3.1.46 \(\int \frac {x^3}{(a+b \sec (c+d \sqrt {x}))^2} \, dx\) [46]

3.1.46.1 Optimal result

Integrand size = 20, antiderivative size = 3123 \[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx =\text {Too large to display} \]

output

2*b^2*x^(7/2)*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^(7/2)*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-84*I*b^2*x^(5/2)*polylog(2,-a*exp(I*(c+d*x^(1/2)))/(b-I*(a^2-b^2) 
^(1/2)))/a^2/(a^2-b^2)/d^3-84*I*b^2*x^(5/2)*polylog(2,-a*exp(I*(c+d*x^(1/2 
)))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3-84*I*b^3*x^(5/2)*polylog(3,-a 
*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3-1680* 
I*b^3*x^(3/2)*polylog(5,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/ 
(-a^2+b^2)^(3/2)/d^5-4*I*b*x^(7/2)*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)-168*I*b*x^(5/2)*polylog(3,-a*exp(I*(c+d* 
x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/(-a^2+b^2)^(1/2)-3360*I*b*x^(3/2)* 
polylog(5,-a*exp(I*(c+d*x^(1/2)))/(b-(-a^2+b^2)^(1/2)))/a^2/d^5/(-a^2+b^2) 
^(1/2)-10080*I*b^2*polylog(6,-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^7-10080*I*b^2*polylog(6,-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^7-10080*I*b^3*polylog(7,-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^ 
7-20160*I*b*polylog(7,-a*exp(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))*x^(1/2 
)/a^2/d^7/(-a^2+b^2)^(1/2)+2*I*b^3*x^(7/2)*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+84*I*b^3*x^(5/2)*polylog(3,-a*ex 
p(I*(c+d*x^(1/2)))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3+1680*I*b 
^2*x^(3/2)*polylog(4,-a*exp(I*(c+d*x^(1/2)))/(b-I*(a^2-b^2)^(1/2)))/a^2...
 

3.1.46.2 Mathematica [A] (verified)

Time = 15.37 (sec) , antiderivative size = 3702, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Result too large to show} \]

input

Integrate[x^3/(a + b*Sec[c + d*Sqrt[x]])^2,x]
 

output

(x^4*(b + a*Cos[c + d*Sqrt[x]])^2*Sec[c + d*Sqrt[x]]^2)/(4*a^2*(a + b*Sec[ 
c + d*Sqrt[x]])^2) + (2*b*E^(I*c)*(b + a*Cos[c + d*Sqrt[x]])^2*((-2*I)*b*E 
^(I*c)*x^(7/2) + ((1 + E^((2*I)*c))*(7*b*d^6*Sqrt[(-a^2 + b^2)*E^((2*I)*c) 
]*x^3*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^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqr 
t[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - I*b^2*d^7*E^(I*c)* 
x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2) 
*E^((2*I)*c)])] + 7*b*d^6*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^3*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^7*E^(I*c)*x^(7/2)*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^7*E^(I*c)*x^(7/2)*Log[1 + (a* 
E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2)*E^((2*I)*c)])] - 7 
*d^5*((6*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])*x^(5/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b* 
E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c)]))] + 7*d^5*((-6*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])*x^(5 
/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + Sqrt[(-a^2 + b^2 
)*E^((2*I)*c)]))] + 210*b*d^4*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*x^2*PolyLog[3 
, -((a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) - Sqrt[(-a^2 + b^2)*E^((2*I)*c) 
]))] + (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqr...
 

3.1.46.3 Rubi [A] (verified)

Time = 4.35 (sec) , antiderivative size = 3124, 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.

\(\displaystyle \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4692

\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x^{7/2}}{\left (a+b \csc \left (c+d \sqrt {x}+\frac {\pi }{2}\right )\right )^2}d\sqrt {x}\)

