3.211 \(\int \frac{\sin ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=119 \[ -\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{a^4 d}+\frac{b^2 \left (a^2-b^2\right )}{a^5 d (a \cos (c+d x)+b)}+\frac{2 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b)}{a^5 d}-\frac{b \cos ^2(c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.228387, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2837, 12, 894} \[ -\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{a^4 d}+\frac{b^2 \left (a^2-b^2\right )}{a^5 d (a \cos (c+d x)+b)}+\frac{2 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b)}{a^5 d}-\frac{b \cos ^2(c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]

[Out]

-(((a^2 - 3*b^2)*Cos[c + d*x])/(a^4*d)) - (b*Cos[c + d*x]^2)/(a^3*d) + Cos[c + d*x]^3/(3*a^2*d) + (b^2*(a^2 -
b^2))/(a^5*d*(b + a*Cos[c + d*x])) + (2*b*(a^2 - 2*b^2)*Log[b + a*Cos[c + d*x]])/(a^5*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 2837

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.426407, size = 167, normalized size = 1.4 \[ \frac{-8 \left (a^4-3 a^2 b^2\right ) \cos (2 (c+d x))+48 a^2 b^2 \log (a \cos (c+d x)+b)+24 a b \cos (c+d x) \left (2 \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b)-a^2+3 b^2\right )+60 a^2 b^2-4 a^3 b \cos (3 (c+d x))+a^4 \cos (4 (c+d x))-9 a^4-96 b^4 \log (a \cos (c+d x)+b)-24 b^4}{24 a^5 d (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]

[Out]

(-9*a^4 + 60*a^2*b^2 - 24*b^4 - 8*(a^4 - 3*a^2*b^2)*Cos[2*(c + d*x)] - 4*a^3*b*Cos[3*(c + d*x)] + a^4*Cos[4*(c
+ d*x)] + 48*a^2*b^2*Log[b + a*Cos[c + d*x]] - 96*b^4*Log[b + a*Cos[c + d*x]] + 24*a*b*Cos[c + d*x]*(-a^2 + 3
*b^2 + 2*(a^2 - 2*b^2)*Log[b + a*Cos[c + d*x]]))/(24*a^5*d*(b + a*Cos[c + d*x]))

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Maple [A]  time = 0.063, size = 153, normalized size = 1.3 \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{2}d}}-{\frac{b \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{{a}^{3}d}}-{\frac{\cos \left ( dx+c \right ) }{{a}^{2}d}}+3\,{\frac{{b}^{2}\cos \left ( dx+c \right ) }{d{a}^{4}}}+2\,{\frac{b\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{{a}^{3}d}}-4\,{\frac{{b}^{3}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d{a}^{5}}}+{\frac{{b}^{2}}{{a}^{3}d \left ( b+a\cos \left ( dx+c \right ) \right ) }}-{\frac{{b}^{4}}{d{a}^{5} \left ( b+a\cos \left ( dx+c \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^3/(a+b*sec(d*x+c))^2,x)

[Out]

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

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Maxima [A]  time = 1.07083, size = 151, normalized size = 1.27 \begin{align*} \frac{\frac{3 \,{\left (a^{2} b^{2} - b^{4}\right )}}{a^{6} \cos \left (d x + c\right ) + a^{5} b} + \frac{a^{2} \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )^{2} - 3 \,{\left (a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )}{a^{4}} + \frac{6 \,{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{5}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.84538, size = 346, normalized size = 2.91 \begin{align*} \frac{2 \, a^{4} \cos \left (d x + c\right )^{4} - 4 \, a^{3} b \cos \left (d x + c\right )^{3} + 9 \, a^{2} b^{2} - 6 \, b^{4} - 6 \,{\left (a^{4} - 2 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (a^{3} b - 6 \, a b^{3}\right )} \cos \left (d x + c\right ) + 12 \,{\left (a^{2} b^{2} - 2 \, b^{4} +{\left (a^{3} b - 2 \, a b^{3}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{6 \,{\left (a^{6} d \cos \left (d x + c\right ) + a^{5} b d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

1/6*(2*a^4*cos(d*x + c)^4 - 4*a^3*b*cos(d*x + c)^3 + 9*a^2*b^2 - 6*b^4 - 6*(a^4 - 2*a^2*b^2)*cos(d*x + c)^2 -
3*(a^3*b - 6*a*b^3)*cos(d*x + c) + 12*(a^2*b^2 - 2*b^4 + (a^3*b - 2*a*b^3)*cos(d*x + c))*log(a*cos(d*x + c) +
b))/(a^6*d*cos(d*x + c) + a^5*b*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(sin(d*x+c)**3/(a+b*sec(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.34274, size = 188, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{5} d} + \frac{a^{2} b^{2} - b^{4}}{{\left (a \cos \left (d x + c\right ) + b\right )} a^{5} d} + \frac{a^{4} d^{5} \cos \left (d x + c\right )^{3} - 3 \, a^{3} b d^{5} \cos \left (d x + c\right )^{2} - 3 \, a^{4} d^{5} \cos \left (d x + c\right ) + 9 \, a^{2} b^{2} d^{5} \cos \left (d x + c\right )}{3 \, a^{6} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^3/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

2*(a^2*b - 2*b^3)*log(abs(-a*cos(d*x + c) - b))/(a^5*d) + (a^2*b^2 - b^4)/((a*cos(d*x + c) + b)*a^5*d) + 1/3*(
a^4*d^5*cos(d*x + c)^3 - 3*a^3*b*d^5*cos(d*x + c)^2 - 3*a^4*d^5*cos(d*x + c) + 9*a^2*b^2*d^5*cos(d*x + c))/(a^
6*d^6)