + I*c))) /d^3
\[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \csc \left (d \sqrt {x} + c\right ) + a\right )} \sqrt {x} \,d x } \]
Timed out. \[ \int \sqrt {x} \left (a+b \csc \left (c+d \sqrt {x}\right )\right ) \, dx=\int \sqrt {x}\,\left (a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}\right ) \,d x \]