Integrand size = 20, antiderivative size = 1041 \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx =\text {Too large to display} \] Output:
1/4*x^4/a-84*I*b*x^(5/2)*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^( 1/2)))/a/(-a^2+b^2)^(1/2)/d^3+1680*I*b*x^(3/2)*polylog(5,-a*exp(I*(c+d*x^( 1/2)))/(b+(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^5+14*b*x^3*polylog(2,-a* exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^2-14*b*x^3 *polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2 )/d^2-10080*I*b*x^(1/2)*polylog(7,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1 /2)))/a/(-a^2+b^2)^(1/2)/d^7-2*I*b*x^(7/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b+ (-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d-420*b*x^2*polylog(4,-a*exp(I*(c+d* x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^4+420*b*x^2*polylog(4 ,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^4+1008 0*I*b*x^(1/2)*polylog(7,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/(- a^2+b^2)^(1/2)/d^7+2*I*b*x^(7/2)*ln(1+a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2) ^(1/2)))/a/(-a^2+b^2)^(1/2)/d+5040*b*x*polylog(6,-a*exp(I*(c+d*x^(1/2)))/( b-(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^6-5040*b*x*polylog(6,-a*exp(I*(c +d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^6-1680*I*b*x^(3/2) *polylog(5,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2 )/d^5+84*I*b*x^(5/2)*polylog(3,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2) ))/a/(-a^2+b^2)^(1/2)/d^3-10080*b*polylog(8,-a*exp(I*(c+d*x^(1/2)))/(b-(-a ^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^8+10080*b*polylog(8,-a*exp(I*(c+d*x^( 1/2)))/(b+(-a^2+b^2)^(1/2)))/a/(-a^2+b^2)^(1/2)/d^8
Time = 1.07 (sec) , antiderivative size = 802, normalized size of antiderivative = 0.77 \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\frac {\sqrt {-a^2+b^2} d^8 x^4+8 i b d^7 x^{7/2} \log \left (1-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-8 i b d^7 x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+56 b d^6 x^3 \operatorname {PolyLog}\left (2,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-56 b d^6 x^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+336 i b d^5 x^{5/2} \operatorname {PolyLog}\left (3,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-336 i b d^5 x^{5/2} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )-1680 b d^4 x^2 \operatorname {PolyLog}\left (4,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )+1680 b d^4 x^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )-6720 i b d^3 x^{3/2} \operatorname {PolyLog}\left (5,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )+6720 i b d^3 x^{3/2} \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+20160 b d^2 x \operatorname {PolyLog}\left (6,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-20160 b d^2 x \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )+40320 i b d \sqrt {x} \operatorname {PolyLog}\left (7,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )-40320 i b d \sqrt {x} \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )-40320 b \operatorname {PolyLog}\left (8,\frac {a e^{i \left (c+d \sqrt {x}\right )}}{-b+\sqrt {-a^2+b^2}}\right )+40320 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{4 a \sqrt {-a^2+b^2} d^8} \] Input:
Integrate[x^3/(a + b*Sec[c + d*Sqrt[x]]),x]
Output:
(Sqrt[-a^2 + b^2]*d^8*x^4 + (8*I)*b*d^7*x^(7/2)*Log[1 - (a*E^(I*(c + d*Sqr t[x])))/(-b + Sqrt[-a^2 + b^2])] - (8*I)*b*d^7*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])] + 56*b*d^6*x^3*PolyLog[2, (a*E^(I*( c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - 56*b*d^6*x^3*PolyLog[2, -((a*E ^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] + (336*I)*b*d^5*x^(5/2)*Pol yLog[3, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - (336*I)*b*d^5 *x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] - 1680*b*d^4*x^2*PolyLog[4, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2 ])] + 1680*b*d^4*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] - (6720*I)*b*d^3*x^(3/2)*PolyLog[5, (a*E^(I*(c + d*Sqrt[x])))/( -b + Sqrt[-a^2 + b^2])] + (6720*I)*b*d^3*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] + 20160*b*d^2*x*PolyLog[6, (a*E^(I* (c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - 20160*b*d^2*x*PolyLog[6, -((a *E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] + (40320*I)*b*d*Sqrt[x]*P olyLog[7, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] - (40320*I)*b *d*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))] - 40320*b*PolyLog[8, (a*E^(I*(c + d*Sqrt[x])))/(-b + Sqrt[-a^2 + b^2])] + 40320*b*PolyLog[8, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/ (4*a*Sqrt[-a^2 + b^2]*d^8)
Time = 1.75 (sec) , antiderivative size = 1043, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx\) |
\(\Big \downarrow \) 4692 |
\(\displaystyle 2 \int \frac {x^{7/2}}{a+b \sec \left (c+d \sqrt {x}\right )}d\sqrt {x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 \int \frac {x^{7/2}}{a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )}d\sqrt {x}\) |
