3.108 \(\int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\)

Optimal. Leaf size=242 \[ \frac{b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac{10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac{b \left (20 a^2 b^2+5 a^4+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac{a \left (20 a^2 b^2+a^4+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac{5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac{2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^9(c+d x)}{9 d}+\frac{b^5 \tan ^{10}(c+d x)}{10 d} \]

[Out]

(a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (2*a^3*(a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*
(a^2 + b^2)*Tan[c + d*x]^4)/(2*d) + (a*(a^4 + 20*a^2*b^2 + 5*b^4)*Tan[c + d*x]^5)/(5*d) + (b*(5*a^4 + 20*a^2*b
^2 + b^4)*Tan[c + d*x]^6)/(6*d) + (10*a*b^2*(a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (b^3*(5*a^2 + b^2)*Tan[c + d*x
]^8)/(4*d) + (5*a*b^4*Tan[c + d*x]^9)/(9*d) + (b^5*Tan[c + d*x]^10)/(10*d)

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Rubi [A]  time = 0.221595, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 948} \[ \frac{b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac{10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac{b \left (20 a^2 b^2+5 a^4+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac{a \left (20 a^2 b^2+a^4+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac{5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac{2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^9(c+d x)}{9 d}+\frac{b^5 \tan ^{10}(c+d x)}{10 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(a^5*Tan[c + d*x])/d + (5*a^4*b*Tan[c + d*x]^2)/(2*d) + (2*a^3*(a^2 + 5*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*
(a^2 + b^2)*Tan[c + d*x]^4)/(2*d) + (a*(a^4 + 20*a^2*b^2 + 5*b^4)*Tan[c + d*x]^5)/(5*d) + (b*(5*a^4 + 20*a^2*b
^2 + b^4)*Tan[c + d*x]^6)/(6*d) + (10*a*b^2*(a^2 + b^2)*Tan[c + d*x]^7)/(7*d) + (b^3*(5*a^2 + b^2)*Tan[c + d*x
]^8)/(4*d) + (5*a*b^4*Tan[c + d*x]^9)/(9*d) + (b^5*Tan[c + d*x]^10)/(10*d)

Rule 3088

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb
ol] :> -Dist[d^(-1), Subst[Int[(x^m*(b + a*x)^n)/(1 + x^2)^((m + n + 2)/2), x], x, Cot[c + d*x]], x] /; FreeQ[
{a, b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] &&  !(GtQ[n, 0] && GtQ[m, 1])

Rule 948

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && (IGtQ[m, 0] || (EqQ[m, -2] && EqQ[p, 1] && EqQ[d, 0]))

Rubi steps

\begin{align*} \int \sec ^{11}(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^5 \left (1+x^2\right )^2}{x^{11}} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{b^5}{x^{11}}+\frac{5 a b^4}{x^{10}}+\frac{2 \left (5 a^2 b^3+b^5\right )}{x^9}+\frac{10 a b^2 \left (a^2+b^2\right )}{x^8}+\frac{5 a^4 b+20 a^2 b^3+b^5}{x^7}+\frac{a^5+20 a^3 b^2+5 a b^4}{x^6}+\frac{10 a^2 b \left (a^2+b^2\right )}{x^5}+\frac{2 \left (a^5+5 a^3 b^2\right )}{x^4}+\frac{5 a^4 b}{x^3}+\frac{a^5}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a^5 \tan (c+d x)}{d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{2 a^3 \left (a^2+5 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{5 a^2 b \left (a^2+b^2\right ) \tan ^4(c+d x)}{2 d}+\frac{a \left (a^4+20 a^2 b^2+5 b^4\right ) \tan ^5(c+d x)}{5 d}+\frac{b \left (5 a^4+20 a^2 b^2+b^4\right ) \tan ^6(c+d x)}{6 d}+\frac{10 a b^2 \left (a^2+b^2\right ) \tan ^7(c+d x)}{7 d}+\frac{b^3 \left (5 a^2+b^2\right ) \tan ^8(c+d x)}{4 d}+\frac{5 a b^4 \tan ^9(c+d x)}{9 d}+\frac{b^5 \tan ^{10}(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 1.20733, size = 115, normalized size = 0.48 \[ \frac{\frac{1}{4} \left (3 a^2+b^2\right ) (a+b \tan (c+d x))^8-\frac{4}{7} a \left (a^2+b^2\right ) (a+b \tan (c+d x))^7+\frac{1}{6} \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^6+\frac{1}{10} (a+b \tan (c+d x))^{10}-\frac{4}{9} a (a+b \tan (c+d x))^9}{b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^11*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(((a^2 + b^2)^2*(a + b*Tan[c + d*x])^6)/6 - (4*a*(a^2 + b^2)*(a + b*Tan[c + d*x])^7)/7 + ((3*a^2 + b^2)*(a + b
*Tan[c + d*x])^8)/4 - (4*a*(a + b*Tan[c + d*x])^9)/9 + (a + b*Tan[c + d*x])^10/10)/(b^5*d)

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Maple [A]  time = 0.256, size = 299, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( -{a}^{5} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) +{\frac{5\,{a}^{4}b}{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+10\,{a}^{3}{b}^{2} \left ( 1/7\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +10\,{a}^{2}{b}^{3} \left ( 1/8\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+1/12\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+1/24\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) +5\,a{b}^{4} \left ( 1/9\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{63\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{315\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) +{b}^{5} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{10\, \left ( \cos \left ( dx+c \right ) \right ) ^{10}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{20\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{60\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)

[Out]

