3.1.83 \(\int \frac {(e x)^{-1+3 n}}{(a+b \sec (c+d x^n))^2} \, dx\) [83]

3.1.83.1 Optimal result

Integrand size = 24, antiderivative size = 1384 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{3 n}}{3 a^2 e n}-\frac {i b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2-b^2\right ) d e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}+\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^2 e n}-\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}+\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}+\frac {i b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d e n}-\frac {2 i b x^{-n} (e x)^{3 n} \log \left (1+\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 i b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3 e n}-\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}+\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}+\frac {2 b^3 x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^2 e n}-\frac {4 b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^2 e n}-\frac {2 i b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}+\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {2 i b^3 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} d^3 e n}-\frac {4 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (c+d x^n\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \sin \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cos \left (c+d x^n\right )\right )} \]

output

1/3*(e*x)^(3*n)/a^2/e/n-2*I*b^2*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x^n))/ 
(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))+2*b^2*(e*x)^(3*n)*l 
n(1+a*exp(I*(c+d*x^n))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(2* 
n))+2*b^2*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b+I*(a^2-b^2)^(1/2)))/a^2/( 
a^2-b^2)/d^2/e/n/(x^(2*n))-2*I*b^3*(e*x)^(3*n)*polylog(3,-a*exp(I*(c+d*x^n 
))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))-4*I*b*(e*x 
)^(3*n)*polylog(3,-a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x 
^(3*n))/(-a^2+b^2)^(1/2)+2*I*b*(e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b-(-a^ 
2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-2*I*b*(e*x)^(3*n)*ln(1+a*e 
xp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-2*b 
^3*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a 
^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))+2*b^3*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d* 
x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))+I*b^3*( 
e*x)^(3*n)*ln(1+a*exp(I*(c+d*x^n))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3 
/2)/d/e/n/(x^n)+4*I*b*(e*x)^(3*n)*polylog(3,-a*exp(I*(c+d*x^n))/(b-(-a^2+b 
^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)+b^2*(e*x)^(3*n)*sin(c+d 
*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(b+a*cos(c+d*x^n))-I*b^2*(e*x)^(3*n)/a^2/(a^ 
2-b^2)/d/e/n/(x^n)-2*I*b^2*(e*x)^(3*n)*polylog(2,-a*exp(I*(c+d*x^n))/(b-I* 
(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))+4*b*(e*x)^(3*n)*polylog( 
2,-a*exp(I*(c+d*x^n))/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2...
 

3.1.83.2 Mathematica [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx \]

input

Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2,x]
 

output

Integrate[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2, x]
 

3.1.83.3 Rubi [A] (verified)

Time = 2.38 (sec) , antiderivative size = 1119, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4696, 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.

\(\displaystyle \int \frac {(e x)^{3 n-1}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx\)

\(\Big \downarrow \) 4696

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{3 n-1}}{\left (a+b \sec \left (d x^n+c\right )\right )^2}dx}{e}\)

\(\Big \downarrow \) 4692

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \frac {x^{2 n}}{\left (a+b \csc \left (d x^n+c+\frac {\pi }{2}\right )\right )^2}dx^n}{e n}\)

\(\Big \downarrow \) 4679

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \int \left (-\frac {2 b x^{2 n}}{a^2 \left (b+a \cos \left (d x^n+c\right )\right )}+\frac {x^{2 n}}{a^2}+\frac {b^2 x^{2 n}}{a^2 \left (b+a \cos \left (d x^n+c\right )\right )^2}\right )dx^n}{e n}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{-3 n} (e x)^{3 n} \left (\frac {2 b^2 \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b-i \sqrt {a^2-b^2}}+1\right ) x^n}{a^2 \left (a^2-b^2\right ) d^2}+\frac {2 b^2 \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b+i \sqrt {a^2-b^2}}+1\right ) x^n}{a^2 \left (a^2-b^2\right ) d^2}+\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^n}{a^2 \sqrt {b^2-a^2} d^2}-\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right ) x^n}{a^2 \left (b^2-a^2\right )^{3/2} d^2}-\frac {4 b \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^n}{a^2 \sqrt {b^2-a^2} d^2}+\frac {2 b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right ) x^n}{a^2 \left (b^2-a^2\right )^{3/2} d^2}+\frac {2 i b \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \sqrt {b^2-a^2} d}-\frac {i b^3 \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d}-\frac {2 i b \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \sqrt {b^2-a^2} d}+\frac {i b^3 \log \left (\frac {e^{i \left (d x^n+c\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d}+\frac {b^2 \sin \left (d x^n+c\right ) x^{2 n}}{a \left (a^2-b^2\right ) d \left (b+a \cos \left (d x^n+c\right )\right )}-\frac {i b^2 x^{2 n}}{a^2 \left (a^2-b^2\right ) d}+\frac {x^{3 n}}{3 a^2}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}-\frac {2 i b^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i \left (d x^n+c\right )}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) d^3}+\frac {4 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}-\frac {2 i b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}-\frac {4 i b \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} d^3}+\frac {2 i b^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i \left (d x^n+c\right )}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} d^3}\right )}{e n}\)

