3.1.40 \(\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \, dx\) [40]

Optimal. Leaf size=147 \[ \frac {11 a^4 x}{4}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {11 a^4 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d} \]

[Out]

11/4*a^4*x+8*a^4*sin(d*x+c)/d+11/4*a^4*cos(d*x+c)*sin(d*x+c)/d+11/6*a^4*cos(d*x+c)^3*sin(d*x+c)/d+2/3*a^4*cos(
d*x+c)^5*sin(d*x+c)/d-16/3*a^4*sin(d*x+c)^3/d+9/5*a^4*sin(d*x+c)^5/d-1/7*a^4*sin(d*x+c)^7/d

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Rubi [A]
time = 0.12, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3876, 2713, 2715, 8} \begin {gather*} -\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {2 a^4 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac {11 a^4 \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac {11 a^4 \sin (c+d x) \cos (c+d x)}{4 d}+\frac {11 a^4 x}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4,x]

[Out]

(11*a^4*x)/4 + (8*a^4*Sin[c + d*x])/d + (11*a^4*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (11*a^4*Cos[c + d*x]^3*Sin[
c + d*x])/(6*d) + (2*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(3*d) - (16*a^4*Sin[c + d*x]^3)/(3*d) + (9*a^4*Sin[c + d
*x]^5)/(5*d) - (a^4*Sin[c + d*x]^7)/(7*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3876

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]

Rubi steps

\begin {align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (a^4 \cos ^3(c+d x)+4 a^4 \cos ^4(c+d x)+6 a^4 \cos ^5(c+d x)+4 a^4 \cos ^6(c+d x)+a^4 \cos ^7(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^3(c+d x) \, dx+a^4 \int \cos ^7(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^4(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^6(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^5(c+d x) \, dx\\ &=\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}+\left (3 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{3} \left (10 a^4\right ) \int \cos ^4(c+d x) \, dx-\frac {a^4 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {a^4 \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (6 a^4\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {8 a^4 \sin (c+d x)}{d}+\frac {3 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {1}{2} \left (3 a^4\right ) \int 1 \, dx+\frac {1}{2} \left (5 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a^4 x}{2}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {11 a^4 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {1}{4} \left (5 a^4\right ) \int 1 \, dx\\ &=\frac {11 a^4 x}{4}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {11 a^4 \cos (c+d x) \sin (c+d x)}{4 d}+\frac {11 a^4 \cos ^3(c+d x) \sin (c+d x)}{6 d}+\frac {2 a^4 \cos ^5(c+d x) \sin (c+d x)}{3 d}-\frac {16 a^4 \sin ^3(c+d x)}{3 d}+\frac {9 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 83, normalized size = 0.56 \begin {gather*} \frac {a^4 (18480 d x+33915 \sin (c+d x)+13020 \sin (2 (c+d x))+5495 \sin (3 (c+d x))+2100 \sin (4 (c+d x))+651 \sin (5 (c+d x))+140 \sin (6 (c+d x))+15 \sin (7 (c+d x)))}{6720 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4,x]

[Out]

(a^4*(18480*d*x + 33915*Sin[c + d*x] + 13020*Sin[2*(c + d*x)] + 5495*Sin[3*(c + d*x)] + 2100*Sin[4*(c + d*x)]
+ 651*Sin[5*(c + d*x)] + 140*Sin[6*(c + d*x)] + 15*Sin[7*(c + d*x)]))/(6720*d)

