3.31 \(\int (a+a \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \sec (c+d x) \, dx\)

Optimal. Leaf size=177 \[ \frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{(5 A+7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{15 d}+\frac{(8 A+7 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^4 x (12 A+7 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]

[Out]

(a^4*(12*A + 7*C)*x)/2 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(10*A + 7*C)*Sin[c + d*x])/(2*d) + (a*C*(a + a
*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((5*A + 7*C)*(a^2 + a^2
*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + ((8*A + 7*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d)

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Rubi [A]  time = 0.538812, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {3046, 2976, 2968, 3023, 2735, 3770} \[ \frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{(5 A+7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{15 d}+\frac{(8 A+7 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{6 d}+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^4 x (12 A+7 C)+\frac{a C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d}+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]

[Out]

(a^4*(12*A + 7*C)*x)/2 + (a^4*A*ArcTanh[Sin[c + d*x]])/d + (a^4*(10*A + 7*C)*Sin[c + d*x])/(2*d) + (a*C*(a + a
*Cos[c + d*x])^3*Sin[c + d*x])/(5*d) + (C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) + ((5*A + 7*C)*(a^2 + a^2
*Cos[c + d*x])^2*Sin[c + d*x])/(15*d) + ((8*A + 7*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(6*d)

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*
sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n
+ 1))/(d*f*(m + n + 2)), x] + Dist[1/(b*d*(m + n + 2)), Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp
[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b
, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[m, -2^(-
1)] && NeQ[m + n + 2, 0]

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 2968

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{\int (a+a \cos (c+d x))^4 (5 a A+4 a C \cos (c+d x)) \sec (c+d x) \, dx}{5 a}\\ &=\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{\int (a+a \cos (c+d x))^3 \left (20 a^2 A+4 a^2 (5 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{20 a}\\ &=\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac{\int (a+a \cos (c+d x))^2 \left (60 a^3 A+20 a^3 (8 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{60 a}\\ &=\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac{(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{\int (a+a \cos (c+d x)) \left (120 a^4 A+60 a^4 (10 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac{(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{\int \left (120 a^5 A+\left (120 a^5 A+60 a^5 (10 A+7 C)\right ) \cos (c+d x)+60 a^5 (10 A+7 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac{(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac{\int \left (120 a^5 A+60 a^5 (12 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{120 a}\\ &=\frac{1}{2} a^4 (12 A+7 C) x+\frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac{(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\left (a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac{1}{2} a^4 (12 A+7 C) x+\frac{a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 (10 A+7 C) \sin (c+d x)}{2 d}+\frac{a C (a+a \cos (c+d x))^3 \sin (c+d x)}{5 d}+\frac{C (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{(5 A+7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{15 d}+\frac{(8 A+7 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.468441, size = 147, normalized size = 0.83 \[ \frac{a^4 \left (30 (54 A+49 C) \sin (c+d x)+240 (A+2 C) \sin (2 (c+d x))+20 A \sin (3 (c+d x))-240 A \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+240 A \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+1440 A d x+145 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+3 C \sin (5 (c+d x))+840 C d x\right )}{240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]

[Out]

(a^4*(1440*A*d*x + 840*C*d*x - 240*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 240*A*Log[Cos[(c + d*x)/2] + S
in[(c + d*x)/2]] + 30*(54*A + 49*C)*Sin[c + d*x] + 240*(A + 2*C)*Sin[2*(c + d*x)] + 20*A*Sin[3*(c + d*x)] + 14
5*C*Sin[3*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 3*C*Sin[5*(c + d*x)]))/(240*d)

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Maple [A]  time = 0.066, size = 221, normalized size = 1.3 \begin{align*}{\frac{A\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}{a}^{4}}{3\,d}}+{\frac{20\,A{a}^{4}\sin \left ( dx+c \right ) }{3\,d}}+{\frac{83\,{a}^{4}C\sin \left ( dx+c \right ) }{15\,d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}+{\frac{34\,{a}^{4}C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{15\,d}}+2\,{\frac{A{a}^{4}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}}+6\,A{a}^{4}x+6\,{\frac{A{a}^{4}c}{d}}+{\frac{{a}^{4}C\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{7\,{a}^{4}C\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2\,d}}+{\frac{7\,{a}^{4}Cx}{2}}+{\frac{7\,{a}^{4}Cc}{2\,d}}+{\frac{A{a}^{4}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c),x)

