\(\int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx\) [226]

Optimal result

Integrand size = 21, antiderivative size = 106 \[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\frac {2^{3+n} \operatorname {AppellF1}\left (\frac {3}{2},2+n,1,\frac {5}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac {1}{1+\sec (c+d x)}\right )^{3+n} (a+a \sec (c+d x))^n \tan ^3(c+d x)}{3 d} \] Output:

1/3*2^(3+n)*AppellF1(3/2,1,2+n,5/2,(a-a*sec(d*x+c))/(a+a*sec(d*x+c)),-(a-a 
*sec(d*x+c))/(a+a*sec(d*x+c)))*(1/(1+sec(d*x+c)))^(3+n)*(a+a*sec(d*x+c))^n 
*tan(d*x+c)^3/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(910\) vs. \(2(106)=212\).

Time = 6.12 (sec) , antiderivative size = 910, normalized size of antiderivative = 8.58 \[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx =\text {Too large to display} \] Input:

Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^2,x]
 

Output:

((a*(1 + Sec[c + d*x]))^n*((-4*Hypergeometric2F1[-1 - n, n, -n, (1 - Tan[( 
c + d*x)/2])/2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(1 + Tan[(c + d*x)/2]) 
^n)/((1 + n)*(1 + Sec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])) - (Hypergeometr 
ic2F1[1 - n, 2 + n, 2 - n, (1 - Tan[(c + d*x)/2])/2]*(Cos[(c + d*x)/2]^2*S 
ec[c + d*x])^n*(-1 + Tan[(c + d*x)/2])*(1 + Tan[(c + d*x)/2])^n)/((-1 + n) 
*(1 + Sec[c + d*x])^n) - (120*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, 
 -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*Cos[c + d*x]*Sin[c + d*x]*(3*Appe 
llF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF 
1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[3/ 
2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/ 
2]^2))/(45*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^ 
2]^2*Cos[(c + d*x)/2]^2*(1 + 2*n - 2*n*Cos[c + d*x] + Cos[2*(c + d*x)]) + 
6*AppellF1[1/2, n, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c 
 + d*x)/2]^2*(-5*AppellF1[3/2, n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d* 
x)/2]^2]*(1 + 2*n - 2*(2 + n)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*n*Appel 
lF1[3/2, 1 + n, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*n 
- 2*(2 + n)*Cos[c + d*x] + Cos[2*(c + d*x)]) - 48*(2*AppellF1[5/2, n, 3, 7 
/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*n*AppellF1[5/2, 1 + n, 2, 
 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*(1 + n)*AppellF1[5/2, 2 
 + n, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cot[c + d*x]*Cs...
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3042, 4377}

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.

\(\displaystyle \int \tan ^2(c+d x) (a \sec (c+d x)+a)^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cot \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^ndx\)

\(\Big \downarrow \) 4377

\(\displaystyle \frac {2^{n+3} \tan ^3(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n+3} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (\frac {3}{2},n+2,1,\frac {5}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d}\)

Input:

Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^2,x]
 

Output:

(2^(3 + n)*AppellF1[3/2, 2 + n, 1, 5/2, -((a - a*Sec[c + d*x])/(a + a*Sec[ 
c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*((1 + Sec[c + d*x]) 
^(-1))^(3 + n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3)/(3*d)
 

Defintions of rubi rules used

rule 3042

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

rule 4377

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( 
a_))^(n_), x_Symbol] :> Simp[(-2^(m + n + 1))*(e*Cot[c + d*x])^(m + 1)*((a 
+ b*Csc[c + d*x])^n/(d*e*(m + 1)))*(a/(a + b*Csc[c + d*x]))^(m + n + 1)*App 
ellF1[(m + 1)/2, m + n, 1, (m + 3)/2, -(a - b*Csc[c + d*x])/(a + b*Csc[c + 
d*x]), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])], x] /; FreeQ[{a, b, c, d, 
 e, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[n]
 

Maple [F]

\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \tan \left (d x +c \right )^{2}d x\]

Input:

int((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x)
 

Output:

int((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x)
 

Fricas [F]

\[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2} \,d x } \] Input:

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x, algorithm="fricas")
 

Output:

integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^2, x)
 

Sympy [F]

\[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \tan ^{2}{\left (c + d x \right )}\, dx \] Input:

integrate((a+a*sec(d*x+c))**n*tan(d*x+c)**2,x)
 

Output:

Integral((a*(sec(c + d*x) + 1))**n*tan(c + d*x)**2, x)
 

Maxima [F]

\[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2} \,d x } \] Input:

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x, algorithm="maxima")
 

Output:

integrate((a*sec(d*x + c) + a)^n*tan(d*x + c)^2, x)
 

Giac [F]

\[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2} \,d x } \] Input:

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x, algorithm="giac")
 

Output:

integrate((a*sec(d*x + c) + a)^n*tan(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \] Input:

int(tan(c + d*x)^2*(a + a/cos(c + d*x))^n,x)
 

Output:

int(tan(c + d*x)^2*(a + a/cos(c + d*x))^n, x)
 

Reduce [F]

\[ \int (a+a \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int \left (\sec \left (d x +c \right ) a +a \right )^{n} \tan \left (d x +c \right )^{2}d x \] Input:

int((a+a*sec(d*x+c))^n*tan(d*x+c)^2,x)
 

Output:

int((sec(c + d*x)*a + a)**n*tan(c + d*x)**2,x)