\(\int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [103]

Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {11 x}{2 a^3}+\frac {3 \sin (c+d x)}{a^3 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))^2}+\frac {19 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))} \]

[Out]

-11/2*x/a^3+3*sin(d*x+c)/a^3/d-1/2*cos(d*x+c)*sin(d*x+c)/a^3/d-2/3*sin(d*x+c)/a^3/d/(1+cos(d*x+c))^2+19/3*sin(
d*x+c)/a^3/d/(1+cos(d*x+c))

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2953, 3045, 2717, 2715, 8, 2729, 2727} \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \sin (c+d x)}{a^3 d}-\frac {\sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac {19 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)}-\frac {2 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)^2}-\frac {11 x}{2 a^3} \]

[In]

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

[Out]

(-11*x)/(2*a^3) + (3*Sin[c + d*x])/(a^3*d) - (Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - (2*Sin[c + d*x])/(3*a^3*d
*(1 + Cos[c + d*x])^2) + (19*Sin[c + d*x])/(3*a^3*d*(1 + Cos[c + d*x]))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2953

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Dist[1/b^2, Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^(m + 1)*(a - b*Sin[e + f*x]), x], x] /;
FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && (ILtQ[m, 0] ||  !IGtQ[n, 0])

Rule 3045

Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; Fr
eeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos ^3(c+d x) \sin ^2(c+d x)}{(-a-a \cos (c+d x))^3} \, dx \\ & = -\frac {\int \frac {\cos ^3(c+d x) (-a+a \cos (c+d x))}{(-a-a \cos (c+d x))^2} \, dx}{a^2} \\ & = -\frac {\int \left (\frac {5}{a}-\frac {3 \cos (c+d x)}{a}+\frac {\cos ^2(c+d x)}{a}+\frac {2}{a (1+\cos (c+d x))^2}-\frac {7}{a (1+\cos (c+d x))}\right ) \, dx}{a^2} \\ & = -\frac {5 x}{a^3}-\frac {\int \cos ^2(c+d x) \, dx}{a^3}-\frac {2 \int \frac {1}{(1+\cos (c+d x))^2} \, dx}{a^3}+\frac {3 \int \cos (c+d x) \, dx}{a^3}+\frac {7 \int \frac {1}{1+\cos (c+d x)} \, dx}{a^3} \\ & = -\frac {5 x}{a^3}+\frac {3 \sin (c+d x)}{a^3 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))^2}+\frac {7 \sin (c+d x)}{a^3 d (1+\cos (c+d x))}-\frac {\int 1 \, dx}{2 a^3}-\frac {2 \int \frac {1}{1+\cos (c+d x)} \, dx}{3 a^3} \\ & = -\frac {11 x}{2 a^3}+\frac {3 \sin (c+d x)}{a^3 d}-\frac {\cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac {2 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))^2}+\frac {19 \sin (c+d x)}{3 a^3 d (1+\cos (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.82 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (1980 d x \cos \left (\frac {d x}{2}\right )+1980 d x \cos \left (c+\frac {d x}{2}\right )+660 d x \cos \left (c+\frac {3 d x}{2}\right )+660 d x \cos \left (2 c+\frac {3 d x}{2}\right )-3216 \sin \left (\frac {d x}{2}\right )+1326 \sin \left (c+\frac {d x}{2}\right )-2012 \sin \left (c+\frac {3 d x}{2}\right )-498 \sin \left (2 c+\frac {3 d x}{2}\right )-135 \sin \left (2 c+\frac {5 d x}{2}\right )-135 \sin \left (3 c+\frac {5 d x}{2}\right )+15 \sin \left (3 c+\frac {7 d x}{2}\right )+15 \sin \left (4 c+\frac {7 d x}{2}\right )\right )}{960 a^3 d} \]

[In]

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

[Out]

