3.229 \(\int \frac{\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=333 \[ \frac{b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}+\frac{\left (2 a^2-7 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac{\left (a^2-b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}-\frac{\left (4 a^2-15 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}+\frac{\left (3 a^2-10 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac{3 \left (7 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^5 d}-\frac{3 b \left (-11 a^2 b^2+2 a^4+10 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^7 d \sqrt{a-b} \sqrt{a+b}}+\frac{3 x \left (-24 a^2 b^2+a^4+40 b^4\right )}{8 a^7} \]

[Out]

(3*(a^4 - 24*a^2*b^2 + 40*b^4)*x)/(8*a^7) - (3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d
*x)/2])/Sqrt[a + b]])/(a^7*Sqrt[a - b]*Sqrt[a + b]*d) + (b*(13*a^2 - 30*b^2)*Sin[c + d*x])/(2*a^6*d) - (3*(7*a
^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^5*d) + ((3*a^2 - 10*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^4*b*d)
- ((4*a^2 - 15*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2
*a^2*b*d*(b + a*Cos[c + d*x])^2) + ((2*a^2 - 7*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2*a^2*b^2*d*(b + a*Cos[c + d
*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.13839, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3872, 2891, 3049, 3023, 2735, 2659, 208} \[ \frac{b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}+\frac{\left (2 a^2-7 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac{\left (a^2-b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}-\frac{\left (4 a^2-15 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}+\frac{\left (3 a^2-10 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac{3 \left (7 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^5 d}-\frac{3 b \left (-11 a^2 b^2+2 a^4+10 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^7 d \sqrt{a-b} \sqrt{a+b}}+\frac{3 x \left (-24 a^2 b^2+a^4+40 b^4\right )}{8 a^7} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]

[Out]

(3*(a^4 - 24*a^2*b^2 + 40*b^4)*x)/(8*a^7) - (3*b*(2*a^4 - 11*a^2*b^2 + 10*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d
*x)/2])/Sqrt[a + b]])/(a^7*Sqrt[a - b]*Sqrt[a + b]*d) + (b*(13*a^2 - 30*b^2)*Sin[c + d*x])/(2*a^6*d) - (3*(7*a
^2 - 20*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*a^5*d) + ((3*a^2 - 10*b^2)*Cos[c + d*x]^2*Sin[c + d*x])/(2*a^4*b*d)
- ((4*a^2 - 15*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a^3*b^2*d) - ((a^2 - b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2
*a^2*b*d*(b + a*Cos[c + d*x])^2) + ((2*a^2 - 7*b^2)*Cos[c + d*x]^4*Sin[c + d*x])/(2*a^2*b^2*d*(b + a*Cos[c + d
*x]))

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 2891

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[((a^2 - b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^(n + 1))/(a*b^2*d*
f*(m + 1)), x] + (-Dist[1/(a^2*b^2*(m + 1)*(m + 2)), Int[(a + b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^n*Simp[
a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m +
n + 2)*(m + n + 4))*Sin[e + f*x]^2, x], x], x] + Simp[((a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a +
b*Sin[e + f*x])^(m + 2)*(d*Sin[e + f*x])^(n + 1))/(a^2*b^2*d*f*(m + 1)*(m + 2)), x]) /; FreeQ[{a, b, d, e, f,
n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m, -2] || EqQ[m + n +
4, 0])

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

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]

Rule 2735

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 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac{\cos ^3(c+d x) \sin ^4(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac{\int \frac{\cos ^3(c+d x) \left (-6 \left (a^2-4 b^2\right )-a b \cos (c+d x)+2 \left (4 a^2-15 b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{2 a^2 b^2}\\ &=-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}-\frac{\int \frac{\cos ^2(c+d x) \left (-6 b \left (4 a^2-15 b^2\right )-6 a b^2 \cos (c+d x)+12 b \left (3 a^2-10 b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{8 a^3 b^2}\\ &=\frac{\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac{\int \frac{\cos (c+d x) \left (-24 b^2 \left (3 a^2-10 b^2\right )-30 a b^3 \cos (c+d x)+18 b^2 \left (7 a^2-20 b^2\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{24 a^4 b^2}\\ &=-\frac{3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac{\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}-\frac{\int \frac{-18 b^3 \left (7 a^2-20 b^2\right )+6 a b^2 \left (3 a^2-20 b^2\right ) \cos (c+d x)+24 b^3 \left (13 a^2-30 b^2\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{48 a^5 b^2}\\ &=\frac{b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac{3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac{\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac{\int \frac{18 a b^3 \left (7 a^2-20 b^2\right )-18 b^2 \left (a^4-24 a^2 b^2+40 b^4\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{48 a^6 b^2}\\ &=\frac{3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}+\frac{b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac{3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac{\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac{\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\right )\right ) \int \frac{1}{-b-a \cos (c+d x)} \, dx}{2 a^7}\\ &=\frac{3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}+\frac{b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac{3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac{\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}+\frac{\left (3 b \left (2 a^4-11 a^2 b^2+10 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^7 d}\\ &=\frac{3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}-\frac{3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^7 \sqrt{a-b} \sqrt{a+b} d}+\frac{b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac{3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac{\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac{\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac{\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac{\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))}\\ \end{align*}

