3.650 \(\int \cos ^4(c+d x) (a+b \tan (c+d x))^n \, dx\)
Optimal. Leaf size=434 \[ \frac {\cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{4 d \left (a^2+b^2\right )}+\frac {b \cos ^2(c+d x) \left (a b \left (\frac {3 a^2}{b^2}-2 n+5\right ) \tan (c+d x)+a^2 (n+1)+b^2 (3-n)\right ) (a+b \tan (c+d x))^{n+1}}{8 d \left (a^2+b^2\right )^2}+\frac {b \left (\frac {a n \left (\frac {3 a^2}{b^2}-2 n+5\right )}{b^2}-\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (-n^2-2 n+6\right )+b^4 \left (n^2-4 n+3\right )\right )}{b^6}\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{16 d (n+1) \left (\frac {a^2}{b^2}+1\right )^2 \left (a-\sqrt {-b^2}\right )}+\frac {b \left (\frac {a n \left (\frac {3 a^2}{b^2}-2 n+5\right )}{b^2}+\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (-n^2-2 n+6\right )+b^4 \left (n^2-4 n+3\right )\right )}{b^6}\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{16 d (n+1) \left (\frac {a^2}{b^2}+1\right )^2 \left (a+\sqrt {-b^2}\right )} \]
[Out]
1/16*b*hypergeom([1, 1+n],[2+n],(a+b*tan(d*x+c))/(a-(-b^2)^(1/2)))*(a*(5+3*a^2/b^2-2*n)*n/b^2-(3*a^4+a^2*b^2*(
-n^2-2*n+6)+b^4*(n^2-4*n+3))*(-b^2)^(1/2)/b^6)*(a+b*tan(d*x+c))^(1+n)/(1+a^2/b^2)^2/d/(1+n)/(a-(-b^2)^(1/2))+1
/16*b*hypergeom([1, 1+n],[2+n],(a+b*tan(d*x+c))/(a+(-b^2)^(1/2)))*(a*(5+3*a^2/b^2-2*n)*n/b^2+(3*a^4+a^2*b^2*(-
n^2-2*n+6)+b^4*(n^2-4*n+3))*(-b^2)^(1/2)/b^6)*(a+b*tan(d*x+c))^(1+n)/(1+a^2/b^2)^2/d/(1+n)/(a+(-b^2)^(1/2))+1/
4*cos(d*x+c)^4*(b+a*tan(d*x+c))*(a+b*tan(d*x+c))^(1+n)/(a^2+b^2)/d+1/8*b*cos(d*x+c)^2*(a+b*tan(d*x+c))^(1+n)*(
b^2*(3-n)+a^2*(1+n)+a*b*(5+3*a^2/b^2-2*n)*tan(d*x+c))/(a^2+b^2)^2/d
________________________________________________________________________________________
Rubi [A] time = 0.69, antiderivative size = 434, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used =
{3506, 741, 823, 831, 68} \[ \frac {b \left (\frac {a n \left (\frac {3 a^2}{b^2}-2 n+5\right )}{b^2}-\frac {\sqrt {-b^2} \left (a^2 b^2 \left (-n^2-2 n+6\right )+3 a^4+b^4 \left (n^2-4 n+3\right )\right )}{b^6}\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{16 d (n+1) \left (\frac {a^2}{b^2}+1\right )^2 \left (a-\sqrt {-b^2}\right )}+\frac {b \left (\frac {\sqrt {-b^2} \left (a^2 b^2 \left (-n^2-2 n+6\right )+3 a^4+b^4 \left (n^2-4 n+3\right )\right )}{b^6}+\frac {a n \left (\frac {3 a^2}{b^2}-2 n+5\right )}{b^2}\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{16 d (n+1) \left (\frac {a^2}{b^2}+1\right )^2 \left (a+\sqrt {-b^2}\right )}+\frac {\cos ^4(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{4 d \left (a^2+b^2\right )}+\frac {b \cos ^2(c+d x) \left (a b \left (\frac {3 a^2}{b^2}-2 n+5\right ) \tan (c+d x)+a^2 (n+1)+b^2 (3-n)\right ) (a+b \tan (c+d x))^{n+1}}{8 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]
