\(\int \cos (a+b x) (d \tan (a+b x))^n \, dx\) [372]

Optimal result

Integrand size = 17, antiderivative size = 72 \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2697} \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} (d \tan (a+b x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {n+1}{2},\frac {n+3}{2},\sin ^2(a+b x)\right )}{b d (n+1)} \]

[In]

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

[Out]

(Cos[a + b*x]*(Cos[a + b*x]^2)^(n/2)*Hypergeometric2F1[n/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(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*} \text {integral}& = \frac {\cos (a+b x) \cos ^2(a+b x)^{n/2} \operatorname {Hypergeometric2F1}\left (\frac {n}{2},\frac {1+n}{2},\frac {3+n}{2},\sin ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 2.02 (sec) , antiderivative size = 452, normalized size of antiderivative = 6.28 \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=-\frac {2 \left (\operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-2 \operatorname {AppellF1}\left (\frac {1+n}{2},n,2,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) \cos \left (\frac {1}{2} (a+b x)\right ) \cos (a+b x) \sin \left (\frac {1}{2} (a+b x)\right ) (d \tan (a+b x))^n}{b (1+n) \left (-\operatorname {AppellF1}\left (\frac {1+n}{2},n,1,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+\frac {\left (-\left (\left (\operatorname {AppellF1}\left (\frac {3+n}{2},n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-4 \operatorname {AppellF1}\left (\frac {3+n}{2},n,3,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )-n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,1,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+2 n \operatorname {AppellF1}\left (\frac {3+n}{2},1+n,2,\frac {5+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )\right ) (-1+\cos (a+b x))\right )+(3+n) \operatorname {AppellF1}\left (\frac {1+n}{2},n,2,\frac {3+n}{2},\tan ^2\left (\frac {1}{2} (a+b x)\right ),-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) (1+\cos (a+b x))\right ) \sec ^2\left (\frac {1}{2} (a+b x)\right )}{3+n}\right )} \]

[In]

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

[Out]

(-2*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 2*AppellF1[(1 + n)/2, n,
2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[(a + b*x)/2]*Cos[a + b*x]*Sin[(a + b*x)/2]*(d*Tan[
a + b*x])^n)/(b*(1 + n)*(-AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + ((-(
(AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(3 + n)/2, n, 3, (
5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/
2]^2, -Tan[(a + b*x)/2]^2] + 2*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2
]^2])*(-1 + Cos[a + b*x])) + (3 + n)*AppellF1[(1 + n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2
]^2]*(1 + Cos[a + b*x]))*Sec[(a + b*x)/2]^2)/(3 + n)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int \left (d \tan {\left (a + b x \right )}\right )^{n} \cos {\left (a + b x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int { \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \cos (a+b x) (d \tan (a+b x))^n \, dx=\int \cos \left (a+b\,x\right )\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n \,d x \]

[In]

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

[Out]

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