\(\int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [229]

Optimal result

Integrand size = 21, antiderivative size = 333 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {3 \left (a^4-24 a^2 b^2+40 b^4\right ) x}{8 a^7}-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 \sqrt {a-b} \sqrt {a+b} d}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \cos ^2(c+d x) \sin (c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{4 a^3 b^2 d}-\frac {\left (a^2-b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b d (b+a \cos (c+d x))^2}+\frac {\left (2 a^2-7 b^2\right ) \cos ^4(c+d x) \sin (c+d x)}{2 a^2 b^2 d (b+a \cos (c+d x))} \]

[Out]

3/8*(a^4-24*a^2*b^2+40*b^4)*x/a^7+1/2*b*(13*a^2-30*b^2)*sin(d*x+c)/a^6/d-3/8*(7*a^2-20*b^2)*cos(d*x+c)*sin(d*x
+c)/a^5/d+1/2*(3*a^2-10*b^2)*cos(d*x+c)^2*sin(d*x+c)/a^4/b/d-1/4*(4*a^2-15*b^2)*cos(d*x+c)^3*sin(d*x+c)/a^3/b^
2/d-1/2*(a^2-b^2)*cos(d*x+c)^4*sin(d*x+c)/a^2/b/d/(b+a*cos(d*x+c))^2+1/2*(2*a^2-7*b^2)*cos(d*x+c)^4*sin(d*x+c)
/a^2/b^2/d/(b+a*cos(d*x+c))-3*b*(2*a^4-11*a^2*b^2+10*b^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/
a^7/d/(a-b)^(1/2)/(a+b)^(1/2)

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3957, 2970, 3128, 3102, 2814, 2738, 214} \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (2 a^2-7 b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac {\left (a^2-b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}+\frac {b \left (13 a^2-30 b^2\right ) \sin (c+d x)}{2 a^6 d}-\frac {3 \left (7 a^2-20 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^5 d}+\frac {\left (3 a^2-10 b^2\right ) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac {\left (4 a^2-15 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}-\frac {3 b \left (2 a^4-11 a^2 b^2+10 b^4\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^7 d \sqrt {a-b} \sqrt {a+b}}+\frac {3 x \left (a^4-24 a^2 b^2+40 b^4\right )}{8 a^7} \]

[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 214

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

Rule 2738

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 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 2970

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 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]

Rule 3128

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 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 ^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 ) \text {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 ) \text {arctanh}\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1178\) vs. \(2(333)=666\).

Time = 6.86 (sec) , antiderivative size = 1178, normalized size of antiderivative = 3.54 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {-\frac {6 \left (8 (c+d x)+\frac {2 b \left (15 a^4-20 a^2 b^2+8 b^4\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\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}-\frac {3 a \left (2 a^4-7 a^2 b^2+4 b^4\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}\right )}{a^3}+\frac {6 \left (\frac {6 a b \text {arctanh}\left (\frac {(-a+b) \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 (-24 \left (a^2-8 b^2\right ) (c+d x)+\frac {6 b \left (-35 a^6+140 a^4 b^2-168 a^2 b^4+64 b^6\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-96 a b \sin (c+d x)+\frac {a b \left (-5 a^4+20 a^2 b^2-16 b^4\right ) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}+\frac {a \left (10 a^6-115 a^4 b^2+220 a^2 b^4-112 b^6\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+8 a^2 \sin (2 (c+d x))\right )}{a^5}+\frac {\frac {12 b \left (105 a^8-840 a^6 b^2+2016 a^4 b^4-1920 a^2 b^6+640 b^8\right ) \text {arctanh}\left (\frac {(-a+b) \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 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 \left (a^2-b^2\right )^2 \left (a^4-20 a^2 b^2+40 b^4\right ) (c+d x) \cos (c+d x)+48 \left (a^3-a b^2\right )^2 \left (a^4-20 a^2 b^2+40 b^4\right ) (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 \sin (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 \sin (6 (c+d x))}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{a^7}}{256 d} \]

[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)

