3.28 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \, dx\)

Optimal. Leaf size=105 \[ \frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 x}{8} \]

[Out]

(13*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]^3*Sin[c
+ d*x])/(4*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.111869, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3791, 2635, 8, 2633} \[ \frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]

[Out]

(13*a^3*x)/8 + (4*a^3*Sin[c + d*x])/d + (13*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (3*a^3*Cos[c + d*x]^3*Sin[c
+ d*x])/(4*d) - (5*a^3*Sin[c + d*x]^3)/(3*d) + (a^3*Sin[c + d*x]^5)/(5*d)

Rule 3791

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*csc[e + f*x])^m*(d*csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0]
&& IGtQ[m, 0] && RationalQ[n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \, dx &=\int \left (a^3 \cos ^2(c+d x)+3 a^3 \cos ^3(c+d x)+3 a^3 \cos ^4(c+d x)+a^3 \cos ^5(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^2(c+d x) \, dx+a^3 \int \cos ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=\frac{a^3 \cos (c+d x) \sin (c+d x)}{2 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac{1}{2} a^3 \int 1 \, dx+\frac{1}{4} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{a^3 x}{2}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{13 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}+\frac{1}{8} \left (9 a^3\right ) \int 1 \, dx\\ &=\frac{13 a^3 x}{8}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{13 a^3 \cos (c+d x) \sin (c+d x)}{8 d}+\frac{3 a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{a^3 \sin ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.156687, size = 63, normalized size = 0.6 \[ \frac{a^3 (1380 \sin (c+d x)+480 \sin (2 (c+d x))+170 \sin (3 (c+d x))+45 \sin (4 (c+d x))+6 \sin (5 (c+d x))+780 d x)}{480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]

[Out]

(a^3*(780*d*x + 1380*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 170*Sin[3*(c + d*x)] + 45*Sin[4*(c + d*x)] + 6*Sin[
5*(c + d*x)]))/(480*d)

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Maple [A]  time = 0.076, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{3}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+3\,{a}^{3} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{a}^{3} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) +{a}^{3} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sec(d*x+c))^3,x)

[Out]

1/d*(1/5*a^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+3*a^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c
)+3/8*d*x+3/8*c)+a^3*(cos(d*x+c)^2+2)*sin(d*x+c)+a^3*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 1.12754, size = 158, normalized size = 1.5 \begin{align*} \frac{32 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{3} - 480 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} + 45 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^3 - 480*(sin(d*x + c)^3 - 3*sin(d*x + c))
*a^3 + 45*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^3 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*a
^3)/d

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Fricas [A]  time = 1.69822, size = 194, normalized size = 1.85 \begin{align*} \frac{195 \, a^{3} d x +{\left (24 \, a^{3} \cos \left (d x + c\right )^{4} + 90 \, a^{3} \cos \left (d x + c\right )^{3} + 152 \, a^{3} \cos \left (d x + c\right )^{2} + 195 \, a^{3} \cos \left (d x + c\right ) + 304 \, a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/120*(195*a^3*d*x + (24*a^3*cos(d*x + c)^4 + 90*a^3*cos(d*x + c)^3 + 152*a^3*cos(d*x + c)^2 + 195*a^3*cos(d*x
+ c) + 304*a^3)*sin(d*x + c))/d

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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(cos(d*x+c)**5*(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.37286, size = 151, normalized size = 1.44 \begin{align*} \frac{195 \,{\left (d x + c\right )} a^{3} + \frac{2 \,{\left (195 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 910 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1664 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1330 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 765 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/120*(195*(d*x + c)*a^3 + 2*(195*a^3*tan(1/2*d*x + 1/2*c)^9 + 910*a^3*tan(1/2*d*x + 1/2*c)^7 + 1664*a^3*tan(1
/2*d*x + 1/2*c)^5 + 1330*a^3*tan(1/2*d*x + 1/2*c)^3 + 765*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 +
1)^5)/d