3.373 \(\int \cos ^3(a+b x) (d \tan (a+b x))^n \, dx\)
Optimal. Leaf size=78 \[ \frac{\cos ^3(a+b x) \cos ^2(a+b x)^{\frac{n-2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n-2}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
[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))
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Rubi [A] time = 0.037228, antiderivative size = 78, normalized size of antiderivative = 1.,
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 =
{2617} \[ \frac{\cos ^3(a+b x) \cos ^2(a+b x)^{\frac{n-2}{2}} (d \tan (a+b x))^{n+1} \, _2F_1\left (\frac{n-2}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right )}{b d (n+1)} \]
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 2617
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)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2,
(n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), 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*}
Mathematica [C] time = 6.27976, size = 1313, normalized size = 16.83 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[a + b*x]^3*(d*Tan[a + b*x])^n,x]
[Out]
(4*(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]^3*Sin[(a + b*x)/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*App
ellF1[(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)*Appe
llF1[(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]))))
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Maple [F] time = 0.826, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( bx+a \right ) \right ) ^{3} \left ( d\tan \left ( bx+a \right ) \right ) ^{n}\, dx \end{align*}
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., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{3}\,{d x} \end{align*}
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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{3}, x\right ) \end{align*}
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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{3}\,{d x} \end{align*}
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)