[Out]
(b*((a*(5 + (3*a^2)/b^2 - 2*n)*n)/b^2 - (Sqrt[-b^2]*(3*a^4 + a^2*b^2*(6 - 2*n - n^2) + b^4*(3 - 4*n + n^2)))/b
^6)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(1
6*(1 + a^2/b^2)^2*(a - Sqrt[-b^2])*d*(1 + n)) + (b*((a*(5 + (3*a^2)/b^2 - 2*n)*n)/b^2 + (Sqrt[-b^2]*(3*a^4 + a
^2*b^2*(6 - 2*n - n^2) + b^4*(3 - 4*n + n^2)))/b^6)*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a
+ Sqrt[-b^2])]*(a + b*Tan[c + d*x])^(1 + n))/(16*(1 + a^2/b^2)^2*(a + Sqrt[-b^2])*d*(1 + n)) + (Cos[c + d*x]^
4*(b + a*Tan[c + d*x])*(a + b*Tan[c + d*x])^(1 + n))/(4*(a^2 + b^2)*d) + (b*Cos[c + d*x]^2*(a + b*Tan[c + d*x]
)^(1 + n)*(b^2*(3 - n) + a^2*(1 + n) + a*b*(5 + (3*a^2)/b^2 - 2*n)*Tan[c + d*x]))/(8*(a^2 + b^2)^2*d)
Rule 68
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && IntegerQ[n]
Rule 741
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 823
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 831
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && !Ration
alQ[m]
Rule 3506
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^n}{\left (1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^n \left (-3-\frac {3 a^2}{b^2}+n-\frac {a (2-n) x}{b^2}\right )}{\left (1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 \left (a^2+b^2\right ) d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) d}+\frac {b \cos ^2(c+d x) (a+b \tan (c+d x))^{1+n} \left (b^2 (3-n)+a^2 (1+n)+a b \left (5+\frac {3 a^2}{b^2}-2 n\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\frac {b^5 \operatorname {Subst}\left (\int \frac {(a+x)^n \left (\frac {3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )}{b^6}-\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n x}{b^4}\right )}{1+\frac {x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) d}+\frac {b \cos ^2(c+d x) (a+b \tan (c+d x))^{1+n} \left (b^2 (3-n)+a^2 (1+n)+a b \left (5+\frac {3 a^2}{b^2}-2 n\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\frac {b^5 \operatorname {Subst}\left (\int \left (\frac {\left (\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n}{b^2}+\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )\right )}{b^6}\right ) (a+x)^n}{2 \left (\sqrt {-b^2}-x\right )}+\frac {\left (-\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n}{b^2}+\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )\right )}{b^6}\right ) (a+x)^n}{2 \left (\sqrt {-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) d}+\frac {b \cos ^2(c+d x) (a+b \tan (c+d x))^{1+n} \left (b^2 (3-n)+a^2 (1+n)+a b \left (5+\frac {3 a^2}{b^2}-2 n\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}-\frac {\left (b^5 \left (\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n}{b^2}-\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )\right )}{b^6}\right )\right ) \operatorname {Subst}\left (\int \frac {(a+x)^n}{\sqrt {-b^2}+x} \, dx,x,b \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^2 d}+\frac {\left (b^5 \left (\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n}{b^2}+\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )\right )}{b^6}\right )\right ) \operatorname {Subst}\left (\int \frac {(a+x)^n}{\sqrt {-b^2}-x} \, dx,x,b \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^2 d}\\ &=\frac {b^5 \left (\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n}{b^2}-\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )\right )}{b^6}\right ) \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{16 \left (a^2+b^2\right )^2 \left (a-\sqrt {-b^2}\right ) d (1+n)}+\frac {b^5 \left (\frac {a \left (5+\frac {3 a^2}{b^2}-2 n\right ) n}{b^2}+\frac {\sqrt {-b^2} \left (3 a^4+a^2 b^2 \left (6-2 n-n^2\right )+b^4 \left (3-4 n+n^2\right )\right )}{b^6}\right ) \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{16 \left (a^2+b^2\right )^2 \left (a+\sqrt {-b^2}\right ) d (1+n)}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) d}+\frac {b \cos ^2(c+d x) (a+b \tan (c+d x))^{1+n} \left (b^2 (3-n)+a^2 (1+n)+a b \left (5+\frac {3 a^2}{b^2}-2 n\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.58, size = 360, normalized size = 0.83 \[ \frac {(a+b \tan (c+d x))^{n+1} \left (-\frac {2 b \cos ^2(c+d x) \left (-a \left (3 a^2+b^2 (5-2 n)\right ) \tan (c+d x)-a^2 b (n+1)+b^3 (n-3)\right )}{a^2+b^2}+\frac {\frac {\left (a b^2 n \left (3 a^2+b^2 (5-2 n)\right )+\sqrt {-b^2} \left (-3 a^4+a^2 b^2 \left (n^2+2 n-6\right )-b^4 \left (n^2-4 n+3\right )\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-\sqrt {-b^2}}\right )}{a-\sqrt {-b^2}}+\frac {\left (a b^2 n \left (3 a^2+b^2 (5-2 n)\right )+\sqrt {-b^2} \left (3 a^4-a^2 b^2 \left (n^2+2 n-6\right )+b^4 \left (n^2-4 n+3\right )\right )\right ) \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+\sqrt {-b^2}}\right )}{a+\sqrt {-b^2}}}{(n+1) \left (a^2+b^2\right )}+4 b \cos ^4(c+d x) (a \tan (c+d x)+b)\right )}{16 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x])^n,x]
[Out]
((a + b*Tan[c + d*x])^(1 + n)*((((a*b^2*(3*a^2 + b^2*(5 - 2*n))*n + Sqrt[-b^2]*(-3*a^4 - b^4*(3 - 4*n + n^2) +
a^2*b^2*(-6 + 2*n + n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a - Sqrt[-b^2])])/(a - Sq
rt[-b^2]) + ((a*b^2*(3*a^2 + b^2*(5 - 2*n))*n + Sqrt[-b^2]*(3*a^4 + b^4*(3 - 4*n + n^2) - a^2*b^2*(-6 + 2*n +
n^2)))*Hypergeometric2F1[1, 1 + n, 2 + n, (a + b*Tan[c + d*x])/(a + Sqrt[-b^2])])/(a + Sqrt[-b^2]))/((a^2 + b^
2)*(1 + n)) + 4*b*Cos[c + d*x]^4*(b + a*Tan[c + d*x]) - (2*b*Cos[c + d*x]^2*(b^3*(-3 + n) - a^2*b*(1 + n) - a*
(3*a^2 + b^2*(5 - 2*n))*Tan[c + d*x]))/(a^2 + b^2)))/(16*b*(a^2 + b^2)*d)
