3.1.38 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \, dx\) [38]

Optimal. Leaf size=102 \[ \frac {7 a^4 x}{2}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {7 a^4 \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}-\frac {8 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3876, 2717, 2715, 8, 2713} \begin {gather*} \frac {a^4 \sin ^5(c+d x)}{5 d}-\frac {8 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac {7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {7 a^4 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4,x]

[Out]

(7*a^4*x)/2 + (8*a^4*Sin[c + d*x])/d + (7*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Cos[c + d*x]^3*Sin[c + d
*x])/d - (8*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^5)/(5*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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

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Mathematica [A]
time = 0.16, size = 63, normalized size = 0.62 \begin {gather*} \frac {a^4 (840 d x+1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x)))}{240 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4,x]

[Out]

(a^4*(840*d*x + 1470*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 145*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] + 3*Sin[
5*(c + d*x)]))/(240*d)

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Maple [A]
time = 0.15, size = 133, normalized size = 1.30

method	result	size
risch	\(\frac {7 a^{4} x}{2}+\frac {49 a^{4} \sin \left (d x +c \right )}{8 d}+\frac {a^{4} \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{4} \sin \left (4 d x +4 c \right )}{8 d}+\frac {29 a^{4} \sin \left (3 d x +3 c \right )}{48 d}+\frac {2 a^{4} \sin \left (2 d x +2 c \right )}{d}\)	\(90\)
derivativedivides	\(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \sin \left (d x +c \right )}{d}\)	\(133\)
default	\(\frac {\frac {a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} \sin \left (d x +c \right )}{d}\)	\(133\)
norman	\(\frac {-\frac {7 a^{4} x}{2}-\frac {25 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {67 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {349 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {203 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {533 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {259 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {35 a^{4} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {7 a^{4} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-7 a^{4} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 a^{4} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+21 a^{4} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-21 a^{4} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7 a^{4} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 a^{4} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 a^{4} x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+a*sec(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 128, normalized size = 1.25 \begin {gather*} \frac {8 \, {\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} - 240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} + 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 )} a^{4} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, a^{4} \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)^5*(a+a*sec(d*x+c))^4,x, algorithm="maxima")

[Out]

1/120*(8*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^4 - 240*(sin(d*x + c)^3 - 3*sin(d*x + c))*
a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^
4 + 120*a^4*sin(d*x + c))/d

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Fricas [A]
time = 2.74, size = 76, normalized size = 0.75 \begin {gather*} \frac {105 \, a^{4} d x + {\left (6 \, a^{4} \cos \left (d x + c\right )^{4} + 30 \, a^{4} \cos \left (d x + c\right )^{3} + 68 \, a^{4} \cos \left (d x + c\right )^{2} + 105 \, a^{4} \cos \left (d x + c\right ) + 166 \, a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^4,x, algorithm="fricas")

[Out]

1/30*(105*a^4*d*x + (6*a^4*cos(d*x + c)^4 + 30*a^4*cos(d*x + c)^3 + 68*a^4*cos(d*x + c)^2 + 105*a^4*cos(d*x +
c) + 166*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)**5*(a+a*sec(d*x+c))**4,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [A]
time = 0.47, size = 112, normalized size = 1.10 \begin {gather*} \frac {105 \, {\left (d x + c\right )} a^{4} + \frac {2 \, {\left (105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 490 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 896 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 790 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 375 \, 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 )}^{5}}}{30 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^4,x, algorithm="giac")

[Out]

1/30*(105*(d*x + c)*a^4 + 2*(105*a^4*tan(1/2*d*x + 1/2*c)^9 + 490*a^4*tan(1/2*d*x + 1/2*c)^7 + 896*a^4*tan(1/2
*d*x + 1/2*c)^5 + 790*a^4*tan(1/2*d*x + 1/2*c)^3 + 375*a^4*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 = 4.43, size = 105, normalized size = 1.03 \begin {gather*} \frac {7\,a^4\,x}{2}+\frac {7\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {98\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {896\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\frac {158\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+25\,a^4\,\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)^5*(a + a/cos(c + d*x))^4,x)

[Out]

(7*a^4*x)/2 + ((158*a^4*tan(c/2 + (d*x)/2)^3)/3 + (896*a^4*tan(c/2 + (d*x)/2)^5)/15 + (98*a^4*tan(c/2 + (d*x)/
2)^7)/3 + 7*a^4*tan(c/2 + (d*x)/2)^9 + 25*a^4*tan(c/2 + (d*x)/2))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^5)

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