[Out]

1/3/d*A*sin(d*x+c)*cos(d*x+c)^2*a^4+20/3/d*A*a^4*sin(d*x+c)+83/15/d*a^4*C*sin(d*x+c)+1/5/d*a^4*C*sin(d*x+c)*co
s(d*x+c)^4+34/15/d*a^4*C*sin(d*x+c)*cos(d*x+c)^2+2/d*A*a^4*cos(d*x+c)*sin(d*x+c)+6*A*a^4*x+6/d*A*a^4*c+1/d*a^4
*C*sin(d*x+c)*cos(d*x+c)^3+7/2/d*a^4*C*cos(d*x+c)*sin(d*x+c)+7/2*a^4*C*x+7/2/d*a^4*C*c+1/d*A*a^4*ln(sec(d*x+c)
+tan(d*x+c))

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Maxima [A]  time = 1.02808, size = 300, normalized size = 1.69 \begin{align*} -\frac{40 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 480 \,{\left (d x + c\right )} A a^{4} - 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 )} C a^{4} + 240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C 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 )} C a^{4} - 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 120 \, A a^{4} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) - 720 \, A a^{4} \sin \left (d x + c\right ) - 120 \, C a^{4} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="maxima")

[Out]

-1/120*(40*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 480*(d*x + c
)*A*a^4 - 8*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 240*(sin(d*x + c)^3 - 3*sin(d*x +
c))*C*a^4 - 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 - 120*(2*d*x + 2*c + sin(2*d*x +
2*c))*C*a^4 - 120*A*a^4*log(sec(d*x + c) + tan(d*x + c)) - 720*A*a^4*sin(d*x + c) - 120*C*a^4*sin(d*x + c))/d

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Fricas [A]  time = 1.5962, size = 351, normalized size = 1.98 \begin{align*} \frac{15 \,{\left (12 \, A + 7 \, C\right )} a^{4} d x + 15 \, A a^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, A a^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) +{\left (6 \, C a^{4} \cos \left (d x + c\right )^{4} + 30 \, C a^{4} \cos \left (d x + c\right )^{3} + 2 \,{\left (5 \, A + 34 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \,{\left (4 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right ) + 2 \,{\left (100 \, A + 83 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="fricas")

[Out]

1/30*(15*(12*A + 7*C)*a^4*d*x + 15*A*a^4*log(sin(d*x + c) + 1) - 15*A*a^4*log(-sin(d*x + c) + 1) + (6*C*a^4*co
s(d*x + c)^4 + 30*C*a^4*cos(d*x + c)^3 + 2*(5*A + 34*C)*a^4*cos(d*x + c)^2 + 15*(4*A + 7*C)*a^4*cos(d*x + c) +
2*(100*A + 83*C)*a^4)*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((a+a*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c),x)

[Out]

Timed out

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Giac [A]  time = 1.32295, size = 335, normalized size = 1.89 \begin{align*} \frac{30 \, A a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 30 \, A a^{4} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) + 15 \,{\left (12 \, A a^{4} + 7 \, C a^{4}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (150 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 105 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 680 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 490 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 1180 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 896 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 920 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 790 \, C a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 270 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 375 \, C 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c),x, algorithm="giac")

[Out]

1/30*(30*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 30*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 15*(12*A*a^4
+ 7*C*a^4)*(d*x + c) + 2*(150*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 105*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 680*A*a^4*tan
(1/2*d*x + 1/2*c)^7 + 490*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 1180*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 896*C*a^4*tan(1/2
*d*x + 1/2*c)^5 + 920*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 790*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 270*A*a^4*tan(1/2*d*x
+ 1/2*c) + 375*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d