________________________________________________________________________________________
fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*(a+b*tan(d*x+c))^n,x, algorithm="fricas")
[Out]
integral((b*tan(d*x + c) + a)^n*cos(d*x + c)^4, x)
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*(a+b*tan(d*x+c))^n,x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.45Unable to divide, perhaps due to rounding error%%%{1048576,[2,0,10,17,20,20]%%%}+%%%{-1048576
0,[2,0,10,17,20,18]%%%}+%%%{47185920,[2,0,10,17,20,16]%%%}+%%%{-125829120,[2,0,10,17,20,14]%%%}+%%%{220200960,
[2,0,10,17,20,12]%%%}+%%%{-264241152,[2,0,10,17,20,10]%%%}+%%%{220200960,[2,0,10,17,20,8]%%%}+%%%{-125829120,[
2,0,10,17,20,6]%%%}+%%%{47185920,[2,0,10,17,20,4]%%%}+%%%{-10485760,[2,0,10,17,20,2]%%%}+%%%{1048576,[2,0,10,1
7,20,0]%%%}+%%%{3145728*i,[2,0,10,16,21,20]%%%}+%%%{-31457280*i,[2,0,10,16,21,18]%%%}+%%%{141557760*i,[2,0,10,
16,21,16]%%%}+%%%{-377487360*i,[2,0,10,16,21,14]%%%}+%%%{660602880*i,[2,0,10,16,21,12]%%%}+%%%{-792723456*i,[2
,0,10,16,21,10]%%%}+%%%{660602880*i,[2,0,10,16,21,8]%%%}+%%%{-377487360*i,[2,0,10,16,21,6]%%%}+%%%{141557760*i
,[2,0,10,16,21,4]%%%}+%%%{-31457280*i,[2,0,10,16,21,2]%%%}+%%%{3145728*i,[2,0,10,16,21,0]%%%}+%%%{4194304,[2,0
,10,15,22,20]%%%}+%%%{-41943040,[2,0,10,15,22,18]%%%}+%%%{188743680,[2,0,10,15,22,16]%%%}+%%%{-503316480,[2,0,
10,15,22,14]%%%}+%%%{880803840,[2,0,10,15,22,12]%%%}+%%%{-1056964608,[2,0,10,15,22,10]%%%}+%%%{880803840,[2,0,
10,15,22,8]%%%}+%%%{-503316480,[2,0,10,15,22,6]%%%}+%%%{188743680,[2,0,10,15,22,4]%%%}+%%%{-41943040,[2,0,10,1
5,22,2]%%%}+%%%{4194304,[2,0,10,15,22,0]%%%}+%%%{20971520*i,[2,0,10,14,23,20]%%%}+%%%{-209715200*i,[2,0,10,14,
23,18]%%%}+%%%{943718400*i,[2,0,10,14,23,16]%%%}+%%%{-2516582400*i,[2,0,10,14,23,14]%%%}+%%%{4404019200*i,[2,0
,10,14,23,12]%%%}+%%%{-5284823040*i,[2,0,10,14,23,10]%%%}+%%%{4404019200*i,[2,0,10,14,23,8]%%%}+%%%{-251658240
0*i,[2,0,10,14,23,6]%%%}+%%%{943718400*i,[2,0,10,14,23,4]%%%}+%%%{-209715200*i,[2,0,10,14,23,2]%%%}+%%%{209715
20*i,[2,0,10,14,23,0]%%%}+%%%{58720256*i,[2,0,10,12,25,20]%%%}+%%%{-587202560*i,[2,0,10,12,25,18]%%%}+%%%{2642
411520*i,[2,0,10,12,25,16]%%%}+%%%{-7046430720*i,[2,0,10,12,25,14]%%%}+%%%{12331253760*i,[2,0,10,12,25,12]%%%}
+%%%{-14797504512*i,[2,0,10,12,25,10]%%%}+%%%{12331253760*i,[2,0,10,12,25,8]%%%}+%%%{-7046430720*i,[2,0,10,12,
25,6]%%%}+%%%{2642411520*i,[2,0,10,12,25,4]%%%}+%%%{-587202560*i,[2,0,10,12,25,2]%%%}+%%%{58720256*i,[2,0,10,1
2,25,0]%%%}+%%%{-29360128,[2,0,10,11,26,20]%%%}+%%%{293601280,[2,0,10,11,26,18]%%%}+%%%{-1321205760,[2,0,10,11
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11,26,10]%%%}+%%%{-6165626880,[2,0,10,11,26,8]%%%}+%%%{3523215360,[2,0,10,11,26,6]%%%}+%%%{-1321205760,[2,0,10
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27,20]%%%}+%%%{-880803840*i,[2,0,10,10,27,18]%%%}+%%%{3963617280*i,[2,0,10,10,27,16]%%%}+%%%{-10569646080*i,[2
