3.104 \(\int \sec ^7(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\)

Optimal. Leaf size=30 \[ \frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d} \]

[Out]

((b + a*Cot[c + d*x])^6*Tan[c + d*x]^6)/(6*b*d)

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Rubi [A]  time = 0.0477584, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 37} \[ \frac{\tan ^6(c+d x) (a \cot (c+d x)+b)^6}{6 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

((b + a*Cot[c + d*x])^6*Tan[c + d*x]^6)/(6*b*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \sec ^7(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}{x^7} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{(b+a \cot (c+d x))^6 \tan ^6(c+d x)}{6 b d}\\ \end{align*}

Mathematica [B]  time = 0.479837, size = 89, normalized size = 2.97 \[ \frac{\tan (c+d x) \left (20 a^3 b^2 \tan ^2(c+d x)+15 a^2 b^3 \tan ^3(c+d x)+15 a^4 b \tan (c+d x)+6 a^5+6 a b^4 \tan ^4(c+d x)+b^5 \tan ^5(c+d x)\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^7*(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]

[Out]

(Tan[c + d*x]*(6*a^5 + 15*a^4*b*Tan[c + d*x] + 20*a^3*b^2*Tan[c + d*x]^2 + 15*a^2*b^3*Tan[c + d*x]^3 + 6*a*b^4
*Tan[c + d*x]^4 + b^5*Tan[c + d*x]^5))/(6*d)

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Maple [B]  time = 0.257, size = 120, normalized size = 4. \begin{align*}{\frac{1}{d} \left ({a}^{5}\tan \left ( dx+c \right ) +{\frac{5\,{a}^{4}b}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{10\,{a}^{3}{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{5\,{a}^{2}{b}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{a{b}^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{{b}^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^7*(a*cos(d*x+c)+b*sin(d*x+c))^5,x)

[Out]

1/d*(a^5*tan(d*x+c)+5/2*a^4*b/cos(d*x+c)^2+10/3*a^3*b^2*sin(d*x+c)^3/cos(d*x+c)^3+5/2*a^2*b^3*sin(d*x+c)^4/cos
(d*x+c)^4+a*b^4*sin(d*x+c)^5/cos(d*x+c)^5+1/6*b^5*sin(d*x+c)^6/cos(d*x+c)^6)

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Maxima [B]  time = 1.20596, size = 224, normalized size = 7.47 \begin{align*} \frac{6 \, a b^{4} \tan \left (d x + c\right )^{5} + 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 6 \, a^{5} \tan \left (d x + c\right ) + \frac{15 \,{\left (2 \, \sin \left (d x + c\right )^{2} - 1\right )} a^{2} b^{3}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - \frac{{\left (3 \, \sin \left (d x + c\right )^{4} - 3 \, \sin \left (d x + c\right )^{2} + 1\right )} b^{5}}{\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{15 \, a^{4} b}{\sin \left (d x + c\right )^{2} - 1}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")

[Out]

1/6*(6*a*b^4*tan(d*x + c)^5 + 20*a^3*b^2*tan(d*x + c)^3 + 6*a^5*tan(d*x + c) + 15*(2*sin(d*x + c)^2 - 1)*a^2*b
^3/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - (3*sin(d*x + c)^4 - 3*sin(d*x + c)^2 + 1)*b^5/(sin(d*x + c)^6 - 3
*sin(d*x + c)^4 + 3*sin(d*x + c)^2 - 1) - 15*a^4*b/(sin(d*x + c)^2 - 1))/d

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Fricas [B]  time = 0.507109, size = 329, normalized size = 10.97 \begin{align*} \frac{b^{5} + 3 \,{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a b^{4} \cos \left (d x + c\right ) +{\left (3 \, a^{5} - 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 2 \,{\left (5 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")

[Out]

1/6*(b^5 + 3*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^4 + 3*(5*a^2*b^3 - b^5)*cos(d*x + c)^2 + 2*(3*a*b^4*cos
(d*x + c) + (3*a^5 - 10*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^5 + 2*(5*a^3*b^2 - 3*a*b^4)*cos(d*x + c)^3)*sin(d*x +
c))/(d*cos(d*x + c)^6)

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

[Out]

Timed out

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Giac [B]  time = 1.31267, size = 120, normalized size = 4. \begin{align*} \frac{b^{5} \tan \left (d x + c\right )^{6} + 6 \, a b^{4} \tan \left (d x + c\right )^{5} + 15 \, a^{2} b^{3} \tan \left (d x + c\right )^{4} + 20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 15 \, a^{4} b \tan \left (d x + c\right )^{2} + 6 \, a^{5} \tan \left (d x + c\right )}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7*(a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")

[Out]

1/6*(b^5*tan(d*x + c)^6 + 6*a*b^4*tan(d*x + c)^5 + 15*a^2*b^3*tan(d*x + c)^4 + 20*a^3*b^2*tan(d*x + c)^3 + 15*
a^4*b*tan(d*x + c)^2 + 6*a^5*tan(d*x + c))/d