-1/960*(Sec[c/2]*Sec[(c + d*x)/2]^3*(1980*d*x*Cos[(d*x)/2] + 1980*d*x*Cos[c + (d*x)/2] + 660*d*x*Cos[c + (3*d*
x)/2] + 660*d*x*Cos[2*c + (3*d*x)/2] - 3216*Sin[(d*x)/2] + 1326*Sin[c + (d*x)/2] - 2012*Sin[c + (3*d*x)/2] - 4
98*Sin[2*c + (3*d*x)/2] - 135*Sin[2*c + (5*d*x)/2] - 135*Sin[3*c + (5*d*x)/2] + 15*Sin[3*c + (7*d*x)/2] + 15*S
in[4*c + (7*d*x)/2]))/(a^3*d)

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(\frac {\frac {275 \left (\cos \left (d x +c \right )+\frac {24 \cos \left (2 d x +2 c \right )}{275}-\frac {3 \cos \left (3 d x +3 c \right )}{275}+\frac {232}{275}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{48}-\frac {11 d x}{2}}{a^{3} d}\)	\(66\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 \left (-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\)	\(87\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 \left (-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}\)	\(87\)
risch	\(-\frac {11 x}{2 a^{3}}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )}}{2 a^{3} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{3} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {2 i \left (21 \,{\mathrm e}^{2 i \left (d x +c \right )}+36 \,{\mathrm e}^{i \left (d x +c \right )}+19\right )}{3 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\)	\(126\)
norman	\(\frac {-\frac {11 x}{2 a}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {56 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a d}+\frac {16 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 a d}-\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a}-\frac {11 x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} a^{2}}\)	\(135\)

[In]

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

[Out]

11/48*(25*(cos(d*x+c)+24/275*cos(2*d*x+2*c)-3/275*cos(3*d*x+3*c)+232/275)*tan(1/2*d*x+1/2*c)*sec(1/2*d*x+1/2*c
)^2-24*d*x)/a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {33 \, d x \cos \left (d x + c\right )^{2} + 66 \, d x \cos \left (d x + c\right ) + 33 \, d x + {\left (3 \, \cos \left (d x + c\right )^{3} - 12 \, \cos \left (d x + c\right )^{2} - 71 \, \cos \left (d x + c\right ) - 52\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \]

[In]

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

[Out]

-1/6*(33*d*x*cos(d*x + c)^2 + 66*d*x*cos(d*x + c) + 33*d*x + (3*cos(d*x + c)^3 - 12*cos(d*x + c)^2 - 71*cos(d*
x + c) - 52)*sin(d*x + c))/(a^3*d*cos(d*x + c)^2 + 2*a^3*d*cos(d*x + c) + a^3*d)

Sympy [F]

\[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\sin ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]

[In]

integrate(sin(d*x+c)**2/(a+a*sec(d*x+c))**3,x)

[Out]

Integral(sin(c + d*x)**2/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.69 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {18 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}} - \frac {33 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{3 \, d} \]

[In]

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

[Out]

1/3*(3*(5*sin(d*x + c)/(cos(d*x + c) + 1) + 7*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^3 + 2*a^3*sin(d*x + c)^2
/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (18*sin(d*x + c)/(cos(d*x + c) + 1) - sin(d
*x + c)^3/(cos(d*x + c) + 1)^3)/a^3 - 33*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {33 \, {\left (d x + c\right )}}{a^{3}} - \frac {6 \, {\left (7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} a^{3}} + \frac {2 \, {\left (a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 18 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{9}}}{6 \, d} \]

[In]

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

[Out]

-1/6*(33*(d*x + c)/a^3 - 6*(7*tan(1/2*d*x + 1/2*c)^3 + 5*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2
*a^3) + 2*(a^6*tan(1/2*d*x + 1/2*c)^3 - 18*a^6*tan(1/2*d*x + 1/2*c))/a^9)/d

Mupad [B] (verification not implemented)

Time = 13.32 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.19 \[ \int \frac {\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-38\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-42\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+33\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (c+d\,x\right )}{6\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]

[In]

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

[Out]

-(2*sin(c/2 + (d*x)/2) - 38*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2) - 42*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/
2) + 12*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2) + 33*cos(c/2 + (d*x)/2)^3*(c + d*x))/(6*a^3*d*cos(c/2 + (d*x)/
2)^3)