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+3 a^{3} b -3 a^{2} b^{2}-10 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (13 a^{3} b -3 a^{2} b^{2}-30 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+3 a^{2} b^{2}+13 a^{3} b -30 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{3} b -10 a \,b^{3}-\frac {3}{8} a^{4}+3 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (a^{4}-24 a^{2} b^{2}+40 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{7}}+\frac {2 b \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}-3 a^{4} b +\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}+3 a^{4} b -\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{7}}}{d}\)	\(392\)
default	\(\frac {\frac {\frac {2 \left (\left (\frac {3}{8} a^{4}+3 a^{3} b -3 a^{2} b^{2}-10 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (13 a^{3} b -3 a^{2} b^{2}-30 a \,b^{3}+\frac {11}{8} a^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {11}{8} a^{4}+3 a^{2} b^{2}+13 a^{3} b -30 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (3 a^{3} b -10 a \,b^{3}-\frac {3}{8} a^{4}+3 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {3 \left (a^{4}-24 a^{2} b^{2}+40 b^{4}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{7}}+\frac {2 b \left (\frac {\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}-3 a^{4} b +\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{2} a^{3} b^{2}-5 a \,b^{4}+3 a^{4} b -\frac {11}{2} a^{2} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{2}}-\frac {3 \left (2 a^{4}-11 a^{2} b^{2}+10 b^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{7}}}{d}\)	\(392\)
risch	\(\frac {3 x}{8 a^{3}}-\frac {9 x \,b^{2}}{a^{5}}+\frac {15 x \,b^{4}}{a^{7}}-\frac {i b \,{\mathrm e}^{-3 i \left (d x +c \right )}}{8 a^{4} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{4 a^{5} d}-\frac {15 i b \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{4} d}-\frac {i b^{2} \left (-7 a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+12 a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+22 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-17 a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}+32 b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}-6 a^{4}+11 a^{2} b^{2}\right )}{a^{7} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2}}-\frac {5 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{a^{6} d}+\frac {15 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{4} d}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {i b \,{\mathrm e}^{3 i \left (d x +c \right )}}{8 a^{4} d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{4 a^{5} d}+\frac {5 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{a^{6} d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{3} d}+\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}-\frac {33 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {15 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{7}}-\frac {3 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{3}}+\frac {33 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{2 \sqrt {a^{2}-b^{2}}\, d \,a^{5}}-\frac {15 b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right )}{\sqrt {a^{2}-b^{2}}\, d \,a^{7}}+\frac {\sin \left (4 d x +4 c \right )}{32 d \,a^{3}}\)	\(843\)

[In]

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

[Out]

1/d*(2/a^7*(((3/8*a^4+3*a^3*b-3*a^2*b^2-10*a*b^3)*tan(1/2*d*x+1/2*c)^7+(13*a^3*b-3*a^2*b^2-30*a*b^3+11/8*a^4)*
tan(1/2*d*x+1/2*c)^5+(-11/8*a^4+3*a^2*b^2+13*a^3*b-30*a*b^3)*tan(1/2*d*x+1/2*c)^3+(3*a^3*b-10*a*b^3-3/8*a^4+3*
a^2*b^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^4+3/8*(a^4-24*a^2*b^2+40*b^4)*arctan(tan(1/2*d*x+1/2*c))
)+2*b/a^7*(((5/2*a^3*b^2-5*a*b^4-3*a^4*b+11/2*a^2*b^3)*tan(1/2*d*x+1/2*c)^3+(5/2*a^3*b^2-5*a*b^4+3*a^4*b-11/2*
a^2*b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^2-3/2*(2*a^4-11*a^2*b^2+10*b^
4)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 1041, normalized size of antiderivative = 3.13 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]

[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)]

