3.4.73 \(\int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx\) [373]

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

[Out]

cos(b*x+a)^3*(cos(b*x+a)^2)^(-1+1/2*n)*hypergeom([-1+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)

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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 {\cos ^3(a+b x) \cos ^2(a+b x)^{\frac {n-2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac {n-2}{2},\frac {n+1}{2};\frac {n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[a + b*x]^3*(Cos[a + b*x]^2)^((-2 + n)/2)*Hypergeometric2F1[(-2 + 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*} \int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\cos ^3(a+b x) \cos ^2(a+b x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (\frac {1}{2} (-2+n),\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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 8.35, size = 1340, normalized size = 17.18 \begin {gather*} \frac {8 (3+n) \left (F_1\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 )-6 F_1\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 )+12 F_1\left (\frac {1+n}{2};n,3;\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 )-8 F_1\left (\frac {1+n}{2};n,4;\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 ^3\left (\frac {1}{2} (a+b x)\right ) \cos ^4(a+b x) \sin ^2\left (\frac {1}{2} (a+b x)\right ) (d \tan (a+b x))^n}{b (1+n) \left ((3+n) F_1\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 ) (1+\cos (a+b x))-2 \left (F_1\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 )-12 F_1\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 )+36 F_1\left (\frac {3+n}{2};n,4;\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 )-32 F_1\left (\frac {3+n}{2};n,5;\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 F_1\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 )+6 n F_1\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 )-12 n F_1\left (\frac {3+n}{2};1+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 )+8 n F_1\left (\frac {3+n}{2};1+n,4;\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 )+18 F_1\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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )+6 n F_1\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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )+8 (3+n) F_1\left (\frac {1+n}{2};n,4;\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 ) \cos ^2\left (\frac {1}{2} (a+b x)\right )-F_1\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 ) \cos (a+b x)+12 F_1\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 ) \cos (a+b x)-36 F_1\left (\frac {3+n}{2};n,4;\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 ) \cos (a+b x)+32 F_1\left (\frac {3+n}{2};n,5;\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 ) \cos (a+b x)+n F_1\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 ) \cos (a+b x)-6 n F_1\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 ) \cos (a+b x)+12 n F_1\left (\frac {3+n}{2};1+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 ) \cos (a+b x)-8 n F_1\left (\frac {3+n}{2};1+n,4;\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 ) \cos (a+b x)-6 (3+n) F_1\left (\frac {1+n}{2};n,3;\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 )\right ) \left (-\sin \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {3}{2} (a+b x)\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*(3 + n)*(AppellF1[(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 6*AppellF1[(1 + n)
/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 12*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a
+ b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 8*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)
/2]^2])*Cos[(a + b*x)/2]^3*Cos[a + b*x]^4*Sin[(a + b*x)/2]^2*(d*Tan[a + b*x])^n)/(b*(1 + n)*((3 + n)*AppellF1[
(1 + n)/2, n, 1, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]) - 2*(AppellF1[(3 + n)/
2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan[(a
+ b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 36*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)
/2]^2] - 32*AppellF1[(3 + n)/2, n, 5, (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] + 6*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/
2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2
, -Tan[(a + b*x)/2]^2] + 8*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]
+ 18*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]^2 + 6*n*A
ppellF1[(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]^2 + 8*(3 + n)*Ap
pellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(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]*Cos[a + b*x] + 12*AppellF1[(3 + n)/2, n, 3, (
5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] - 36*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[
(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 32*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2
, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + n*AppellF1[(3 + n)/2, 1 + n, 1, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a +
b*x)/2]^2]*Cos[a + b*x] - 6*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^
2]*Cos[a + b*x] + 12*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a
+ b*x] - 8*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] -
6*(3 + n)*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x])))*
(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))

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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \left (\cos ^{3}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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