3.368 \(\int \cos ^4(a+b x) (d \tan (a+b x))^n \, dx\)
Optimal. Leaf size=50 \[ \frac {(d \tan (a+b x))^{n+1} \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(a+b x)\right )}{b d (n+1)} \]
[Out]
hypergeom([3, 1/2+1/2*n],[3/2+1/2*n],-tan(b*x+a)^2)*(d*tan(b*x+a))^(1+n)/b/d/(1+n)
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Rubi [A] time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used =
{2607, 364} \[ \frac {(d \tan (a+b x))^{n+1} \, _2F_1\left (3,\frac {n+1}{2};\frac {n+3}{2};-\tan ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[a + b*x]^4*(d*Tan[a + b*x])^n,x]
[Out]
(Hypergeometric2F1[3, (1 + n)/2, (3 + n)/2, -Tan[a + b*x]^2]*(d*Tan[a + b*x])^(1 + n))/(b*d*(1 + n))
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 2607
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])
Rubi steps
\begin {align*} \int \cos ^4(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(d x)^n}{\left (1+x^2\right )^3} \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\, _2F_1\left (3,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [C] time = 12.63, size = 1712, normalized size = 34.24 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Cos[a + b*x]^4*(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] - 8*(AppellF1[(1 +
n)/2, n, 2, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 3*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(
a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 4*AppellF1[(1 + n)/2, n, 4, (3 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x
)/2]^2] - 2*AppellF1[(1 + n)/2, n, 5, (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]^5*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*(16*AppellF1[(3 + n)/2, n, 3, (5 + n)/2, Tan
[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 72*AppellF1[(3 + n)/2, n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a +
b*x)/2]^2] + 128*AppellF1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 80*AppellF1[(
3 + n)/2, n, 6, (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] - 8*n*AppellF1[(3 + n)/2, 1 + n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2,
-Tan[(a + b*x)/2]^2] + 24*n*AppellF1[(3 + n)/2, 1 + n, 3, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]
- 32*n*AppellF1[(3 + n)/2, 1 + n, 4, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 16*n*AppellF1[(3 +
n)/2, 1 + n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 24*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 - 8*n*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 + 72*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/
2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 24*n*AppellF1[(1 + n)/2, n, 3, (3 + n)/2, Tan[(a + b*x)/2]^2,
-Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - 96*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]^2 - 32*n*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]^2 + AppellF1[(3 + n)/2, n, 2, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(-1
+ Cos[a + b*x]) - 16*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
] + 72*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] - 128*Appell
F1[(3 + n)/2, n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[a + b*x] + 80*AppellF1[(3 + n)/2,
n, 6, (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] + 8*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] - 24*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] + 32*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] - 16*n*AppellF1[(3 + n)/2, 1 + n, 5, (5 + n)/2, Tan[(a + b*x)/2]^2, -Tan[(a
+ b*x)/2]^2]*Cos[a + b*x] + 8*(3 + n)*AppellF1[(1 + n)/2, n, 5, (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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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^4*(d*tan(b*x+a))^n,x, algorithm="fricas")
[Out]
integral((d*tan(b*x + a))^n*cos(b*x + a)^4, x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^4*(d*tan(b*x+a))^n,x, algorithm="giac")
[Out]
integrate((d*tan(b*x + a))^n*cos(b*x + a)^4, x)
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maple [F] time = 1.52, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{4}\left (b x +a \right )\right ) \left (d \tan \left (b x +a \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(b*x+a)^4*(d*tan(b*x+a))^n,x)
[Out]
int(cos(b*x+a)^4*(d*tan(b*x+a))^n,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \cos \left (b x + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)^4*(d*tan(b*x+a))^n,x, algorithm="maxima")
[Out]
integrate((d*tan(b*x + a))^n*cos(b*x + a)^4, x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\cos \left (a+b\,x\right )}^4\,{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(a + b*x)^4*(d*tan(a + b*x))^n,x)
[Out]
int(cos(a + b*x)^4*(d*tan(a + b*x))^n, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{n} \cos ^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(b*x+a)**4*(d*tan(b*x+a))**n,x)
[Out]
Integral((d*tan(a + b*x))**n*cos(a + b*x)**4, x)
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