input

Int[(e*x)^(-1 + 3*n)/(a + b*Sec[c + d*x^n])^2,x]
 

output

((e*x)^(3*n)*(((-I)*b^2*x^(2*n))/(a^2*(a^2 - b^2)*d) + x^(3*n)/(3*a^2) + ( 
2*b^2*x^n*Log[1 + (a*E^(I*(c + d*x^n)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^ 
2 - b^2)*d^2) + (2*b^2*x^n*Log[1 + (a*E^(I*(c + d*x^n)))/(b + I*Sqrt[a^2 - 
 b^2])])/(a^2*(a^2 - b^2)*d^2) - (I*b^3*x^(2*n)*Log[1 + (a*E^(I*(c + d*x^n 
)))/(b - Sqrt[-a^2 + b^2])])/(a^2*(-a^2 + b^2)^(3/2)*d) + ((2*I)*b*x^(2*n) 
*Log[1 + (a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b 
^2]*d) + (I*b^3*x^(2*n)*Log[1 + (a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + b^2 
])])/(a^2*(-a^2 + b^2)^(3/2)*d) - ((2*I)*b*x^(2*n)*Log[1 + (a*E^(I*(c + d* 
x^n)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 + b^2]*d) - ((2*I)*b^2*Poly 
Log[2, -((a*E^(I*(c + d*x^n)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2) 
*d^3) - ((2*I)*b^2*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + I*Sqrt[a^2 - b^ 
2]))])/(a^2*(a^2 - b^2)*d^3) - (2*b^3*x^n*PolyLog[2, -((a*E^(I*(c + d*x^n) 
))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) + (4*b*x^n*PolyL 
og[2, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b 
^2]*d^2) + (2*b^3*x^n*PolyLog[2, -((a*E^(I*(c + d*x^n)))/(b + Sqrt[-a^2 + 
b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*d^2) - (4*b*x^n*PolyLog[2, -((a*E^(I*(c + 
 d*x^n)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^2) - ((2*I)*b^ 
3*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 
+ b^2)^(3/2)*d^3) + ((4*I)*b*PolyLog[3, -((a*E^(I*(c + d*x^n)))/(b - Sqrt[ 
-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*d^3) + ((2*I)*b^3*PolyLog[3, -((a...
 

3.1.83.3.1 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]
 

rule 4696

Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x 
_Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m])   Int[x^m*( 
a + b*Sec[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
 

3.1.83.4 Maple [F]

\[\int \frac {\left (e x \right )^{3 n -1}}{{\left (a +b \sec \left (c +d \,x^{n}\right )\right )}^{2}}d x\]

input

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

output

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

3.1.83.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3831 vs. \(2 (1282) = 2564\).

Time = 0.68 (sec) , antiderivative size = 3831, normalized size of antiderivative = 2.77 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]

input

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

output

1/6*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d^3*e^(3*n - 1)*x^(3*n)*cos(d*x^n + c) + 
2*(a^4*b - 2*a^2*b^3 + b^5)*d^3*e^(3*n - 1)*x^(3*n) + 6*(a^3*b^2 - a*b^4)* 
d^2*e^(3*n - 1)*x^(2*n)*sin(d*x^n + c) - 6*((2*a^3*b^2 - a*b^4)*d*e^(3*n - 
 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + (-I*a^2*b^3 + I*b^5)*e^(3*n - 1) + ((2*a^ 
4*b - a^2*b^3)*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + (-I*a^3*b^2 + I* 
a*b^4)*e^(3*n - 1))*cos(d*x^n + c))*dilog(-((a*sqrt(-(a^2 - b^2)/a^2) + b) 
*cos(d*x^n + c) - (I*a*sqrt(-(a^2 - b^2)/a^2) + I*b)*sin(d*x^n + c) + a)/a 
 + 1) - 6*((2*a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) + 
(I*a^2*b^3 - I*b^5)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d*e^(3*n - 1)*x^n*s 
qrt(-(a^2 - b^2)/a^2) + (I*a^3*b^2 - I*a*b^4)*e^(3*n - 1))*cos(d*x^n + c)) 
*dilog(-((a*sqrt(-(a^2 - b^2)/a^2) + b)*cos(d*x^n + c) - (-I*a*sqrt(-(a^2 
- b^2)/a^2) - I*b)*sin(d*x^n + c) + a)/a + 1) + 6*((2*a^3*b^2 - a*b^4)*d*e 
^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - (I*a^2*b^3 - I*b^5)*e^(3*n - 1) + 
((2*a^4*b - a^2*b^3)*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2) - (I*a^3*b^2 
 - I*a*b^4)*e^(3*n - 1))*cos(d*x^n + c))*dilog(((a*sqrt(-(a^2 - b^2)/a^2) 
- b)*cos(d*x^n + c) + (I*a*sqrt(-(a^2 - b^2)/a^2) - I*b)*sin(d*x^n + c) - 
a)/a + 1) + 6*((2*a^3*b^2 - a*b^4)*d*e^(3*n - 1)*x^n*sqrt(-(a^2 - b^2)/a^2 
) - (-I*a^2*b^3 + I*b^5)*e^(3*n - 1) + ((2*a^4*b - a^2*b^3)*d*e^(3*n - 1)* 
x^n*sqrt(-(a^2 - b^2)/a^2) - (-I*a^3*b^2 + I*a*b^4)*e^(3*n - 1))*cos(d*x^n 
 + c))*dilog(((a*sqrt(-(a^2 - b^2)/a^2) - b)*cos(d*x^n + c) + (-I*a*sqr...
 