1/d*(-a^5*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+5/6*a^4*b/cos(d*x+c)^6+10*a^3*b^2*(1/7*sin(d*x
+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3/cos(d*x+c)^3)+10*a^2*b^3*(1/8*sin(d*x+c)^
4/cos(d*x+c)^8+1/12*sin(d*x+c)^4/cos(d*x+c)^6+1/24*sin(d*x+c)^4/cos(d*x+c)^4)+5*a*b^4*(1/9*sin(d*x+c)^5/cos(d*
x+c)^9+4/63*sin(d*x+c)^5/cos(d*x+c)^7+8/315*sin(d*x+c)^5/cos(d*x+c)^5)+b^5*(1/10*sin(d*x+c)^6/cos(d*x+c)^10+1/
20*sin(d*x+c)^6/cos(d*x+c)^8+1/60*sin(d*x+c)^6/cos(d*x+c)^6))

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Maxima [A]  time = 1.24709, size = 371, normalized size = 1.53 \begin{align*} \frac{84 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{5} + 120 \,{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{3} b^{2} + 20 \,{\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a b^{4} + \frac{525 \,{\left (4 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1} - \frac{21 \,{\left (10 \, \sin \left (d x + c\right )^{4} - 5 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\sin \left (d x + c\right )^{10} - 5 \, \sin \left (d x + c\right )^{8} + 10 \, \sin \left (d x + c\right )^{6} - 10 \, \sin \left (d x + c\right )^{4} + 5 \, \sin \left (d x + c\right )^{2} - 1} - \frac{1050 \, a^{4} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{1260 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

1/1260*(84*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*a^5 + 120*(15*tan(d*x + c)^7 + 42*tan(d*x
+ c)^5 + 35*tan(d*x + c)^3)*a^3*b^2 + 20*(35*tan(d*x + c)^9 + 90*tan(d*x + c)^7 + 63*tan(d*x + c)^5)*a*b^4 + 5
25*(4*sin(d*x + c)^2 - 1)*a^2*b^3/(sin(d*x + c)^8 - 4*sin(d*x + c)^6 + 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1
) - 21*(10*sin(d*x + c)^4 - 5*sin(d*x + c)^2 + 1)*b^5/(sin(d*x + c)^10 - 5*sin(d*x + c)^8 + 10*sin(d*x + c)^6
- 10*sin(d*x + c)^4 + 5*sin(d*x + c)^2 - 1) - 1050*a^4*b/(sin(d*x + c)^2 - 1)^3)/d

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Fricas [A]  time = 0.598797, size = 491, normalized size = 2.03 \begin{align*} \frac{126 \, b^{5} + 210 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 315 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (8 \,{\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{9} + 4 \,{\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 175 \, a b^{4} \cos \left (d x + c\right ) + 3 \,{\left (21 \, a^{5} - 30 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 50 \,{\left (9 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{1260 \, d \cos \left (d x + c\right )^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/1260*(126*b^5 + 210*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 315*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 4*(
8*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^9 + 4*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^7 + 175*a*b^
4*cos(d*x + c) + 3*(21*a^5 - 30*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^5 + 50*(9*a^3*b^2 - 5*a*b^4)*cos(d*x + c)^3)*s
in(d*x + c))/(d*cos(d*x + c)^10)

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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(sec(d*x+c)**11*(a*cos(d*x+c)+b*sin(d*x+c))**5,x)

[Out]

Timed out

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Giac [A]  time = 1.32152, size = 354, normalized size = 1.46 \begin{align*} \frac{126 \, b^{5} \tan \left (d x + c\right )^{10} + 700 \, a b^{4} \tan \left (d x + c\right )^{9} + 1575 \, a^{2} b^{3} \tan \left (d x + c\right )^{8} + 315 \, b^{5} \tan \left (d x + c\right )^{8} + 1800 \, a^{3} b^{2} \tan \left (d x + c\right )^{7} + 1800 \, a b^{4} \tan \left (d x + c\right )^{7} + 1050 \, a^{4} b \tan \left (d x + c\right )^{6} + 4200 \, a^{2} b^{3} \tan \left (d x + c\right )^{6} + 210 \, b^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 5040 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 1260 \, a b^{4} \tan \left (d x + c\right )^{5} + 3150 \, a^{4} b \tan \left (d x + c\right )^{4} + 3150 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 840 \, a^{5} \tan \left (d x + c\right )^{3} + 4200 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 3150 \, a^{4} b \tan \left (d x + c\right )^{2} + 1260 \, a^{5} \tan \left (d x + c\right )}{1260 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/1260*(126*b^5*tan(d*x + c)^10 + 700*a*b^4*tan(d*x + c)^9 + 1575*a^2*b^3*tan(d*x + c)^8 + 315*b^5*tan(d*x + c
)^8 + 1800*a^3*b^2*tan(d*x + c)^7 + 1800*a*b^4*tan(d*x + c)^7 + 1050*a^4*b*tan(d*x + c)^6 + 4200*a^2*b^3*tan(d
*x + c)^6 + 210*b^5*tan(d*x + c)^6 + 252*a^5*tan(d*x + c)^5 + 5040*a^3*b^2*tan(d*x + c)^5 + 1260*a*b^4*tan(d*x
+ c)^5 + 3150*a^4*b*tan(d*x + c)^4 + 3150*a^2*b^3*tan(d*x + c)^4 + 840*a^5*tan(d*x + c)^3 + 4200*a^3*b^2*tan(
d*x + c)^3 + 3150*a^4*b*tan(d*x + c)^2 + 1260*a^5*tan(d*x + c))/d