\(\Big \downarrow \) 4679 |
\(\displaystyle 2 \int \left (\frac {x^{7/2}}{a}-\frac {b x^{7/2}}{a \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {x^4}{8 a}+\frac {i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}-\frac {i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}+\frac {7 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}-\frac {7 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}+\frac {42 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {42 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {210 b \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}+\frac {210 b \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}-\frac {840 i b \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {840 i b \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {2520 b \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}-\frac {2520 b \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}+\frac {5040 i b \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {5040 i b \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {5040 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}+\frac {5040 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}\right )\) |
Input:
Int[x^3/(a + b*Sec[c + d*Sqrt[x]]),x]
Output:
2*(x^4/(8*a) + (I*b*x^(7/2)*Log[1 + (a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a ^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) - (I*b*x^(7/2)*Log[1 + (a*E^(I*(c + d* Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (7*b*x^3*Poly Log[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) - (7*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a ^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^2) + ((42*I)*b*x^(5/2)*PolyLog[3, -((a *E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - ((42*I)*b*x^(5/2)*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^3) - (210*b*x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^4) + (210*b *x^2*PolyLog[4, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*S qrt[-a^2 + b^2]*d^4) - ((840*I)*b*x^(3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt [x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^5) + ((840*I)*b*x^( 3/2)*PolyLog[5, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*S qrt[-a^2 + b^2]*d^5) + (2520*b*x*PolyLog[6, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^6) - (2520*b*x*PolyLog[6, -( (a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^ 6) + ((5040*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[ -a^2 + b^2]))])/(a*Sqrt[-a^2 + b^2]*d^7) - ((5040*I)*b*Sqrt[x]*PolyLog[7, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2]))])/(a*Sqrt[-a^2 + b^...
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]
\[\int \frac {x^{3}}{a +b \sec \left (c +d \sqrt {x}\right )}d x\]
Input:
int(x^3/(a+b*sec(c+d*x^(1/2))),x)
Output:
int(x^3/(a+b*sec(c+d*x^(1/2))),x)
\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{3}}{b \sec \left (d \sqrt {x} + c\right ) + a} \,d x } \] Input:
integrate(x^3/(a+b*sec(c+d*x^(1/2))),x, algorithm="fricas")
Output:
integral(x^3/(b*sec(d*sqrt(x) + c) + a), x)
\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^{3}}{a + b \sec {\left (c + d \sqrt {x} \right )}}\, dx \] Input:
integrate(x**3/(a+b*sec(c+d*x**(1/2))),x)
Output:
Integral(x**3/(a + b*sec(c + d*sqrt(x))), x)
Exception generated. \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^3/(a+b*sec(c+d*x^(1/2))),x, algorithm="maxima")
Output:
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
\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int { \frac {x^{3}}{b \sec \left (d \sqrt {x} + c\right ) + a} \,d x } \] Input:
integrate(x^3/(a+b*sec(c+d*x^(1/2))),x, algorithm="giac")
Output:
integrate(x^3/(b*sec(d*sqrt(x) + c) + a), x)
Timed out. \[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\int \frac {x^3}{a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}} \,d x \] Input:
int(x^3/(a + b/cos(c + d*x^(1/2))),x)
Output:
int(x^3/(a + b/cos(c + d*x^(1/2))), x)
\[ \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx=\frac {8 \left (\int \frac {\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right )^{2} x^{3}}{\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right )^{2} a^{2}-\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right )^{2} b^{2}-a^{2}-2 a b -b^{2}}d x \right ) a b +8 \left (\int \frac {\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right )^{2} x^{3}}{\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right )^{2} a^{2}-\tan \left (\frac {\sqrt {x}\, d}{2}+\frac {c}{2}\right )^{2} b^{2}-a^{2}-2 a b -b^{2}}d x \right ) b^{2}+x^{4}}{4 a +4 b} \] Input:
int(x^3/(a+b*sec(c+d*x^(1/2))),x)
Output:
(8*int((tan((sqrt(x)*d + c)/2)**2*x**3)/(tan((sqrt(x)*d + c)/2)**2*a**2 - tan((sqrt(x)*d + c)/2)**2*b**2 - a**2 - 2*a*b - b**2),x)*a*b + 8*int((tan( (sqrt(x)*d + c)/2)**2*x**3)/(tan((sqrt(x)*d + c)/2)**2*a**2 - tan((sqrt(x) *d + c)/2)**2*b**2 - a**2 - 2*a*b - b**2),x)*b**2 + x**4)/(4*(a + b))