3.41 \(\int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx\)

Optimal. Leaf size=98 \[ \frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{2 a^3 \log (\cos (c+d x))}{d} \]

[Out]

(2*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (2*a^3*Log[Cos[c + d*x]])
/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)

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Rubi [A]  time = 0.0959318, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3872, 2707, 75} \[ \frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{2 a^3 \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^3,x]

[Out]

(2*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/(2*d) + (a^3*Cos[c + d*x]^3)/(3*d) - (2*a^3*Log[Cos[c + d*x]])
/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[c + d*x]^2)/(2*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2707

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^(m - (p + 1)/2))/(a - x)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& EqQ[a^2 - b^2, 0] && IntegerQ[(p + 1)/2]

Rule 75

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps

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

Mathematica [A]  time = 0.194758, size = 86, normalized size = 0.88 \[ \frac{a^3 \sec ^2(c+d x) (226 \cos (c+d x)+29 \cos (3 (c+d x))+9 \cos (4 (c+d x))+\cos (5 (c+d x))-48 \log (\cos (c+d x))-8 \cos (2 (c+d x)) (6 \log (\cos (c+d x))+7)-41)}{48 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^3,x]

[Out]

(a^3*(-41 + 226*Cos[c + d*x] + 29*Cos[3*(c + d*x)] + 9*Cos[4*(c + d*x)] + Cos[5*(c + d*x)] - 48*Log[Cos[c + d*
x]] - 8*Cos[2*(c + d*x)]*(7 + 6*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/(48*d)

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Maple [A]  time = 0.045, size = 109, normalized size = 1.1 \begin{align*}{\frac{8\,{a}^{3}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{16\,{a}^{3}\cos \left ( dx+c \right ) }{3\,d}}-{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-2\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d\cos \left ( dx+c \right ) }}+{\frac{{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^3*sin(d*x+c)^3,x)

[Out]

8/3/d*a^3*cos(d*x+c)*sin(d*x+c)^2+16/3*a^3*cos(d*x+c)/d-3/2/d*a^3*sin(d*x+c)^2-2*a^3*ln(cos(d*x+c))/d+3/d*a^3*
sin(d*x+c)^4/cos(d*x+c)+1/2/d*a^3*tan(d*x+c)^2

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Maxima [A]  time = 1.01041, size = 108, normalized size = 1.1 \begin{align*} \frac{2 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \cos \left (d x + c\right )^{2} + 12 \, a^{3} \cos \left (d x + c\right ) - 12 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) + \frac{3 \,{\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a^3*cos(d*x + c)^3 + 9*a^3*cos(d*x + c)^2 + 12*a^3*cos(d*x + c) - 12*a^3*log(cos(d*x + c)) + 3*(6*a^3*c
os(d*x + c) + a^3)/cos(d*x + c)^2)/d

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Fricas [A]  time = 1.85308, size = 259, normalized size = 2.64 \begin{align*} \frac{4 \, a^{3} \cos \left (d x + c\right )^{5} + 18 \, a^{3} \cos \left (d x + c\right )^{4} + 24 \, a^{3} \cos \left (d x + c\right )^{3} - 24 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) - 9 \, a^{3} \cos \left (d x + c\right )^{2} + 36 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3}}{12 \, d \cos \left (d x + c\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*a^3*cos(d*x + c)^5 + 18*a^3*cos(d*x + c)^4 + 24*a^3*cos(d*x + c)^3 - 24*a^3*cos(d*x + c)^2*log(-cos(d*
x + c)) - 9*a^3*cos(d*x + c)^2 + 36*a^3*cos(d*x + c) + 6*a^3)/(d*cos(d*x + c)^2)

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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*sec(d*x+c))**3*sin(d*x+c)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.31223, size = 138, normalized size = 1.41 \begin{align*} -\frac{2 \, a^{3} \log \left (\frac{{\left | \cos \left (d x + c\right ) \right |}}{{\left | d \right |}}\right )}{d} + \frac{6 \, a^{3} \cos \left (d x + c\right ) + a^{3}}{2 \, d \cos \left (d x + c\right )^{2}} + \frac{2 \, a^{3} d^{8} \cos \left (d x + c\right )^{3} + 9 \, a^{3} d^{8} \cos \left (d x + c\right )^{2} + 12 \, a^{3} d^{8} \cos \left (d x + c\right )}{6 \, d^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^3*log(abs(cos(d*x + c))/abs(d))/d + 1/2*(6*a^3*cos(d*x + c) + a^3)/(d*cos(d*x + c)^2) + 1/6*(2*a^3*d^8*co
s(d*x + c)^3 + 9*a^3*d^8*cos(d*x + c)^2 + 12*a^3*d^8*cos(d*x + c))/d^9