Mathematica [B]  time = 9.26501, size = 1178, normalized size = 3.54 \[ \frac{-\frac{6 \left (8 (c+d x)+\frac{2 b \left (15 a^4-20 b^2 a^2+8 b^4\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac{3 a \left (2 a^4-7 b^2 a^2+4 b^4\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+\frac{a b \left (3 a^2-4 b^2\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}\right )}{a^3}+\frac{6 \left (\frac{6 a b \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+\frac{\left (b \left (a^2+2 b^2\right )+a \left (2 a^2+b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{(b+a \cos (c+d x))^2}\right )}{(a-b)^2 (a+b)^2}-\frac{2 \left (8 \sin (2 (c+d x)) a^2-96 b \sin (c+d x) a+\frac{\left (10 a^6-115 b^2 a^4+220 b^4 a^2-112 b^6\right ) \sin (c+d x) a}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+\frac{b \left (-5 a^4+20 b^2 a^2-16 b^4\right ) \sin (c+d x) a}{(a-b) (a+b) (b+a \cos (c+d x))^2}-24 \left (a^2-8 b^2\right ) (c+d x)+\frac{6 b \left (-35 a^6+140 b^2 a^4-168 b^4 a^2+64 b^6\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}\right )}{a^5}+\frac{\frac{12 b \left (105 a^8-840 b^2 a^6+2016 b^4 a^4-1920 b^6 a^2+640 b^8\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac{48 c a^{10}+48 d x a^{10}-36 \sin (2 (c+d x)) a^{10}-8 \sin (4 (c+d x)) a^{10}+2 \sin (6 (c+d x)) a^{10}+114 b \sin (c+d x) a^9+120 b \sin (3 (c+d x)) a^9-8 b \sin (5 (c+d x)) a^9-960 b^2 c a^8-960 b^2 d x a^8+1221 b^2 \sin (2 (c+d x)) a^8+56 b^2 \sin (4 (c+d x)) a^8-4 b^2 \sin (6 (c+d x)) a^8+788 b^3 \sin (c+d x) a^7-560 b^3 \sin (3 (c+d x)) a^7+16 b^3 \sin (5 (c+d x)) a^7+1776 b^4 c a^6+1776 b^4 d x a^6-5182 b^4 \sin (2 (c+d x)) a^6-88 b^4 \sin (4 (c+d x)) a^6+2 b^4 \sin (6 (c+d x)) a^6-5696 b^5 \sin (c+d x) a^5+760 b^5 \sin (3 (c+d x)) a^5-8 b^5 \sin (5 (c+d x)) a^5+2976 b^6 c a^4+2976 b^6 d x a^4+6880 b^6 \sin (2 (c+d x)) a^4+40 b^6 \sin (4 (c+d x)) a^4+8640 b^7 \sin (c+d x) a^3-320 b^7 \sin (3 (c+d x)) a^3-7680 b^8 c a^2-7680 b^8 d x a^2-2880 b^8 \sin (2 (c+d x)) a^2+192 b \left (a^2-b^2\right )^2 \left (a^4-20 b^2 a^2+40 b^4\right ) (c+d x) \cos (c+d x) a-3840 b^9 \sin (c+d x) a+3840 b^{10} c+3840 b^{10} d x+48 \left (a^3-a b^2\right )^2 \left (a^4-20 b^2 a^2+40 b^4\right ) (c+d x) \cos (2 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{a^7}}{256 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]

[Out]