3.1.83.6 Sympy [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \sec {\left (c + d x^{n} \right )}\right )^{2}}\, dx \]

input

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

output

Integral((e*x)**(3*n - 1)/(a + b*sec(c + d*x**n))**2, x)
 

3.1.83.7 Maxima [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]

input

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

output

1/3*(6*a*b^3*e^(3*n)*x^(2*n)*sin(d*x^n + c) + (a^4 - a^2*b^2)*d*e^(3*n)*x^ 
(3*n)*cos(2*d*x^n + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*e^(3*n)*x^(3*n)*cos(d*x^n 
 + c)^2 + (a^4 - a^2*b^2)*d*e^(3*n)*x^(3*n)*sin(2*d*x^n + 2*c)^2 + 4*(a^2* 
b^2 - b^4)*d*e^(3*n)*x^(3*n)*sin(d*x^n + c)^2 + 4*(a^3*b - a*b^3)*d*e^(3*n 
)*x^(3*n)*cos(d*x^n + c) + (a^4 - a^2*b^2)*d*e^(3*n)*x^(3*n) - 2*(3*a*b^3* 
e^(3*n)*x^(2*n)*sin(d*x^n + c) - 2*(a^3*b - a*b^3)*d*e^(3*n)*x^(3*n)*cos(d 
*x^n + c) - (a^4 - a^2*b^2)*d*e^(3*n)*x^(3*n))*cos(2*d*x^n + 2*c) + 3*((a^ 
6 - a^4*b^2)*d*e*n*cos(2*d*x^n + 2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*e*n*cos( 
d*x^n + c)^2 + (a^6 - a^4*b^2)*d*e*n*sin(2*d*x^n + 2*c)^2 + 4*(a^5*b - a^3 
*b^3)*d*e*n*sin(2*d*x^n + 2*c)*sin(d*x^n + c) + 4*(a^4*b^2 - a^2*b^4)*d*e* 
n*sin(d*x^n + c)^2 + 4*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 - a^4 
*b^2)*d*e*n + 2*(2*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 - a^4*b^2 
)*d*e*n)*cos(2*d*x^n + 2*c))*integrate(2*(2*a^2*b^4*e^(3*n)*x^(2*n)*cos(2* 
c)*sin(2*d*x^n) + 2*a^2*b^4*e^(3*n)*x^(2*n)*cos(2*d*x^n)*sin(2*c) - 4*(a^3 
*b^3 - a*b^5)*e^(3*n)*x^(2*n)*cos(c)*sin(d*x^n) - 4*(a^3*b^3 - a*b^5)*e^(3 
*n)*x^(2*n)*cos(d*x^n)*sin(c) + (2*a^3*b^3*e^(3*n)*x^(2*n)*sin(d*x^n + c) 
- (2*a^5*b - a^3*b^3)*d*e^(3*n)*x^(3*n)*cos(d*x^n + c))*cos(2*d*x^n + 2*c) 
 - ((2*a^5*b - 3*a^3*b^3 + a*b^5)*d*e^(3*n)*x^(3*n) - (2*a*b^5*e^(3*n)*x^( 
2*n)*sin(2*c) + (2*a^3*b^3 - a*b^5)*d*e^(3*n)*x^(3*n)*cos(2*c))*cos(2*d*x^ 
n) + 2*((2*a^4*b^2 - 3*a^2*b^4 + b^6)*d*e^(3*n)*x^(3*n)*cos(c) + 2*(a^2...
 

3.1.83.8 Giac [F]

\[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]

input

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

output

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

3.1.83.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )}^2} \,d x \]

input

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

output

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