3.4.70 \(\int \sec ^3(a+b x) (d \tan (a+b x))^n \, dx\) [370]

Optimal. Leaf size=78 \[ \frac {\cos ^2(a+b x)^{\frac {4+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) \sec ^3(a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)} \]

[Out]

(cos(b*x+a)^2)^(2+1/2*n)*hypergeom([2+1/2*n, 1/2+1/2*n],[3/2+1/2*n],sin(b*x+a)^2)*sec(b*x+a)^3*(d*tan(b*x+a))^
(1+n)/b/d/(1+n)

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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2697} \begin {gather*} \frac {\sec ^3(a+b x) \cos ^2(a+b x)^{\frac {n+4}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+4}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[a + b*x]^2)^((4 + n)/2)*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]^
3*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps

\begin {align*} \int \sec ^3(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\cos ^2(a+b x)^{\frac {4+n}{2}} \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\sin ^2(a+b x)\right ) \sec ^3(a+b x) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}

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Mathematica [A]
time = 0.10, size = 72, normalized size = 0.92 \begin {gather*} \frac {d \, _2F_1\left (\frac {3}{2},\frac {1-n}{2};\frac {5}{2};\sec ^2(a+b x)\right ) \sec ^3(a+b x) (d \tan (a+b x))^{-1+n} \left (-\tan ^2(a+b x)\right )^{\frac {1-n}{2}}}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*Hypergeometric2F1[3/2, (1 - n)/2, 5/2, Sec[a + b*x]^2]*Sec[a + b*x]^3*(d*Tan[a + b*x])^(-1 + n)*(-Tan[a + b
*x]^2)^((1 - n)/2))/(3*b)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (\sec ^{3}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d \tan {\left (a + b x \right )}\right )^{n} \sec ^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n}{{\cos \left (a+b\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(a + b*x))^n/cos(a + b*x)^3,x)

[Out]

int((d*tan(a + b*x))^n/cos(a + b*x)^3, x)

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