Sympy [F]

\[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\int \frac {\sin ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.75 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{4} - 24 \, a^{2} b^{2} + 40 \, b^{4}\right )} {\left (d x + c\right )}}{a^{7}} - \frac {24 \, {\left (2 \, a^{4} b - 11 \, a^{2} b^{3} + 10 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{7}} - \frac {8 \, {\left (6 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 11 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2} a^{6}} + \frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 80 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 104 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 80 \, b^{3} \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 )}^{4} a^{6}}}{8 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 18.29 (sec) , antiderivative size = 3255, normalized size of antiderivative = 9.77 \[ \int \frac {\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((((3*((108*a^19*b - 12*a^20 - 480*a^14*b^6 + 720*a^15*b^5 + 288*a^16*b^4 - 732*a^17*b^3 + 108*a^18*b^2)
/a^18 - (3*tan(c/2 + (d*x)/2)*(a^4*1i + b^4*40i - a^2*b^2*24i)*(128*a^16*b + 128*a^14*b^3 - 256*a^15*b^2))/(16
*a^19))*(a^4*1i + b^4*40i - a^2*b^2*24i))/(8*a^7) + (tan(c/2 + (d*x)/2)*(57600*a*b^10 - 27*a^10*b + 9*a^11 - 2
8800*b^11 + 5760*a^2*b^9 - 69120*a^3*b^8 + 22752*a^4*b^7 + 23616*a^5*b^6 - 10944*a^6*b^5 - 1728*a^7*b^4 + 711*
a^8*b^3 + 171*a^9*b^2))/(2*a^12))*(a^4*1i + b^4*40i - a^2*b^2*24i)*3i)/(8*a^7) - (((3*((108*a^19*b - 12*a^20 -
480*a^14*b^6 + 720*a^15*b^5 + 288*a^16*b^4 - 732*a^17*b^3 + 108*a^18*b^2)/a^18 + (3*tan(c/2 + (d*x)/2)*(a^4*1
i + b^4*40i - a^2*b^2*24i)*(128*a^16*b + 128*a^14*b^3 - 256*a^15*b^2))/(16*a^19))*(a^4*1i + b^4*40i - a^2*b^2*
24i))/(8*a^7) - (tan(c/2 + (d*x)/2)*(57600*a*b^10 - 27*a^10*b + 9*a^11 - 28800*b^11 + 5760*a^2*b^9 - 69120*a^3
*b^8 + 22752*a^4*b^7 + 23616*a^5*b^6 - 10944*a^6*b^5 - 1728*a^7*b^4 + 711*a^8*b^3 + 171*a^9*b^2))/(2*a^12))*(a
^4*1i + b^4*40i - a^2*b^2*24i)*3i)/(8*a^7))/((324000*a*b^13 + 27*a^13*b - 216000*b^14 + 388800*a^2*b^12 - 7182
00*a^3*b^11 - 195480*a^4*b^10 + 576720*a^5*b^9 - 4104*a^6*b^8 - 205119*a^7*b^7 + 24408*a^8*b^6 + (62181*a^9*b^
5)/2 - 4671*a^10*b^4 - (3267*a^11*b^3)/2 + 162*a^12*b^2)/a^18 + (3*((3*((108*a^19*b - 12*a^20 - 480*a^14*b^6 +
720*a^15*b^5 + 288*a^16*b^4 - 732*a^17*b^3 + 108*a^18*b^2)/a^18 - (3*tan(c/2 + (d*x)/2)*(a^4*1i + b^4*40i - a