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Maple [A]
time = 0.12, size = 185, normalized size = 1.26

method	result	size
risch	\(\frac {11 a^{4} x}{4}+\frac {323 a^{4} \sin \left (d x +c \right )}{64 d}+\frac {a^{4} \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{48 d}+\frac {31 a^{4} \sin \left (5 d x +5 c \right )}{320 d}+\frac {5 a^{4} \sin \left (4 d x +4 c \right )}{16 d}+\frac {157 a^{4} \sin \left (3 d x +3 c \right )}{192 d}+\frac {31 a^{4} \sin \left (2 d x +2 c \right )}{16 d}\)	\(124\)
derivativedivides	\(\frac {\frac {a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {6 a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(185\)
default	\(\frac {\frac {a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+4 a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {6 a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+a*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/7*a^4*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)+4*a^4*(1/6*(cos(d*x+c)^5+5/4*cos
(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+6/5*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+
4*a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+1/3*a^4*(2+cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]
time = 0.30, size = 187, normalized size = 1.27 \begin {gather*} -\frac {48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{4} - 672 \, {\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^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} - 210 \, {\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^{4}}{1680 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/1680*(48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a^4 - 672*(3*sin(d*x
+ c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^4 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4
*c) - 48*sin(2*d*x + 2*c))*a^4 + 560*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^4 - 210*(12*d*x + 12*c + sin(4*d*x +
4*c) + 8*sin(2*d*x + 2*c))*a^4)/d

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Fricas [A]
time = 3.50, size = 102, normalized size = 0.69 \begin {gather*} \frac {1155 \, a^{4} d x + {\left (60 \, a^{4} \cos \left (d x + c\right )^{6} + 280 \, a^{4} \cos \left (d x + c\right )^{5} + 576 \, a^{4} \cos \left (d x + c\right )^{4} + 770 \, a^{4} \cos \left (d x + c\right )^{3} + 908 \, a^{4} \cos \left (d x + c\right )^{2} + 1155 \, a^{4} \cos \left (d x + c\right ) + 1816 \, a^{4}\right )} \sin \left (d x + c\right )}{420 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

1/420*(1155*a^4*d*x + (60*a^4*cos(d*x + c)^6 + 280*a^4*cos(d*x + c)^5 + 576*a^4*cos(d*x + c)^4 + 770*a^4*cos(d
*x + c)^3 + 908*a^4*cos(d*x + c)^2 + 1155*a^4*cos(d*x + c) + 1816*a^4)*sin(d*x + c))/d

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+a*sec(d*x+c))**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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Giac [A]
time = 0.51, size = 144, normalized size = 0.98 \begin {gather*} \frac {1155 \, {\left (d x + c\right )} a^{4} + \frac {2 \, {\left (1155 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 7700 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 21791 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 33792 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 31521 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14700 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5565 \, a^{4} \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 )}^{7}}}{420 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/420*(1155*(d*x + c)*a^4 + 2*(1155*a^4*tan(1/2*d*x + 1/2*c)^13 + 7700*a^4*tan(1/2*d*x + 1/2*c)^11 + 21791*a^4
*tan(1/2*d*x + 1/2*c)^9 + 33792*a^4*tan(1/2*d*x + 1/2*c)^7 + 31521*a^4*tan(1/2*d*x + 1/2*c)^5 + 14700*a^4*tan(
1/2*d*x + 1/2*c)^3 + 5565*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d

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Mupad [B]
time = 3.64, size = 137, normalized size = 0.93 \begin {gather*} \frac {11\,a^4\,x}{4}+\frac {\frac {11\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{2}+\frac {110\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}+\frac {3113\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{30}+\frac {5632\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{35}+\frac {1501\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{10}+70\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {53\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7*(a + a/cos(c + d*x))^4,x)

[Out]

(11*a^4*x)/4 + (70*a^4*tan(c/2 + (d*x)/2)^3 + (1501*a^4*tan(c/2 + (d*x)/2)^5)/10 + (5632*a^4*tan(c/2 + (d*x)/2
)^7)/35 + (3113*a^4*tan(c/2 + (d*x)/2)^9)/30 + (110*a^4*tan(c/2 + (d*x)/2)^11)/3 + (11*a^4*tan(c/2 + (d*x)/2)^
13)/2 + (53*a^4*tan(c/2 + (d*x)/2))/2)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^7)

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