\(\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx\) [359]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx \]

[In]

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

[Out]

(Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - n, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*
Sec[c + d*x])^n*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^n) - (Sqrt[2]*a*Appel
lF1[1/2, 1/2, -n, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*
x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^n) - Defer[Int][(a + b*Sec[c + d*x])^n, x]

Rubi steps \begin{align*} \text {integral}& = \int (a+b \sec (c+d x))^n \left (-1+\sec ^2(c+d x)\right ) \, dx \\ & = \frac {\int (-b-a \sec (c+d x)) (a+b \sec (c+d x))^n \, dx}{b}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^{1+n} \, dx}{b} \\ & = -\frac {a \int \sec (c+d x) (a+b \sec (c+d x))^n \, dx}{b}-\frac {\tan (c+d x) \text {Subst}\left (\int \frac {(a+b x)^{1+n}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\int (a+b \sec (c+d x))^n \, dx \\ & = \frac {(a \tan (c+d x)) \text {Subst}\left (\int \frac {(a+b x)^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}+\frac {\left ((-a-b) (a+b \sec (c+d x))^n \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{-n} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{1+n}}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\int (a+b \sec (c+d x))^n \, dx \\ & = \frac {\sqrt {2} (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-n,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)}}+\frac {\left (a (a+b \sec (c+d x))^n \left (-\frac {a+b \sec (c+d x)}{-a-b}\right )^{-n} \tan (c+d x)\right ) \text {Subst}\left (\int \frac {\left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^n}{\sqrt {1-x} \sqrt {1+x}} \, dx,x,\sec (c+d x)\right )}{b d \sqrt {1-\sec (c+d x)} \sqrt {1+\sec (c+d x)}}-\int (a+b \sec (c+d x))^n \, dx \\ & = \frac {\sqrt {2} (a+b) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-1-n,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)}}-\frac {\sqrt {2} a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} \tan (c+d x)}{b d \sqrt {1+\sec (c+d x)}}-\int (a+b \sec (c+d x))^n \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 12.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.98 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 3.98 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \tan ^{2}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 5.80 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 15.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int (a+b \sec (c+d x))^n \tan ^2(c+d x) \, dx=\int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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