^2*b^2*24i)*(128*a^16*b + 128*a^14*b^3 - 256*a^15*b^2))/(16*a^19))*(a^4*1i + b^4*40i - a^2*b^2*24i))/(8*a^7) +
(tan(c/2 + (d*x)/2)*(57600*a*b^10 - 27*a^10*b + 9*a^11 - 28800*b^11 + 5760*a^2*b^9 - 69120*a^3*b^8 + 22752*a^
4*b^7 + 23616*a^5*b^6 - 10944*a^6*b^5 - 1728*a^7*b^4 + 711*a^8*b^3 + 171*a^9*b^2))/(2*a^12))*(a^4*1i + b^4*40i
- a^2*b^2*24i))/(8*a^7) + (3*((3*((108*a^19*b - 12*a^20 - 480*a^14*b^6 + 720*a^15*b^5 + 288*a^16*b^4 - 732*a^
17*b^3 + 108*a^18*b^2)/a^18 + (3*tan(c/2 + (d*x)/2)*(a^4*1i + b^4*40i - a^2*b^2*24i)*(128*a^16*b + 128*a^14*b^
3 - 256*a^15*b^2))/(16*a^19))*(a^4*1i + b^4*40i - a^2*b^2*24i))/(8*a^7) - (tan(c/2 + (d*x)/2)*(57600*a*b^10 -
27*a^10*b + 9*a^11 - 28800*b^11 + 5760*a^2*b^9 - 69120*a^3*b^8 + 22752*a^4*b^7 + 23616*a^5*b^6 - 10944*a^6*b^5
- 1728*a^7*b^4 + 711*a^8*b^3 + 171*a^9*b^2))/(2*a^12))*(a^4*1i + b^4*40i - a^2*b^2*24i))/(8*a^7)))*(a^4*1i +
b^4*40i - a^2*b^2*24i)*3i)/(4*a^7*d) - ((tan(c/2 + (d*x)/2)^5*(180*a*b^4 + 26*a^4*b - 15*a^5 + 600*b^5 - 300*a
^2*b^3 - 73*a^3*b^2))/(2*a^6) - (3*tan(c/2 + (d*x)/2)^11*(60*a*b^4 + 6*a^4*b + a^5 - 40*b^5 + 4*a^2*b^3 - 31*a
^3*b^2))/(4*a^6) + (tan(c/2 + (d*x)/2)^7*(26*a^4*b - 180*a*b^4 + 15*a^5 + 600*b^5 - 300*a^2*b^3 + 73*a^3*b^2))
/(2*a^6) + (tan(c/2 + (d*x)/2)^3*(540*a*b^4 - 34*a^4*b + 5*a^5 + 600*b^5 - 220*a^2*b^3 - 239*a^3*b^2))/(4*a^6)
- (tan(c/2 + (d*x)/2)^9*(540*a*b^4 + 34*a^4*b + 5*a^5 - 600*b^5 + 220*a^2*b^3 - 239*a^3*b^2))/(4*a^6) + (3*ta
n(c/2 + (d*x)/2)*(a + b)*(20*a*b^3 - 7*a^3*b + a^4 + 40*b^4 - 24*a^2*b^2))/(4*a^6))/(d*(2*a*b - tan(c/2 + (d*x
)/2)^6*(4*a^2 - 20*b^2) + tan(c/2 + (d*x)/2)^2*(8*a*b + 2*a^2 + 6*b^2) + tan(c/2 + (d*x)/2)^10*(2*a^2 - 8*a*b
+ 6*b^2) + tan(c/2 + (d*x)/2)^4*(10*a*b - a^2 + 15*b^2) + tan(c/2 + (d*x)/2)^12*(a^2 - 2*a*b + b^2) + a^2 + b^
2 - tan(c/2 + (d*x)/2)^8*(10*a*b + a^2 - 15*b^2))) + (b*atan(((b*((a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*
(57600*a*b^10 - 27*a^10*b + 9*a^11 - 28800*b^11 + 5760*a^2*b^9 - 69120*a^3*b^8 + 22752*a^4*b^7 + 23616*a^5*b^6
- 10944*a^6*b^5 - 1728*a^7*b^4 + 711*a^8*b^3 + 171*a^9*b^2))/(2*a^12) + (3*b*((a + b)*(a - b))^(1/2)*((108*a^
19*b - 12*a^20 - 480*a^14*b^6 + 720*a^15*b^5 + 288*a^16*b^4 - 732*a^17*b^3 + 108*a^18*b^2)/a^18 - (3*b*tan(c/2
+ (d*x)/2)*((a + b)*(a - b))^(1/2)*(2*a^4 + 10*b^4 - 11*a^2*b^2)*(128*a^16*b + 128*a^14*b^3 - 256*a^15*b^2))/
