3.243 \(\int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx\)

Optimal. Leaf size=77 \[ \frac {(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac {2 b (a \cos (c+d x)+b)^5}{5 a^3 d}-\frac {\left (a^2-b^2\right ) (a \cos (c+d x)+b)^4}{4 a^3 d} \]

[Out]

-1/4*(a^2-b^2)*(b+a*cos(d*x+c))^4/a^3/d-2/5*b*(b+a*cos(d*x+c))^5/a^3/d+1/6*(b+a*cos(d*x+c))^6/a^3/d

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Rubi [A]  time = 0.19, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {4397, 2668, 697} \[ -\frac {\left (a^2-b^2\right ) (a \cos (c+d x)+b)^4}{4 a^3 d}+\frac {(a \cos (c+d x)+b)^6}{6 a^3 d}-\frac {2 b (a \cos (c+d x)+b)^5}{5 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]

[Out]

-((a^2 - b^2)*(b + a*Cos[c + d*x])^4)/(4*a^3*d) - (2*b*(b + a*Cos[c + d*x])^5)/(5*a^3*d) + (b + a*Cos[c + d*x]
)^6/(6*a^3*d)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

Rubi steps

\begin {align*} \int \cos ^3(c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx &=\int (b+a \cos (c+d x))^3 \sin ^3(c+d x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int (b+x)^3 \left (a^2-x^2\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\left (a^2-b^2\right ) (b+x)^3+2 b (b+x)^4-(b+x)^5\right ) \, dx,x,a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {\left (a^2-b^2\right ) (b+a \cos (c+d x))^4}{4 a^3 d}-\frac {2 b (b+a \cos (c+d x))^5}{5 a^3 d}+\frac {(b+a \cos (c+d x))^6}{6 a^3 d}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 114, normalized size = 1.48 \[ \frac {-45 \left (a^3+8 a b^2\right ) \cos (2 (c+d x))+5 a^3 \cos (6 (c+d x))-360 b \left (a^2+2 b^2\right ) \cos (c+d x)-60 a^2 b \cos (3 (c+d x))+36 a^2 b \cos (5 (c+d x))+90 a b^2 \cos (4 (c+d x))+80 b^3 \cos (3 (c+d x))}{960 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a*Sin[c + d*x] + b*Tan[c + d*x])^3,x]

[Out]

(-360*b*(a^2 + 2*b^2)*Cos[c + d*x] - 45*(a^3 + 8*a*b^2)*Cos[2*(c + d*x)] - 60*a^2*b*Cos[3*(c + d*x)] + 80*b^3*
Cos[3*(c + d*x)] + 90*a*b^2*Cos[4*(c + d*x)] + 36*a^2*b*Cos[5*(c + d*x)] + 5*a^3*Cos[6*(c + d*x)])/(960*d)

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fricas [A]  time = 1.27, size = 100, normalized size = 1.30 \[ \frac {10 \, a^{3} \cos \left (d x + c\right )^{6} + 36 \, a^{2} b \cos \left (d x + c\right )^{5} - 90 \, a b^{2} \cos \left (d x + c\right )^{2} - 15 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} - 60 \, b^{3} \cos \left (d x + c\right ) - 20 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{3}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*sin(d*x+c)+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/60*(10*a^3*cos(d*x + c)^6 + 36*a^2*b*cos(d*x + c)^5 - 90*a*b^2*cos(d*x + c)^2 - 15*(a^3 - 3*a*b^2)*cos(d*x +
c)^4 - 60*b^3*cos(d*x + c) - 20*(3*a^2*b - b^3)*cos(d*x + c)^3)/d

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*sin(d*x+c)+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.11, size = 109, normalized size = 1.42 \[ \frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )+3 a^{2} b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+\frac {3 a \,b^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {b^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a*sin(d*x+c)+b*tan(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/6*sin(d*x+c)^2*cos(d*x+c)^4-1/12*cos(d*x+c)^4)+3*a^2*b*(-1/5*sin(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d
*x+c)^3)+3/4*a*b^2*sin(d*x+c)^4-1/3*b^3*(2+sin(d*x+c)^2)*cos(d*x+c))

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maxima [A]  time = 0.38, size = 95, normalized size = 1.23 \[ \frac {45 \, a b^{2} \sin \left (d x + c\right )^{4} - 5 \, {\left (2 \, \sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4}\right )} a^{3} + 12 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b + 20 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} b^{3}}{60 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a*sin(d*x+c)+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/60*(45*a*b^2*sin(d*x + c)^4 - 5*(2*sin(d*x + c)^6 - 3*sin(d*x + c)^4)*a^3 + 12*(3*cos(d*x + c)^5 - 5*cos(d*x
+ c)^3)*a^2*b + 20*(cos(d*x + c)^3 - 3*cos(d*x + c))*b^3)/d

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mupad [B]  time = 0.80, size = 149, normalized size = 1.94 \[ \frac {32\,a^3}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}+\frac {4\,{\left (a-b\right )}^3}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {32\,a^2\,\left (5\,a-3\,b\right )}{5\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5}-\frac {8\,{\left (a-b\right )}^2\,\left (7\,a-b\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}+\frac {12\,a\,\left (3\,a^2-4\,a\,b+b^2\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^3*(a*sin(c + d*x) + b*tan(c + d*x))^3,x)

[Out]

(32*a^3)/(3*d*(tan(c/2 + (d*x)/2)^2 + 1)^6) + (4*(a - b)^3)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^2) - (32*a^2*(5*a -
3*b))/(5*d*(tan(c/2 + (d*x)/2)^2 + 1)^5) - (8*(a - b)^2*(7*a - b))/(3*d*(tan(c/2 + (d*x)/2)^2 + 1)^3) + (12*a*
(3*a^2 - 4*a*b + b^2))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^4)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3} \cos ^{3}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**3*(a*sin(d*x+c)+b*tan(d*x+c))**3,x)

[Out]

Integral((a*sin(c + d*x) + b*tan(c + d*x))**3*cos(c + d*x)**3, x)

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