\(\int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx\) [805]

Optimal result

Integrand size = 27, antiderivative size = 86 \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=2 a^2 x-\frac {4 a^2 \cos (c+d x)}{3 d}-\frac {2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+\frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2948, 2844, 3047, 3102, 12, 2814, 2727} \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {a^4 \sin ^2(c+d x) \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {4 a^2 \cos (c+d x)}{3 d}-\frac {2 a^2 \cos (c+d x)}{d (1-\sin (c+d x))}+2 a^2 x \]

[In]

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

[Out]

2*a^2*x - (4*a^2*Cos[c + d*x])/(3*d) - (2*a^2*Cos[c + d*x])/(d*(1 - Sin[c + d*x])) + (a^4*Cos[c + d*x]*Sin[c +
d*x]^2)/(3*d*(a - a*Sin[c + d*x])^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2814

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 2844

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

Rule 2948

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

Rule 3047

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 3102

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]

Rubi steps \begin{align*} \text {integral}& = a^4 \int \frac {\sin ^3(c+d x)}{(a-a \sin (c+d x))^2} \, dx \\ & = \frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} a^2 \int \frac {\sin (c+d x) (-2 a-4 a \sin (c+d x))}{a-a \sin (c+d x)} \, dx \\ & = \frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} a^2 \int \frac {-2 a \sin (c+d x)-4 a \sin ^2(c+d x)}{a-a \sin (c+d x)} \, dx \\ & = -\frac {4 a^2 \cos (c+d x)}{3 d}+\frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {1}{3} a \int \frac {6 a^2 \sin (c+d x)}{a-a \sin (c+d x)} \, dx \\ & = -\frac {4 a^2 \cos (c+d x)}{3 d}+\frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2}-\left (2 a^3\right ) \int \frac {\sin (c+d x)}{a-a \sin (c+d x)} \, dx \\ & = 2 a^2 x-\frac {4 a^2 \cos (c+d x)}{3 d}+\frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2}-\left (2 a^3\right ) \int \frac {1}{a-a \sin (c+d x)} \, dx \\ & = 2 a^2 x-\frac {4 a^2 \cos (c+d x)}{3 d}+\frac {a^4 \cos (c+d x) \sin ^2(c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {2 a^3 \cos (c+d x)}{d (a-a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.52 \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {a^2 (1+\sin (c+d x))^2 \left (6 c+6 d x-3 \cos (c+d x)+\frac {1}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right ) (-7+8 \sin (c+d x))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \]

[In]

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

[Out]

(a^2*(1 + Sin[c + d*x])^2*(6*c + 6*d*x - 3*Cos[c + d*x] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) + (2*Sin[
(c + d*x)/2]*(-7 + 8*Sin[c + d*x]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2])^4)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.03

method	result	size
risch	\(2 a^{2} x -\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {2 a^{2} \left (-15 i {\mathrm e}^{i \left (d x +c \right )}+9 \,{\mathrm e}^{2 i \left (d x +c \right )}-8\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3}}\)	\(89\)
parallelrisch	\(\frac {2 \left (\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) d x +\left (-3 d x +2\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (4 d x -6\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-4 d x +\frac {22}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (3 d x -8\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-d x +\frac {10}{3}\right ) a^{2}}{d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\)	\(127\)
derivativedivides	\(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )}{d}\)	\(162\)
default	\(\frac {a^{2} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+2 a^{2} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+a^{2} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )}{d}\)	\(162\)
norman	\(\frac {-2 a^{2} x +\frac {20 a^{2}}{3 d}+\frac {4 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {16 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {56 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {16 a^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {4 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+2 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {20 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {52 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\)	\(282\)

[In]

int(sec(d*x+c)^4*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

2*a^2*x-1/2*a^2/d*exp(I*(d*x+c))-1/2*a^2/d*exp(-I*(d*x+c))-2/3*a^2*(-15*I*exp(I*(d*x+c))+9*exp(2*I*(d*x+c))-8)
/d/(exp(I*(d*x+c))-I)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (81) = 162\).

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.95 \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=-\frac {3 \, a^{2} \cos \left (d x + c\right )^{3} + 12 \, a^{2} d x - {\left (6 \, a^{2} d x + 11 \, a^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + {\left (6 \, a^{2} d x - 13 \, a^{2}\right )} \cos \left (d x + c\right ) - {\left (12 \, a^{2} d x - 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} + 2 \, {\left (3 \, a^{2} d x - 7 \, a^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3*(3*a^2*cos(d*x + c)^3 + 12*a^2*d*x - (6*a^2*d*x + 11*a^2)*cos(d*x + c)^2 + a^2 + (6*a^2*d*x - 13*a^2)*cos
(d*x + c) - (12*a^2*d*x - 3*a^2*cos(d*x + c)^2 - a^2 + 2*(3*a^2*d*x - 7*a^2)*cos(d*x + c))*sin(d*x + c))/(d*co
s(d*x + c)^2 - d*cos(d*x + c) + (d*cos(d*x + c) + 2*d)*sin(d*x + c) - 2*d)

Sympy [F(-1)]

Timed out. \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**4*sin(d*x+c)**3*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.10 \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{2} - a^{2} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{2}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/3*(2*(tan(d*x + c)^3 + 3*d*x + 3*c - 3*tan(d*x + c))*a^2 - a^2*((6*cos(d*x + c)^2 - 1)/cos(d*x + c)^3 + 3*co
s(d*x + c)) - (3*cos(d*x + c)^2 - 1)*a^2/cos(d*x + c)^3)/d

Giac [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=\frac {2 \, {\left (3 \, {\left (d x + c\right )} a^{2} - \frac {3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{2}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}\right )}}{3 \, d} \]

[In]

integrate(sec(d*x+c)^4*sin(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

2/3*(3*(d*x + c)*a^2 - 3*a^2/(tan(1/2*d*x + 1/2*c)^2 + 1) + (6*a^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2*tan(1/2*d*x
+ 1/2*c) + 7*a^2)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d

Mupad [B] (verification not implemented)

Time = 14.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.12 \[ \int \sec (c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx=2\,a^2\,x+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {2\,a^2\,\left (9\,d\,x-24\right )}{3}-6\,a^2\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {2\,a^2\,\left (9\,d\,x-6\right )}{3}-6\,a^2\,d\,x\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {2\,a^2\,\left (12\,d\,x-18\right )}{3}-8\,a^2\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {2\,a^2\,\left (12\,d\,x-22\right )}{3}-8\,a^2\,d\,x\right )-\frac {2\,a^2\,\left (3\,d\,x-10\right )}{3}+2\,a^2\,d\,x}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

int((sin(c + d*x)^3*(a + a*sin(c + d*x))^2)/cos(c + d*x)^4,x)

[Out]

2*a^2*x + (tan(c/2 + (d*x)/2)*((2*a^2*(9*d*x - 24))/3 - 6*a^2*d*x) - tan(c/2 + (d*x)/2)^4*((2*a^2*(9*d*x - 6))
/3 - 6*a^2*d*x) + tan(c/2 + (d*x)/2)^3*((2*a^2*(12*d*x - 18))/3 - 8*a^2*d*x) - tan(c/2 + (d*x)/2)^2*((2*a^2*(1
2*d*x - 22))/3 - 8*a^2*d*x) - (2*a^2*(3*d*x - 10))/3 + 2*a^2*d*x)/(d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c/2 + (d*
x)/2)^2 + 1))