((-6*(8*(c + d*x) + (2*b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(
a^2 - b^2)^(5/2) + (a*b*(3*a^2 - 4*b^2)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) - (3*a*(2*a^4 -
7*a^2*b^2 + 4*b^4)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x]))))/a^3 + (6*((6*a*b*ArcTanh[((-a +
b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + ((b*(a^2 + 2*b^2) + a*(2*a^2 + b^2)*Cos[c + d*x])*Si
n[c + d*x])/(b + a*Cos[c + d*x])^2))/((a - b)^2*(a + b)^2) - (2*(-24*(a^2 - 8*b^2)*(c + d*x) + (6*b*(-35*a^6 +
140*a^4*b^2 - 168*a^2*b^4 + 64*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) -
96*a*b*Sin[c + d*x] + (a*b*(-5*a^4 + 20*a^2*b^2 - 16*b^4)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])
^2) + (a*(10*a^6 - 115*a^4*b^2 + 220*a^2*b^4 - 112*b^6)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x]
)) + 8*a^2*Sin[2*(c + d*x)]))/a^5 + ((12*b*(105*a^8 - 840*a^6*b^2 + 2016*a^4*b^4 - 1920*a^2*b^6 + 640*b^8)*Arc
Tanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (48*a^10*c - 960*a^8*b^2*c + 1776*a^6*b
^4*c + 2976*a^4*b^6*c - 7680*a^2*b^8*c + 3840*b^10*c + 48*a^10*d*x - 960*a^8*b^2*d*x + 1776*a^6*b^4*d*x + 2976
*a^4*b^6*d*x - 7680*a^2*b^8*d*x + 3840*b^10*d*x + 192*a*b*(a^2 - b^2)^2*(a^4 - 20*a^2*b^2 + 40*b^4)*(c + d*x)*
Cos[c + d*x] + 48*(a^3 - a*b^2)^2*(a^4 - 20*a^2*b^2 + 40*b^4)*(c + d*x)*Cos[2*(c + d*x)] + 114*a^9*b*Sin[c + d
*x] + 788*a^7*b^3*Sin[c + d*x] - 5696*a^5*b^5*Sin[c + d*x] + 8640*a^3*b^7*Sin[c + d*x] - 3840*a*b^9*Sin[c + d*
x] - 36*a^10*Sin[2*(c + d*x)] + 1221*a^8*b^2*Sin[2*(c + d*x)] - 5182*a^6*b^4*Sin[2*(c + d*x)] + 6880*a^4*b^6*S
in[2*(c + d*x)] - 2880*a^2*b^8*Sin[2*(c + d*x)] + 120*a^9*b*Sin[3*(c + d*x)] - 560*a^7*b^3*Sin[3*(c + d*x)] +
760*a^5*b^5*Sin[3*(c + d*x)] - 320*a^3*b^7*Sin[3*(c + d*x)] - 8*a^10*Sin[4*(c + d*x)] + 56*a^8*b^2*Sin[4*(c +
d*x)] - 88*a^6*b^4*Sin[4*(c + d*x)] + 40*a^4*b^6*Sin[4*(c + d*x)] - 8*a^9*b*Sin[5*(c + d*x)] + 16*a^7*b^3*Sin[
5*(c + d*x)] - 8*a^5*b^5*Sin[5*(c + d*x)] + 2*a^10*Sin[6*(c + d*x)] - 4*a^8*b^2*Sin[6*(c + d*x)] + 2*a^6*b^4*S
in[6*(c + d*x)])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/a^7)/(256*d)

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Maple [B]  time = 0.086, size = 1227, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^4/(a+b*sec(d*x+c))^3,x)

[Out]