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880640*i,[2,0,10,10,27,8]%%%}+%%%{-10569646080*i,[2,0,10,10,27,6]%%%}+%%%{3963617280*i,[2,0,10,10,27,4]%%%}+%%
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{734003200,[2,0,10,9,28,18]%%%}+%%%{-3303014400,[2,0,10,9,28,16]%%%}+%%%{8808038400,[2,0,10,9,28,14]%%%}+%%%{-
15414067200,[2,0,10,9,28,12]%%%}+%%%{18496880640,[2,0,10,9,28,10]%%%}+%%%{-15414067200,[2,0,10,9,28,8]%%%}+%%%
{8808038400,[2,0,10,9,28,6]%%%}+%%%{-3303014400,[2,0,10,9,28,4]%%%}+%%%{734003200,[2,0,10,9,28,2]%%%}+%%%{-734
00320,[2,0,10,9,28,0]%%%}+%%%{73400320*i,[2,0,10,8,29,20]%%%}+%%%{-734003200*i,[2,0,10,8,29,18]%%%}+%%%{330301
4400*i,[2,0,10,8,29,16]%%%}+%%%{-8808038400*i,[2,0,10,8,29,14]%%%}+%%%{15414067200*i,[2,0,10,8,29,12]%%%}+%%%{
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}+%%%{3303014400*i,[2,0,10,8,29,4]%%%}+%%%{-734003200*i,[2,0,10,8,29,2]%%%}+%%%{73400320*i,[2,0,10,8,29,0]%%%}
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%{10569646080,[2,0,10,7,30,14]%%%}+%%%{-18496880640,[2,0,10,7,30,12]%%%}+%%%{22196256768,[2,0,10,7,30,10]%%%}+
%%%{-18496880640,[2,0,10,7,30,8]%%%}+%%%{10569646080,[2,0,10,7,30,6]%%%}+%%%{-3963617280,[2,0,10,7,30,4]%%%}+%
%%{880803840,[2,0,10,7,30,2]%%%}+%%%{-88080384,[2,0,10,7,30,0]%%%}+%%%{29360128*i,[2,0,10,6,31,20]%%%}+%%%{-29
3601280*i,[2,0,10,6,31,18]%%%}+%%%{1321205760*i,[2,0,10,6,31,16]%%%}+%%%{-3523215360*i,[2,0,10,6,31,14]%%%}+%%
%{6165626880*i,[2,0,10,6,31,12]%%%}+%%%{-7398752256*i,[2,0,10,6,31,10]%%%}+%%%{6165626880*i,[2,0,10,6,31,8]%%%
}+%%%{-3523215360*i,[2,0,10,6,31,6]%%%}+%%%{1321205760*i,[2,0,10,6,31,4]%%%}+%%%{-293601280*i,[2,0,10,6,31,2]%
%%}+%%%{29360128*i,[2,0,10,6,31,0]%%%}+%%%{-58720256,[2,0,10,5,32,20]%%%}+%%%{587202560,[2,0,10,5,32,18]%%%}+%
%%{-2642411520,[2,0,10,5,32,16]%%%}+%%%{7046430720,[2,0,10,5,32,14]%%%}+%%%{-12331253760,[2,0,10,5,32,12]%%%}+
%%%{14797504512,[2,0,10,5,32,10]%%%}+%%%{-12331253760,[2,0,10,5,32,8]%%%}+%%%{7046430720,[2,0,10,5,32,6]%%%}+%
%%{-2642411520,[2,0,10,5,32,4]%%%}+%%%{587202560,[2,0,10,5,32,2]%%%}+%%%{-58720256,[2,0,10,5,32,0]%%%}+%%%{-20
971520,[2,0,10,3,34,20]%%%}+%%%{209715200,[2,0,10,3,34,18]%%%}+%%%{-943718400,[2,0,10,3,34,16]%%%}+%%%{2516582
400,[2,0,10,3,34,14]%%%}+%%%{-4404019200,[2,0,10,3,34,12]%%%}+%%%{5284823040,[2,0,10,3,34,10]%%%}+%%%{-4404019
200,[2,0,10,3,34,8]%%%}+%%%{2516582400,[2,0,10,3,34,6]%%%}+%%%{-943718400,[2,0,10,3,34,4]%%%}+%%%{209715200,[2
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10,2,35,0]%%%}+%%%{-3145728,[2,0,10,1,36,20]%%%}+%%%{31457280,[2,0,10,1,36,18]%%%}+%%%{-141557760,[2,0,10,1,36
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%%%}+%%%{-660602880,[2,0,10,1,36,8]%%%}+%%%{377487360,[2,0,10,1,36,6]%%%}+%%%{-141557760,[2,0,10,1,36,4]%%%}+%
