\(\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx\) [813]

Optimal result

Integrand size = 29, antiderivative size = 77 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=3 a^3 x-\frac {3 a^3 \cos (c+d x)}{d}-\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d} \]

[Out]

3*a^3*x-3*a^3*cos(d*x+c)/d-2*a^5*cos(d*x+c)^3/d/(a-a*sin(d*x+c))^2+1/3*sec(d*x+c)^3*(a+a*sin(d*x+c))^3/d

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2950, 2749, 2759, 2761, 8} \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=-\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}-\frac {3 a^3 \cos (c+d x)}{d}+3 a^3 x+\frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^3}{3 d} \]

[In]

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

[Out]

3*a^3*x - (3*a^3*Cos[c + d*x])/d - (2*a^5*Cos[c + d*x]^3)/(d*(a - a*Sin[c + d*x])^2) + (Sec[c + d*x]^3*(a + a*
Sin[c + d*x])^3)/(3*d)

Rule 8

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

Rule 2749

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

Rule 2759

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[2*g*(g
*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(2*m + p + 1))), x] + Dist[g^2*((p - 1)/(b^2*(2*m +
p + 1))), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, g}, x] && Eq
Q[a^2 - b^2, 0] && LeQ[m, -2] && GtQ[p, 1] && NeQ[2*m + p + 1, 0] &&  !ILtQ[m + p + 1, 0] && IntegersQ[2*m, 2*
p]

Rule 2761

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*((g*Cos[e
+ f*x])^(p - 1)/(b*f*(p - 1))), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]

Rule 2950

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[b*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] - Dist[1/g^2, Int[(g*Cos
[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && E
qQ[m + p + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = \frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}-a^6 \int \frac {\cos ^4(c+d x)}{(a-a \sin (c+d x))^3} \, dx \\ & = -\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}+\left (3 a^4\right ) \int \frac {\cos ^2(c+d x)}{a-a \sin (c+d x)} \, dx \\ & = -\frac {3 a^3 \cos (c+d x)}{d}-\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d}+\left (3 a^3\right ) \int 1 \, dx \\ & = 3 a^3 x-\frac {3 a^3 \cos (c+d x)}{d}-\frac {2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^3}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.34 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (9 \arctan (\tan (c+d x))-15 \sec (c+d x)+4 \sec ^3(c+d x)+6 \sin ^2\left (\frac {1}{2} (c+d x)\right )-9 \tan (c+d x)+4 \tan ^3(c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \]

[In]

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

[Out]

(a^3*(1 + Sin[c + d*x])^3*(9*ArcTan[Tan[c + d*x]] - 15*Sec[c + d*x] + 4*Sec[c + d*x]^3 + 6*Sin[(c + d*x)/2]^2
- 9*Tan[c + d*x] + 4*Tan[c + d*x]^3))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.25

method	result	size
risch	\(3 a^{3} x -\frac {a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {2 \left (-24 i a^{3} {\mathrm e}^{i \left (d x +c \right )}-13 a^{3}+15 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{3} d}\)	\(96\)
parallelrisch	\(\frac {a^{3} \left (9 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) d x -27 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d +36 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) d x +18 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) x d -54 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+27 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d x +58 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 d x -66 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+28\right )}{3 d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}\)	\(166\)
derivativedivides	\(\frac {a^{3} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\)	\(184\)
default	\(\frac {a^{3} \left (\frac {\sin ^{6}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{6}\left (d x +c \right )}{\cos \left (d x +c \right )}-\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )+3 a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3 \cos \left (d x +c \right )^{3}}}{d}\)	\(184\)
norman	\(\frac {-3 a^{3} x +\frac {28 a^{3}}{3 d}+\frac {6 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {14 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {44 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {128 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {44 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {14 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {6 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+9 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {12 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {40 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\)	\(267\)

[In]

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

[Out]

3*a^3*x-1/2*a^3/d*exp(I*(d*x+c))-1/2*a^3/d*exp(-I*(d*x+c))-2/3*(-24*I*a^3*exp(I*(d*x+c))-13*a^3+15*a^3*exp(2*I
*(d*x+c)))/(exp(I*(d*x+c))-I)^3/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 169 vs. \(2 (76) = 152\).

Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.19 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=-\frac {3 \, a^{3} \cos \left (d x + c\right )^{3} + 18 \, a^{3} d x + 2 \, a^{3} - {\left (9 \, a^{3} d x + 16 \, a^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, a^{3} d x - 17 \, a^{3}\right )} \cos \left (d x + c\right ) - {\left (18 \, a^{3} d x - 3 \, a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3} + {\left (9 \, a^{3} d x - 19 \, a^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]

[In]

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

[Out]

-1/3*(3*a^3*cos(d*x + c)^3 + 18*a^3*d*x + 2*a^3 - (9*a^3*d*x + 16*a^3)*cos(d*x + c)^2 + (9*a^3*d*x - 17*a^3)*c
os(d*x + c) - (18*a^3*d*x - 3*a^3*cos(d*x + c)^2 - 2*a^3 + (9*a^3*d*x - 19*a^3)*cos(d*x + c))*sin(d*x + c))/(d
*cos(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. \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.39 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=\frac {a^{3} \tan \left (d x + c\right )^{3} + 3 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{3} - a^{3} {\left (\frac {6 \, \cos \left (d x + c\right )^{2} - 1}{\cos \left (d x + c\right )^{3}} + 3 \, \cos \left (d x + c\right )\right )} - \frac {3 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} a^{3}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.13 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=\frac {9 \, {\left (d x + c\right )} a^{3} - \frac {6 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {2 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \]

[In]

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

[Out]

1/3*(9*(d*x + c)*a^3 - 6*a^3/(tan(1/2*d*x + 1/2*c)^2 + 1) + 2*(9*a^3*tan(1/2*d*x + 1/2*c)^2 - 24*a^3*tan(1/2*d
*x + 1/2*c) + 11*a^3)/(tan(1/2*d*x + 1/2*c) - 1)^3)/d

Mupad [B] (verification not implemented)

Time = 12.68 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.36 \[ \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^2(c+d x) \, dx=3\,a^3\,x+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3\,\left (27\,d\,x-66\right )}{3}-9\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (27\,d\,x-18\right )}{3}-9\,a^3\,d\,x\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^3\,\left (36\,d\,x-54\right )}{3}-12\,a^3\,d\,x\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (36\,d\,x-58\right )}{3}-12\,a^3\,d\,x\right )-\frac {a^3\,\left (9\,d\,x-28\right )}{3}+3\,a^3\,d\,x}{d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

3*a^3*x + (tan(c/2 + (d*x)/2)*((a^3*(27*d*x - 66))/3 - 9*a^3*d*x) - tan(c/2 + (d*x)/2)^4*((a^3*(27*d*x - 18))/
3 - 9*a^3*d*x) + tan(c/2 + (d*x)/2)^3*((a^3*(36*d*x - 54))/3 - 12*a^3*d*x) - tan(c/2 + (d*x)/2)^2*((a^3*(36*d*
x - 58))/3 - 12*a^3*d*x) - (a^3*(9*d*x - 28))/3 + 3*a^3*d*x)/(d*(tan(c/2 + (d*x)/2) - 1)^3*(tan(c/2 + (d*x)/2)
^2 + 1))