\(\Big \downarrow \) 4679

\(\displaystyle 2 \int \left (-\frac {2 b x^{7/2}}{a^2 \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}+\frac {x^{7/2}}{a^2}+\frac {b^2 x^{7/2}}{a^2 \left (b+a \cos \left (c+d \sqrt {x}\right )\right )^2}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {x^4}{8 a^2}+\frac {2 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^2 \sqrt {b^2-a^2} d}-\frac {i b^3 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 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^2 \sqrt {b^2-a^2} d}+\frac {i b^3 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {b^2 \sin \left (c+d \sqrt {x}\right ) x^{7/2}}{a \left (a^2-b^2\right ) d \left (b+a \cos \left (c+d \sqrt {x}\right )\right )}-\frac {i b^2 x^{7/2}}{a^2 \left (a^2-b^2\right ) d}+\frac {7 b^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-i \sqrt {a^2-b^2}}+1\right ) x^3}{a^2 \left (a^2-b^2\right ) d^2}+\frac {7 b^2 \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+i \sqrt {a^2-b^2}}+1\right ) x^3}{a^2 \left (a^2-b^2\right ) d^2}+\frac {14 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^2 \sqrt {b^2-a^2} d^2}-\frac {7 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {14 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^2 \sqrt {b^2-a^2} d^2}+\frac {7 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {42 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) x^{5/2}}{a^2 \left (a^2-b^2\right ) d^3}-\frac {42 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) x^{5/2}}{a^2 \left (a^2-b^2\right ) d^3}+\frac {84 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^2 \sqrt {b^2-a^2} d^3}-\frac {42 i b^3 \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^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {84 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^2 \sqrt {b^2-a^2} d^3}+\frac {42 i b^3 \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^2 \left (b^2-a^2\right )^{3/2} d^3}+\frac {210 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) x^2}{a^2 \left (a^2-b^2\right ) d^4}+\frac {210 b^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) x^2}{a^2 \left (a^2-b^2\right ) d^4}-\frac {420 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^2 \sqrt {b^2-a^2} d^4}+\frac {210 b^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {420 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^2 \sqrt {b^2-a^2} d^4}-\frac {210 b^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a^2 \left (b^2-a^2\right )^{3/2} d^4}+\frac {840 i b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) x^{3/2}}{a^2 \left (a^2-b^2\right ) d^5}+\frac {840 i b^2 \operatorname {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) x^{3/2}}{a^2 \left (a^2-b^2\right ) d^5}-\frac {1680 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^2 \sqrt {b^2-a^2} d^5}+\frac {840 i b^3 \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^2 \left (b^2-a^2\right )^{3/2} d^5}+\frac {1680 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^2 \sqrt {b^2-a^2} d^5}-\frac {840 i b^3 \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^2 \left (b^2-a^2\right )^{3/2} d^5}-\frac {2520 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) x}{a^2 \left (a^2-b^2\right ) d^6}-\frac {2520 b^2 \operatorname {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) x}{a^2 \left (a^2-b^2\right ) d^6}+\frac {5040 b \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \sqrt {b^2-a^2} d^6}-\frac {2520 b^3 \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {5040 b \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \sqrt {b^2-a^2} d^6}+\frac {2520 b^3 \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a^2 \left (b^2-a^2\right )^{3/2} d^6}-\frac {5040 i b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right ) \sqrt {x}}{a^2 \left (a^2-b^2\right ) d^7}-\frac {5040 i b^2 \operatorname {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right ) \sqrt {x}}{a^2 \left (a^2-b^2\right ) d^7}+\frac {10080 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^2 \sqrt {b^2-a^2} d^7}-\frac {5040 i b^3 \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (b^2-a^2\right )^{3/2} d^7}-\frac {10080 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^2 \sqrt {b^2-a^2} d^7}+\frac {5040 i b^3 \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a^2 \left (b^2-a^2\right )^{3/2} d^7}+\frac {5040 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}+\frac {5040 b^2 \operatorname {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^8}-\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^8}+\frac {5040 b^3 \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^8}+\frac {10080 b \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^8}-\frac {5040 b^3 \operatorname {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^8}\right )\)

input

Int[x^3/(a + b*Sec[c + d*Sqrt[x]])^2,x]
 

output

2*(((-I)*b^2*x^(7/2))/(a^2*(a^2 - b^2)*d) + x^4/(8*a^2) + (7*b^2*x^3*Log[1 
 + (a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*d^ 
2) + (7*b^2*x^3*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^(7/2)*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^(7/2)*Lo 
g[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^(7/2)*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^(7/2)*Log[1 + (a*E^(I*(c 
 + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((42*I 
)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b - I*Sqrt[a^2 - b^2 
]))])/(a^2*(a^2 - b^2)*d^3) - ((42*I)*b^2*x^(5/2)*PolyLog[2, -((a*E^(I*(c 
+ d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*d^3) - (7*b^3*x 
^3*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) + (14*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/ 
(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (7*b^3*x^3*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) - (14*b*x^3*PolyLog[2, -((a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[- 
a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) + (210*b^2*x^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) + ( 
210*b^2*x^2*PolyLog[3, -((a*E^(I*(c + d*Sqrt[x])))/(b + I*Sqrt[a^2 - b^...
 

3.1.46.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

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]
 

rule 4692

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]
 

3.1.46.4 Maple [F]

\[\int \frac {x^{3}}{\left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}d x\]

input

int(x^3/(a+b*sec(c+d*x^(1/2)))^2,x)
 

output

int(x^3/(a+b*sec(c+d*x^(1/2)))^2,x)
 

3.1.46.5 Fricas [F]

\[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^3/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="fricas")
 

output

integral(x^3/(b^2*sec(d*sqrt(x) + c)^2 + 2*a*b*sec(d*sqrt(x) + c) + a^2), 
x)
 

3.1.46.6 Sympy [F]

\[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^{3}}{\left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}}\, dx \]

input

integrate(x**3/(a+b*sec(c+d*x**(1/2)))**2,x)
 

output

Integral(x**3/(a + b*sec(c + d*sqrt(x)))**2, x)
 

3.1.46.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\text {Exception raised: ValueError} \]

input

integrate(x^3/(a+b*sec(c+d*x^(1/2)))^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
 

3.1.46.8 Giac [F]

\[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int { \frac {x^{3}}{{\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2}} \,d x } \]

input

integrate(x^3/(a+b*sec(c+d*x^(1/2)))^2,x, algorithm="giac")
 

output

integrate(x^3/(b*sec(d*sqrt(x) + c) + a)^2, x)
 

3.1.46.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2} \, dx=\int \frac {x^3}{{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2} \,d x \]

input

int(x^3/(a + b/cos(c + d*x^(1/2)))^2,x)
 

output

int(x^3/(a + b/cos(c + d*x^(1/2)))^2, x)