(4*a^12*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4 - 11*a^2*b^2))/(2*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4 - 11*a^2*b^2)*3i
)/(2*(a^9 - a^7*b^2)) + (b*((a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(57600*a*b^10 - 27*a^10*b + 9*a^11 - 2
8800*b^11 + 5760*a^2*b^9 - 69120*a^3*b^8 + 22752*a^4*b^7 + 23616*a^5*b^6 - 10944*a^6*b^5 - 1728*a^7*b^4 + 711*
a^8*b^3 + 171*a^9*b^2))/(2*a^12) - (3*b*((a + b)*(a - b))^(1/2)*((108*a^19*b - 12*a^20 - 480*a^14*b^6 + 720*a^
15*b^5 + 288*a^16*b^4 - 732*a^17*b^3 + 108*a^18*b^2)/a^18 + (3*b*tan(c/2 + (d*x)/2)*((a + b)*(a - b))^(1/2)*(2
*a^4 + 10*b^4 - 11*a^2*b^2)*(128*a^16*b + 128*a^14*b^3 - 256*a^15*b^2))/(4*a^12*(a^9 - a^7*b^2)))*(2*a^4 + 10*
b^4 - 11*a^2*b^2))/(2*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4 - 11*a^2*b^2)*3i)/(2*(a^9 - a^7*b^2)))/((324000*a*b^13
+ 27*a^13*b - 216000*b^14 + 388800*a^2*b^12 - 718200*a^3*b^11 - 195480*a^4*b^10 + 576720*a^5*b^9 - 4104*a^6*b
^8 - 205119*a^7*b^7 + 24408*a^8*b^6 + (62181*a^9*b^5)/2 - 4671*a^10*b^4 - (3267*a^11*b^3)/2 + 162*a^12*b^2)/a^
18 + (3*b*((a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(57600*a*b^10 - 27*a^10*b + 9*a^11 - 28800*b^11 + 5760*
a^2*b^9 - 69120*a^3*b^8 + 22752*a^4*b^7 + 23616*a^5*b^6 - 10944*a^6*b^5 - 1728*a^7*b^4 + 711*a^8*b^3 + 171*a^9
*b^2))/(2*a^12) + (3*b*((a + b)*(a - b))^(1/2)*((108*a^19*b - 12*a^20 - 480*a^14*b^6 + 720*a^15*b^5 + 288*a^16
*b^4 - 732*a^17*b^3 + 108*a^18*b^2)/a^18 - (3*b*tan(c/2 + (d*x)/2)*((a + b)*(a - b))^(1/2)*(2*a^4 + 10*b^4 - 1
1*a^2*b^2)*(128*a^16*b + 128*a^14*b^3 - 256*a^15*b^2))/(4*a^12*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4 - 11*a^2*b^2)
)/(2*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4 - 11*a^2*b^2))/(2*(a^9 - a^7*b^2)) - (3*b*((a + b)*(a - b))^(1/2)*((tan
(c/2 + (d*x)/2)*(57600*a*b^10 - 27*a^10*b + 9*a^11 - 28800*b^11 + 5760*a^2*b^9 - 69120*a^3*b^8 + 22752*a^4*b^7
+ 23616*a^5*b^6 - 10944*a^6*b^5 - 1728*a^7*b^4 + 711*a^8*b^3 + 171*a^9*b^2))/(2*a^12) - (3*b*((a + b)*(a - b)
)^(1/2)*((108*a^19*b - 12*a^20 - 480*a^14*b^6 + 720*a^15*b^5 + 288*a^16*b^4 - 732*a^17*b^3 + 108*a^18*b^2)/a^1
8 + (3*b*tan(c/2 + (d*x)/2)*((a + b)*(a - b))^(1/2)*(2*a^4 + 10*b^4 - 11*a^2*b^2)*(128*a^16*b + 128*a^14*b^3 -
256*a^15*b^2))/(4*a^12*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4 - 11*a^2*b^2))/(2*(a^9 - a^7*b^2)))*(2*a^4 + 10*b^4
- 11*a^2*b^2))/(2*(a^9 - a^7*b^2))))*((a + b)*(a - b))^(1/2)*(2*a^4 + 10*b^4 - 11*a^2*b^2)*3i)/(d*(a^9 - a^7*b
^2))