-18/d/a^5*arctan(tan(1/2*d*x+1/2*c))*b^2+30/d/a^7*arctan(tan(1/2*d*x+1/2*c))*b^4+11/4/d/a^3/(1+tan(1/2*d*x+1/2
*c)^2)^4*tan(1/2*d*x+1/2*c)^5-11/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3-3/4/d/a^3/(1+tan(1/2*
d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)+3/4/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7+11/d*b^4/a^5/(tan
(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*tan(1/2*d*x+1/2*c)^3+6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(
1/2*d*x+1/2*c)^7*b+26/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*b-60/d/a^6/(1+tan(1/2*d*x+1/2*c)^2
)^4*tan(1/2*d*x+1/2*c)^3*b^3+6/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*b-20/d/a^6/(1+tan(1/2*d*x+1
/2*c)^2)^4*tan(1/2*d*x+1/2*c)*b^3+6/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*b^2-6/d/a^5/(1+tan(1/2
*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*b^2-20/d/a^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*b^3+26/d/a^
4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*b-6/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*b^
2-60/d/a^6/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*b^3+6/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+
1/2*c)^3*b^2+3/4/d/a^3*arctan(tan(1/2*d*x+1/2*c))+5/d*b^3/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a
-b)^2*tan(1/2*d*x+1/2*c)^3-10/d*b^5/a^6/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*tan(1/2*d*x+1/2*
c)^3+5/d*b^3/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*tan(1/2*d*x+1/2*c)-10/d*b^5/a^6/(tan(1/
2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*tan(1/2*d*x+1/2*c)+6/d*b^2/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*
d*x+1/2*c)^2*b-a-b)^2*tan(1/2*d*x+1/2*c)-11/d*b^4/a^5/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2*ta
n(1/2*d*x+1/2*c)-6/d*b/a^3/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+33/d*b^3/
a^5/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))-30/d*b^5/a^7/((a+b)*(a-b))^(1/2)
*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))-6/d*b^2/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b-a-b)^2*tan(1/2*d*x+1/2*c)^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.68988, size = 2331, normalized size = 7. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*(a^8 - 25*a^6*b^2 + 64*a^4*b^4 - 40*a^2*b^6)*d*x*cos(d*x + c)^2 + 6*(a^7*b - 25*a^5*b^3 + 64*a^3*b^5 -
40*a*b^7)*d*x*cos(d*x + c) + 3*(a^6*b^2 - 25*a^4*b^4 + 64*a^2*b^6 - 40*b^8)*d*x + 6*(2*a^4*b^3 - 11*a^2*b^5 +
10*b^7 + (2*a^6*b - 11*a^4*b^3 + 10*a^2*b^5)*cos(d*x + c)^2 + 2*(2*a^5*b^2 - 11*a^3*b^4 + 10*a*b^6)*cos(d*x +
c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c
) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (52*a^5*b^3 - 172*a^3*b^
5 + 120*a*b^7 + 2*(a^8 - a^6*b^2)*cos(d*x + c)^5 - 4*(a^7*b - a^5*b^3)*cos(d*x + c)^4 - 5*(a^8 - 3*a^6*b^2 + 2
*a^4*b^4)*cos(d*x + c)^3 + 2*(11*a^7*b - 31*a^5*b^3 + 20*a^3*b^5)*cos(d*x + c)^2 + (83*a^6*b^2 - 263*a^4*b^4 +
180*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - a^9*b^2)*d*cos(d*x + c)^2 + 2*(a^10*b - a^8*b^3)*d*cos(d*x
+ c) + (a^9*b^2 - a^7*b^4)*d), 1/8*(3*(a^8 - 25*a^6*b^2 + 64*a^4*b^4 - 40*a^2*b^6)*d*x*cos(d*x + c)^2 + 6*(a^7
*b - 25*a^5*b^3 + 64*a^3*b^5 - 40*a*b^7)*d*x*cos(d*x + c) + 3*(a^6*b^2 - 25*a^4*b^4 + 64*a^2*b^6 - 40*b^8)*d*x
- 12*(2*a^4*b^3 - 11*a^2*b^5 + 10*b^7 + (2*a^6*b - 11*a^4*b^3 + 10*a^2*b^5)*cos(d*x + c)^2 + 2*(2*a^5*b^2 - 1
1*a^3*b^4 + 10*a*b^6)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2
)*sin(d*x + c))) + (52*a^5*b^3 - 172*a^3*b^5 + 120*a*b^7 + 2*(a^8 - a^6*b^2)*cos(d*x + c)^5 - 4*(a^7*b - a^5*b
^3)*cos(d*x + c)^4 - 5*(a^8 - 3*a^6*b^2 + 2*a^4*b^4)*cos(d*x + c)^3 + 2*(11*a^7*b - 31*a^5*b^3 + 20*a^3*b^5)*c
os(d*x + c)^2 + (83*a^6*b^2 - 263*a^4*b^4 + 180*a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - a^9*b^2)*d*cos(d
*x + c)^2 + 2*(a^10*b - a^8*b^3)*d*cos(d*x + c) + (a^9*b^2 - a^7*b^4)*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)**4/(a+b*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.49874, size = 788, normalized size = 2.37 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*(a^4 - 24*a^2*b^2 + 40*b^4)*(d*x + c)/a^7 - 24*(2*a^4*b - 11*a^2*b^3 + 10*b^5)*(pi*floor(1/2*(d*x + c)/
pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/(sqrt
(-a^2 + b^2)*a^7) - 8*(6*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 - 11*a*b^4*tan(1/2*
d*x + 1/2*c)^3 + 10*b^5*tan(1/2*d*x + 1/2*c)^3 - 6*a^3*b^2*tan(1/2*d*x + 1/2*c) - 5*a^2*b^3*tan(1/2*d*x + 1/2*
c) + 11*a*b^4*tan(1/2*d*x + 1/2*c) + 10*b^5*tan(1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x +
1/2*c)^2 - a - b)^2*a^6) + 2*(3*a^3*tan(1/2*d*x + 1/2*c)^7 + 24*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 24*a*b^2*tan(1
/2*d*x + 1/2*c)^7 - 80*b^3*tan(1/2*d*x + 1/2*c)^7 + 11*a^3*tan(1/2*d*x + 1/2*c)^5 + 104*a^2*b*tan(1/2*d*x + 1/
2*c)^5 - 24*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 240*b^3*tan(1/2*d*x + 1/2*c)^5 - 11*a^3*tan(1/2*d*x + 1/2*c)^3 + 10
4*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 24*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 240*b^3*tan(1/2*d*x + 1/2*c)^3 - 3*a^3*tan(
1/2*d*x + 1/2*c) + 24*a^2*b*tan(1/2*d*x + 1/2*c) + 24*a*b^2*tan(1/2*d*x + 1/2*c) - 80*b^3*tan(1/2*d*x + 1/2*c)
)/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^6))/d