3.221 \(\int (a+a \sec (c+d x))^n \tan ^3(c+d x) \, dx\)

Optimal. Leaf size=69 \[ \frac{(a \sec (c+d x)+a)^{n+2} \text{Hypergeometric2F1}(1,n+2,n+3,\sec (c+d x)+1)}{a^2 d (n+2)}+\frac{(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)} \]

[Out]

(a + a*Sec[c + d*x])^(2 + n)/(a^2*d*(2 + n)) + (Hypergeometric2F1[1, 2 + n, 3 + n, 1 + Sec[c + d*x]]*(a + a*Se
c[c + d*x])^(2 + n))/(a^2*d*(2 + n))

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Rubi [A]  time = 0.0645631, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3880, 80, 65} \[ \frac{(a \sec (c+d x)+a)^{n+2} \, _2F_1(1,n+2;n+3;\sec (c+d x)+1)}{a^2 d (n+2)}+\frac{(a \sec (c+d x)+a)^{n+2}}{a^2 d (n+2)} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3,x]

[Out]

(a + a*Sec[c + d*x])^(2 + n)/(a^2*d*(2 + n)) + (Hypergeometric2F1[1, 2 + n, 3 + n, 1 + Sec[c + d*x]]*(a + a*Se
c[c + d*x])^(2 + n))/(a^2*d*(2 + n))

Rule 3880

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int (a+a \sec (c+d x))^n \tan ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x) (a+a x)^{1+n}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{(a+a \sec (c+d x))^{2+n}}{a^2 d (2+n)}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{1+n}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{(a+a \sec (c+d x))^{2+n}}{a^2 d (2+n)}+\frac{\, _2F_1(1,2+n;3+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{2+n}}{a^2 d (2+n)}\\ \end{align*}

Mathematica [A]  time = 0.0478846, size = 49, normalized size = 0.71 \[ \frac{(\sec (c+d x)+1)^2 (a (\sec (c+d x)+1))^n (\text{Hypergeometric2F1}(1,n+2,n+3,\sec (c+d x)+1)+1)}{d (n+2)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^n*Tan[c + d*x]^3,x]

[Out]

((1 + Hypergeometric2F1[1, 2 + n, 3 + n, 1 + Sec[c + d*x]])*(1 + Sec[c + d*x])^2*(a*(1 + Sec[c + d*x]))^n)/(d*
(2 + n))

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Maple [F]  time = 0.244, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sec \left ( dx+c \right ) \right ) ^{n} \left ( \tan \left ( dx+c \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^n*tan(d*x+c)^3,x)

[Out]

int((a+a*sec(d*x+c))^n*tan(d*x+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^3,x, algorithm="maxima")

[Out]

integrate((a*sec(d*x + c) + a)^n*tan(d*x + c)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^3,x, algorithm="fricas")

[Out]

integral((a*sec(d*x + c) + a)^n*tan(d*x + c)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{n} \tan ^{3}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**n*tan(d*x+c)**3,x)

[Out]

Integral((a*(sec(c + d*x) + 1))**n*tan(c + d*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^n*tan(d*x+c)^3,x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^n*tan(d*x + c)^3, x)