%%{31457280,[2,0,10,1,36,2]%%%}+%%%{-3145728,[2,0,10,1,36,0]%%%}+%%%{-1048576*i,[2,0,10,0,37,20]%%%}+%%%{10485
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00960*i,[2,0,10,0,37,12]%%%}+%%%{264241152*i,[2,0,10,0,37,10]%%%}+%%%{-220200960*i,[2,0,10,0,37,8]%%%}+%%%{125
829120*i,[2,0,10,0,37,6]%%%}+%%%{-47185920*i,[2,0,10,0,37,4]%%%}+%%%{10485760*i,[2,0,10,0,37,2]%%%}+%%%{-10485
76*i,[2,0,10,0,37,0]%%%}+%%%{2097152,[1,0,10,17,20,20]%%%}+%%%{-20971520,[1,0,10,17,20,18]%%%}+%%%{94371840,[1
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16,21,14]%%%}+%%%{880803840*i,[1,0,10,16,21,12]%%%}+%%%{-1056964608*i,[1,0,10,16,21,10]%%%}+%%%{880803840*i,[1
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1,0,10,15,22,6]%%%}+%%%{754974720,[1,0,10,15,22,4]%%%}+%%%{-167772160,[1,0,10,15,22,2]%%%}+%%%{16777216,[1,0,1
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46080*i,[1,0,10,14,23,10]%%%}+%%%{8808038400*i,[1,0,10,14,23,8]%%%}+%%%{-5033164800*i,[1,0,10,14,23,6]%%%}+%%%
{1887436800*i,[1,0,10,14,23,4]%%%}+%%%{-419430400*i,[1,0,10,14,23,2]%%%}+%%%{41943040*i,[1,0,10,14,23,0]%%%}+%
%%{41943040,[1,0,10,13,24,20]%%%}+%%%{-419430400,[1,0,10,13,24,18]%%%}+%%%{1887436800,[1,0,10,13,24,16]%%%}+%%
%{-5033164800,[1,0,10,13,24,14]%%%}+%%%{8808038400,[1,0,10,13,24,12]%%%}+%%%{-10569646080,[1,0,10,13,24,10]%%%
}+%%%{8808038400,[1,0,10,13,24,8]%%%}+%%%{-5033164800,[1,0,10,13,24,6]%%%}+%%%{1887436800,[1,0,10,13,24,4]%%%}
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%%{-1593835520*i,[1,0,10,12,25,18]%%%}+%%%{7172259840*i,[1,0,10,12,25,16]%%%}+%%%{-19126026240*i,[1,0,10,12,25
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20*i,[1,0,10,12,25,2]%%%}+%%%{159383552*i,[1,0,10,12,25,0]%%%}+%%%{16777216,[1,0,10,11,26,20]%%%}+%%%{-1677721
60,[1,0,10,11,26,18]%%%}+%%%{754974720,[1,0,10,11,26,16]%%%}+%%%{-2013265920,[1,0,10,11,26,14]%%%}+%%%{3523215
360,[1,0,10,11,26,12]%%%}+%%%{-4227858432,[1,0,10,11,26,10]%%%}+%%%{3523215360,[1,0,10,11,26,8]%%%}+%%%{-20132
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45801799680,[1,0,10,7,30,12]%%%}+%%%{54962159616,[1,0,10,7,30,10]%%%}+%%%{-45801799680,[1,0,10,7,30,8]%%%}+%%%
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1,6]%%%}+%%%{8682209280*i,[1,0,10,6,31,4]%%%}+%%%{-1929379840*i,[1,0,10,6,31,2]%%%}+%%%{192937984*i,[1,0,10,6,
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4,0]%%%}+%%%{-8388608*i,[1,0,10,2,35,20]%%%}+%%%{83886080*i,[1,0,10,2,35,18]%%%}+%%%{-377487360*i,[1,0,10,2,35
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%{1048576,[0,0,10,17,20,20]%%%}+%%%{-10485760,[0,0,10,17,20,18]%%%}+%%%{47185920,[0,0,10,17,20,16]%%%}+%%%{-12
5829120,[0,0,10,17,20,14]%%%}+%%%{220200960,[0,0,10,17,20,12]%%%}+%%%{-264241152,[0,0,10,17,20,10]%%%}+%%%{220
