3.1.46 \(\int \cos ^6(c+d x) (B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [46]

Optimal. Leaf size=92 \[ \frac {3 C x}{8}+\frac {B \sin (c+d x)}{d}+\frac {3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4132, 2713, 12, 2715, 8} \begin {gather*} \frac {B \sin ^5(c+d x)}{5 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 C x}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(3*C*x)/8 + (B*Sin[c + d*x])/d + (3*C*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (C*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)
- (2*B*Sin[c + d*x]^3)/(3*d) + (B*Sin[c + d*x]^5)/(5*d)

Rule 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2713

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]

Rule 2715

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] && Integ
erQ[2*n]

Rule 4132

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rubi steps

\begin {align*} \int \cos ^6(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \cos ^5(c+d x) \, dx+\int C \cos ^4(c+d x) \, dx\\ &=C \int \cos ^4(c+d x) \, dx-\frac {B \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {B \sin (c+d x)}{d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}+\frac {1}{4} (3 C) \int \cos ^2(c+d x) \, dx\\ &=\frac {B \sin (c+d x)}{d}+\frac {3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}+\frac {1}{8} (3 C) \int 1 \, dx\\ &=\frac {3 C x}{8}+\frac {B \sin (c+d x)}{d}+\frac {3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 92, normalized size = 1.00 \begin {gather*} \frac {3 C (c+d x)}{8 d}+\frac {5 B \sin (c+d x)}{8 d}+\frac {C \sin (2 (c+d x))}{4 d}+\frac {5 B \sin (3 (c+d x))}{48 d}+\frac {C \sin (4 (c+d x))}{32 d}+\frac {B \sin (5 (c+d x))}{80 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(3*C*(c + d*x))/(8*d) + (5*B*Sin[c + d*x])/(8*d) + (C*Sin[2*(c + d*x)])/(4*d) + (5*B*Sin[3*(c + d*x)])/(48*d)
+ (C*Sin[4*(c + d*x)])/(32*d) + (B*Sin[5*(c + d*x)])/(80*d)

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Maple [A]
time = 0.60, size = 70, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {\frac {B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(70\)
default	\(\frac {\frac {B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(70\)
risch	\(\frac {3 C x}{8}+\frac {5 B \sin \left (d x +c \right )}{8 d}+\frac {B \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {5 B \sin \left (3 d x +3 c \right )}{48 d}+\frac {C \sin \left (2 d x +2 c \right )}{4 d}\)	\(78\)
norman	\(\frac {\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 C x}{8}-\frac {15 C x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {27 C x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {15 C x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 C x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {27 C x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 C x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 C x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {\left (8 B -5 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {\left (8 B +5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (16 B -3 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {\left (16 B +3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {\left (344 B -75 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}+\frac {\left (344 B +75 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(294\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 69, normalized size = 0.75 \begin {gather*} \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 )} B + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{480 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 2.47, size = 64, normalized size = 0.70 \begin {gather*} \frac {45 \, C d x + {\left (24 \, B \cos \left (d x + c\right )^{4} + 30 \, C \cos \left (d x + c\right )^{3} + 32 \, B \cos \left (d x + c\right )^{2} + 45 \, C \cos \left (d x + c\right ) + 64 \, B\right )} \sin \left (d x + c\right )}{120 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/120*(45*C*d*x + (24*B*cos(d*x + c)^4 + 30*C*cos(d*x + c)^3 + 32*B*cos(d*x + c)^2 + 45*C*cos(d*x + c) + 64*B)
*sin(d*x + c))/d

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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Giac [A]
time = 0.45, size = 154, normalized size = 1.67 \begin {gather*} \frac {45 \, {\left (d x + c\right )} C + \frac {2 \, {\left (120 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 160 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, C \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/120*(45*(d*x + c)*C + 2*(120*B*tan(1/2*d*x + 1/2*c)^9 - 75*C*tan(1/2*d*x + 1/2*c)^9 + 160*B*tan(1/2*d*x + 1/
2*c)^7 - 30*C*tan(1/2*d*x + 1/2*c)^7 + 464*B*tan(1/2*d*x + 1/2*c)^5 + 160*B*tan(1/2*d*x + 1/2*c)^3 + 30*C*tan(
1/2*d*x + 1/2*c)^3 + 120*B*tan(1/2*d*x + 1/2*c) + 75*C*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d

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Mupad [B]
time = 6.31, size = 113, normalized size = 1.23 \begin {gather*} \frac {3\,C\,x}{8}+\frac {\left (2\,B-\frac {5\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {8\,B}{3}-\frac {C}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {116\,B\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\left (\frac {8\,B}{3}+\frac {C}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,B+\frac {5\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^6*(B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(3*C*x)/8 + (tan(c/2 + (d*x)/2)^3*((8*B)/3 + C/2) + tan(c/2 + (d*x)/2)^9*(2*B - (5*C)/4) + tan(c/2 + (d*x)/2)^
7*((8*B)/3 - C/2) + (116*B*tan(c/2 + (d*x)/2)^5)/15 + tan(c/2 + (d*x)/2)*(2*B + (5*C)/4))/(d*(tan(c/2 + (d*x)/
2)^2 + 1)^5)

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