3.106 \(\int \sec ^9(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\)

Optimal. Leaf size=177 \[ \frac{b^3 \left (10 a^2+b^2\right ) \tan ^6(c+d x)}{6 d}+\frac{a b^2 \left (2 a^2+b^2\right ) \tan ^5(c+d x)}{d}+\frac{5 a^2 b \left (a^2+2 b^2\right ) \tan ^4(c+d x)}{4 d}+\frac{a^3 \left (a^2+10 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^7(c+d x)}{7 d}+\frac{b^5 \tan ^8(c+d x)}{8 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.151652, antiderivative size = 177, 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, 894} \[ \frac{b^3 \left (10 a^2+b^2\right ) \tan ^6(c+d x)}{6 d}+\frac{a b^2 \left (2 a^2+b^2\right ) \tan ^5(c+d x)}{d}+\frac{5 a^2 b \left (a^2+2 b^2\right ) \tan ^4(c+d x)}{4 d}+\frac{a^3 \left (a^2+10 b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{5 a^4 b \tan ^2(c+d x)}{2 d}+\frac{a^5 \tan (c+d x)}{d}+\frac{5 a b^4 \tan ^7(c+d x)}{7 d}+\frac{b^5 \tan ^8(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^9*(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) + (a^3*(a^2 + 10*b^2)*Tan[c + d*x]^3)/(3*d) + (5*a^2*b*(
a^2 + 2*b^2)*Tan[c + d*x]^4)/(4*d) + (a*b^2*(2*a^2 + b^2)*Tan[c + d*x]^5)/d + (b^3*(10*a^2 + b^2)*Tan[c + d*x]
^6)/(6*d) + (5*a*b^4*Tan[c + d*x]^7)/(7*d) + (b^5*Tan[c + d*x]^8)/(8*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 894

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] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.443175, size = 54, normalized size = 0.31 \[ \frac{(a+b \tan (c+d x))^6 \left (a^2-6 a b \tan (c+d x)+21 b^2 \tan ^2(c+d x)+28 b^2\right )}{168 b^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^9*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

((a + b*Tan[c + d*x])^6*(a^2 + 28*b^2 - 6*a*b*Tan[c + d*x] + 21*b^2*Tan[c + d*x]^2))/(168*b^3*d)

________________________________________________________________________________________

Maple [A]  time = 0.267, size = 217, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ( -{a}^{5} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) +{\frac{5\,{a}^{4}b}{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+10\,{a}^{3}{b}^{2} \left ( 1/5\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+2/15\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +10\,{a}^{2}{b}^{3} \left ( 1/6\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+1/12\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \right ) +5\,a{b}^{4} \left ( 1/7\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) +{b}^{5} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{8}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{24\, \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)^9*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.28562, size = 301, normalized size = 1.7 \begin{align*} \frac{56 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{5} + 112 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a^{3} b^{2} + 24 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} a b^{4} - \frac{140 \,{\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} + \frac{7 \,{\left (6 \, \sin \left (d x + c\right )^{4} - 4 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\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{210 \, a^{4} b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{168 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

1/168*(56*(tan(d*x + c)^3 + 3*tan(d*x + c))*a^5 + 112*(3*tan(d*x + c)^5 + 5*tan(d*x + c)^3)*a^3*b^2 + 24*(5*ta
n(d*x + c)^7 + 7*tan(d*x + c)^5)*a*b^4 - 140*(3*sin(d*x + c)^2 - 1)*a^2*b^3/(sin(d*x + c)^6 - 3*sin(d*x + c)^4
+ 3*sin(d*x + c)^2 - 1) + 7*(6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1)*b^5/(sin(d*x + c)^8 - 4*sin(d*x + c)^6
+ 6*sin(d*x + c)^4 - 4*sin(d*x + c)^2 + 1) + 210*a^4*b/(sin(d*x + c)^2 - 1)^2)/d

________________________________________________________________________________________

Fricas [A]  time = 0.546022, size = 408, normalized size = 2.31 \begin{align*} \frac{21 \, b^{5} + 42 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 56 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (2 \,{\left (7 \, a^{5} - 14 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 15 \, a b^{4} \cos \left (d x + c\right ) +{\left (7 \, a^{5} - 14 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 6 \,{\left (7 \, a^{3} b^{2} - 4 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{168 \, d \cos \left (d x + c\right )^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/168*(21*b^5 + 42*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 56*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 8*(2*(7
*a^5 - 14*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^7 + 15*a*b^4*cos(d*x + c) + (7*a^5 - 14*a^3*b^2 + 3*a*b^4)*cos(d*x +
c)^5 + 6*(7*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(d*cos(d*x + c)^8)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.33172, size = 238, normalized size = 1.34 \begin{align*} \frac{21 \, b^{5} \tan \left (d x + c\right )^{8} + 120 \, a b^{4} \tan \left (d x + c\right )^{7} + 280 \, a^{2} b^{3} \tan \left (d x + c\right )^{6} + 28 \, b^{5} \tan \left (d x + c\right )^{6} + 336 \, a^{3} b^{2} \tan \left (d x + c\right )^{5} + 168 \, a b^{4} \tan \left (d x + c\right )^{5} + 210 \, a^{4} b \tan \left (d x + c\right )^{4} + 420 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 56 \, a^{5} \tan \left (d x + c\right )^{3} + 560 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 420 \, a^{4} b \tan \left (d x + c\right )^{2} + 168 \, a^{5} \tan \left (d x + c\right )}{168 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^9*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/168*(21*b^5*tan(d*x + c)^8 + 120*a*b^4*tan(d*x + c)^7 + 280*a^2*b^3*tan(d*x + c)^6 + 28*b^5*tan(d*x + c)^6 +
336*a^3*b^2*tan(d*x + c)^5 + 168*a*b^4*tan(d*x + c)^5 + 210*a^4*b*tan(d*x + c)^4 + 420*a^2*b^3*tan(d*x + c)^4
+ 56*a^5*tan(d*x + c)^3 + 560*a^3*b^2*tan(d*x + c)^3 + 420*a^4*b*tan(d*x + c)^2 + 168*a^5*tan(d*x + c))/d