Integrand size = 18, antiderivative size = 1523 \[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]
12*I*b^3*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^(1/2)/a ^2/(-a^2+b^2)^(3/2)/d^3+24*I*b*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+ b^2)^(1/2)))*x^(1/2)/a^2/d^3/(-a^2+b^2)^(1/2)+2*I*b^3*x^(3/2)*ln(1+a*exp(I *(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d+4*I*b*x^(3/2) *ln(1+a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d/(-a^2+b^2)^(1/2)+ 2*b^2*x^(3/2)*sin(c+d*x^(1/2))/a/(a^2-b^2)/d/(b+a*cos(c+d*x^(1/2)))-2*I*b^ 3*x^(3/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2) ^(3/2)/d-4*I*b*x^(3/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a ^2/d/(-a^2+b^2)^(1/2)-12*I*b^2*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b-I*(a^2 -b^2)^(1/2)))*x^(1/2)/a^2/(a^2-b^2)/d^3-12*I*b^2*polylog(2,-a*exp(I*(c+d*x ^(1/2)))/(b+I*(a^2-b^2)^(1/2)))*x^(1/2)/a^2/(a^2-b^2)/d^3-12*I*b^3*polylog (3,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))*x^(1/2)/a^2/(-a^2+b^2)^(3 /2)/d^3-24*I*b*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^( 1/2)/a^2/d^3/(-a^2+b^2)^(1/2)+1/2*x^2/a^2+12*b^2*polylog(3,-a*exp(I*(c+d*x ^(1/2)))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^4+12*b^2*polylog(3,-a*exp( I*(c+d*x^(1/2)))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^4+12*b^3*polylog(4 ,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^4-12 *b^3*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2 )^(3/2)/d^4-24*b*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a ^2/d^4/(-a^2+b^2)^(1/2)+24*b*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2...
Time = 15.26 (sec) , antiderivative size = 1767, normalized size of antiderivative = 1.16 \[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]
(x^2*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(2*a^2*(a + b*Sec[ c + d*Sqrt[x]])^2) + (2*b*(b + a*Cos[c + d*Sqrt[x]])^2*(((-2*I)*b*d^3*E^(( 2*I)*c)*x^(3/2))/(1 + E^((2*I)*c)) + (3*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c )]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^ ((2*I)*c)])] + (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt [x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^3*E^(I*c)*x ^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)* E^((2*I)*c)])] + 3*b*d^2*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I* (2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - (2*I)* a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + S qrt[(-a^2 + b^2)*E^((2*I)*c)])] + I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^( I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 3*d* ((2*I)*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d* E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I* c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 3*d*((-2*I)*b*Sqrt[(-a^2 + b^2)*E ^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*Pol yLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2 *I)*c)]))] + 6*b*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + (12*I)*a^2 *d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) ...
Time = 2.71 (sec) , antiderivative size = 1524, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4692, 3042, 4679, 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.
\(\displaystyle \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx\) |
\(\Big \downarrow \) 4692 |
\(\displaystyle 2 \int \frac {x^{3/2}}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int \frac {x^{3/2}}{\left (a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )\right )^2}d\sqrt {x}\) |
\(\Big \downarrow \) 4679 |
\(\displaystyle 2 \int \left (\frac {x^{3/2} b^2}{a^2 \left (b+a \cos \left (c+d \sqrt {x}\right )\right )^2}-\frac {2 x^{3/2} b}{a^2 \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}+\frac {x^{3/2}}{a^2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {i x^{3/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {i x^{3/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {3 x \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac {3 x \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {6 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {6 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b^3}{a^2 \left (b^2-a^2\right )^{3/2} d^4}-\frac {i x^{3/2} b^2}{a^2 \left (a^2-b^2\right ) d}+\frac {3 x \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-i \sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}+\frac {3 x \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+i \sqrt {a^2-b^2}}+1\right ) b^2}{a^2 \left (a^2-b^2\right ) d^2}-\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}-\frac {6 i \sqrt {x} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^3}+\frac {6 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {6 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) b^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {x^{3/2} \sin \left (c+d \sqrt {x}\right ) b^2}{a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}+\frac {2 i x^{3/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}-\frac {2 i x^{3/2} \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) b}{a^2 \sqrt {b^2-a^2} d}+\frac {6 x \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}-\frac {6 x \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^2}+\frac {12 i \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {12 i \sqrt {x} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^3}-\frac {12 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {12 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) b}{a^2 \sqrt {b^2-a^2} d^4}+\frac {x^2}{4 a^2}\right )\) |
2*(((-I)*b^2*x^(3/2))/(a^2*(a^2 - b^2)*d) + x^2/(4*a^2) + (3*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^2) + (3*b^2*x*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2])])/(a ^2*(a^2 - b^2)*d^2) - (I*b^3*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((2*I)*b*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2] *d) + (I*b^3*x^(3/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^ 2])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((2*I)*b*x^(3/2)*Log[1 + (a*E^(I*(c + d *Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((6*I)*b^2 *Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2]))]) /(a^2*(a^2 - b^2)*d^3) - ((6*I)*b^2*Sqrt[x]*PolyLog[2, -((a*E^(I*(c + d*Sq rt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (3*b^3*x*PolyL og[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b ^2)^(3/2)*d^2) + (6*b*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[- a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (3*b^3*x*PolyLog[2, -((a*E^(I* (c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (6*b*x*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/( a^2*Sqrt[-a^2 + b^2]*d^2) + (6*b^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/ (b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4) + (6*b^2*PolyLog[3, -((a* E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^4)...
3.1.48.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
\[\int \frac {x}{\left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}d x\]
\[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x}{{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x}{\left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
\[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x}{{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x}{{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]