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2,0]%%%}+%%%{8388608*i,[0,0,10,14,23,20]%%%}+%%%{25165824*i,[0,0,10,14,23,19]%%%}+%%%{-109051904*i,[0,0,10,14,
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170893824*i,[0,0,10,14,23,9]%%%}+%%%{4932501504*i,[0,0,10,14,23,8]%%%}+%%%{2113929216*i,[0,0,10,14,23,7]%%%}+%
%%{-3120562176*i,[0,0,10,14,23,6]%%%}+%%%{-905969664*i,[0,0,10,14,23,5]%%%}+%%%{1283457024*i,[0,0,10,14,23,4]%
%%}+%%%{226492416*i,[0,0,10,14,23,3]%%%}+%%%{-310378496*i,[0,0,10,14,23,2]%%%}+%%%{-25165824*i,[0,0,10,14,23,1
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%}+%%%{905969664,[0,0,10,13,24,15]%%%}+%%%{-4429185024,[0,0,10,13,24,14]%%%}+%%%{-2113929216,[0,0,10,13,24,13]
%%%}+%%%{8279556096,[0,0,10,13,24,12]%%%}+%%%{3170893824,[0,0,10,13,24,11]%%%}+%%%{-10569646080,[0,0,10,13,24,
10]%%%}+%%%{-3170893824,[0,0,10,13,24,9]%%%}+%%%{9336520704,[0,0,10,13,24,8]%%%}+%%%{2113929216,[0,0,10,13,24,
7]%%%}+%%%{-5637144576,[0,0,10,13,24,6]%%%}+%%%{-905969664,[0,0,10,13,24,5]%%%}+%%%{2227175424,[0,0,10,13,24,4
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%%{11978932224*i,[0,0,10,12,25,7]%%%}+%%%{-15502147584*i,[0,0,10,12,25,6]%%%}+%%%{-5133828096*i,[0,0,10,12,25,
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2*i,[0,0,10,10,27,14]%%%}+%%%{-27481079808*i,[0,0,10,10,27,13]%%%}+%%%{39812333568*i,[0,0,10,10,27,12]%%%}+%%%
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165824,[0,0,10,1,36,19]%%%}+%%%{14680064,[0,0,10,1,36,18]%%%}+%%%{226492416,[0,0,10,1,36,17]%%%}+%%%{-17930649
6,[0,0,10,1,36,16]%%%}+%%%{-905969664,[0,0,10,1,36,15]%%%}+%%%{780140544,[0,0,10,1,36,14]%%%}+%%%{2113929216,[
0,0,10,1,36,13]%%%}+%%%{-1893728256,[0,0,10,1,36,12]%%%}+%%%{-3170893824,[0,0,10,1,36,11]%%%}+%%%{2906652672,[
0,0,10,1,36,10]%%%}+%%%{3170893824,[0,0,10,1,36,9]%%%}+%%%{-2950692864,[0,0,10,1,36,8]%%%}+%%%{-2113929216,[0,
0,10,1,36,7]%%%}+%%%{1988100096,[0,0,10,1,36,6]%%%}+%%%{905969664,[0,0,10,1,36,5]%%%}+%%%{-858783744,[0,0,10,1
,36,4]%%%}+%%%{-226492416,[0,0,10,1,36,3]%%%}+%%%{216006656,[0,0,10,1,36,2]%%%}+%%%{25165824,[0,0,10,1,36,1]%%
%}+%%%{-24117248,[0,0,10,1,36,0]%%%}+%%%{1048576*i,[0,0,10,0,37,20]%%%}+%%%{-8388608*i,[0,0,10,0,37,19]%%%}+%%
%{-2097152*i,[0,0,10,0,37,18]%%%}+%%%{75497472*i,[0,0,10,0,37,17]%%%}+%%%{-28311552*i,[0,0,10,0,37,16]%%%}+%%%
{-301989888*i,[0,0,10,0,37,15]%%%}+%%%{176160768*i,[0,0,10,0,37,14]%%%}+%%%{704643072*i,[0,0,10,0,37,13]%%%}+%
%%{-484442112*i,[0,0,10,0,37,12]%%%}+%%%{-1056964608*i,[0,0,10,0,37,11]%%%}+%%%{792723456*i,[0,0,10,0,37,10]%%
%}+%%%{1056964608*i,[0,0,10,0,37,9]%%%}+%%%{-836763648*i,[0,0,10,0,37,8]%%%}+%%%{-704643072*i,[0,0,10,0,37,7]%
%%}+%%%{578813952*i,[0,0,10,0,37,6]%%%}+%%%{301989888*i,[0,0,10,0,37,5]%%%}+%%%{-254803968*i,[0,0,10,0,37,4]%%
%}+%%%{-75497472*i,[0,0,10,0,37,3]%%%}+%%%{65011712*i,[0,0,10,0,37,2]%%%}+%%%{8388608*i,[0,0,10,0,37,1]%%%}+%%
%{-7340032*i,[0,0,10,0,37,0]%%%} / %%%{1024,[0,0,5,9,10,10]%%%}+%%%{-5120,[0,0,5,9,10,8]%%%}+%%%{10240,[0,0,5,
9,10,6]%%%}+%%%{-10240,[0,0,5,9,10,4]%%%}+%%%{5120,[0,0,5,9,10,2]%%%}+%%%{-1024,[0,0,5,9,10,0]%%%}+%%%{1024*i,
[0,0,5,8,11,10]%%%}+%%%{-5120*i,[0,0,5,8,11,8]%%%}+%%%{10240*i,[0,0,5,8,11,6]%%%}+%%%{-10240*i,[0,0,5,8,11,4]%
%%}+%%%{5120*i,[0,0,5,8,11,2]%%%}+%%%{-1024*i,[0,0,5,8,11,0]%%%}+%%%{4096,[0,0,5,7,12,10]%%%}+%%%{-20480,[0,0,
5,7,12,8]%%%}+%%%{40960,[0,0,5,7,12,6]%%%}+%%%{-40960,[0,0,5,7,12,4]%%%}+%%%{20480,[0,0,5,7,12,2]%%%}+%%%{-409
6,[0,0,5,7,12,0]%%%}+%%%{4096*i,[0,0,5,6,13,10]%%%}+%%%{-20480*i,[0,0,5,6,13,8]%%%}+%%%{40960*i,[0,0,5,6,13,6]
%%%}+%%%{-40960*i,[0,0,5,6,13,4]%%%}+%%%{20480*i,[0,0,5,6,13,2]%%%}+%%%{-4096*i,[0,0,5,6,13,0]%%%}+%%%{6144,[0
,0,5,5,14,10]%%%}+%%%{-30720,[0,0,5,5,14,8]%%%}+%%%{61440,[0,0,5,5,14,6]%%%}+%%%{-61440,[0,0,5,5,14,4]%%%}+%%%
{30720,[0,0,5,5,14,2]%%%}+%%%{-6144,[0,0,5,5,14,0]%%%}+%%%{6144*i,[0,0,5,4,15,10]%%%}+%%%{-30720*i,[0,0,5,4,15
,8]%%%}+%%%{61440*i,[0,0,5,4,15,6]%%%}+%%%{-61440*i,[0,0,5,4,15,4]%%%}+%%%{30720*i,[0,0,5,4,15,2]%%%}+%%%{-614
4*i,[0,0,5,4,15,0]%%%}+%%%{4096,[0,0,5,3,16,10]%%%}+%%%{-20480,[0,0,5,3,16,8]%%%}+%%%{40960,[0,0,5,3,16,6]%%%}
+%%%{-40960,[0,0,5,3,16,4]%%%}+%%%{20480,[0,0,5,3,16,2]%%%}+%%%{-4096,[0,0,5,3,16,0]%%%}+%%%{4096*i,[0,0,5,2,1
7,10]%%%}+%%%{-20480*i,[0,0,5,2,17,8]%%%}+%%%{40960*i,[0,0,5,2,17,6]%%%}+%%%{-40960*i,[0,0,5,2,17,4]%%%}+%%%{2
0480*i,[0,0,5,2,17,2]%%%}+%%%{-4096*i,[0,0,5,2,17,0]%%%}+%%%{1024,[0,0,5,1,18,10]%%%}+%%%{-5120,[0,0,5,1,18,8]
%%%}+%%%{10240,[0,0,5,1,18,6]%%%}+%%%{-10240,[0,0,5,1,18,4]%%%}+%%%{5120,[0,0,5,1,18,2]%%%}+%%%{-1024,[0,0,5,1
,18,0]%%%}+%%%{1024*i,[0,0,5,0,19,10]%%%}+%%%{-5120*i,[0,0,5,0,19,8]%%%}+%%%{10240*i,[0,0,5,0,19,6]%%%}+%%%{-1
0240*i,[0,0,5,0,19,4]%%%}+%%%{5120*i,[0,0,5,0,19,2]%%%}+%%%{-1024*i,[0,0,5,0,19,0]%%%} Error: Bad Argument Val
ue
________________________________________________________________________________________
maple [F] time = 1.28, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{4}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^4*(a+b*tan(d*x+c))^n,x)
[Out]
int(cos(d*x+c)^4*(a+b*tan(d*x+c))^n,x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*(a+b*tan(d*x+c))^n,x, algorithm="maxima")
[Out]
integrate((b*tan(d*x + c) + a)^n*cos(d*x + c)^4, x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^4*(a + b*tan(c + d*x))^n,x)
[Out]
int(cos(c + d*x)^4*(a + b*tan(c + d*x))^n, x)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**4*(a+b*tan(d*x+c))**n,x)
[Out]
Timed out
________________________________________________________________________________________