\(\int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [73]

Optimal result

Integrand size = 21, antiderivative size = 366 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right ) x}{8 \left (a^2+b^2\right )^6}+\frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^4 b}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^3}-\frac {a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d} \]

[Out]

1/8*(3*a^8-132*a^6*b^2+370*a^4*b^4-132*a^2*b^6+3*b^8)*x/(a^2+b^2)^6+4*a*b*(a^2-b^2)*(a^4-8*a^2*b^2+b^4)*ln(a*c
os(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^6/d-1/3*a^4*b/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^3-a^3*b*(a^2-2*b^2)/(a^2+b^2)^4
/d/(a+b*tan(d*x+c))^2-3*a^2*b*(a^4-5*a^2*b^2+2*b^4)/(a^2+b^2)^5/d/(a+b*tan(d*x+c))+1/4*cos(d*x+c)^4*(4*a*b*(a^
2-b^2)+(a^4-6*a^2*b^2+b^4)*tan(d*x+c))/(a^2+b^2)^4/d-1/8*cos(d*x+c)^2*(16*a*b*(2*a^4-5*a^2*b^2+b^4)+(5*a^6-65*
a^4*b^2+55*a^2*b^4-3*b^6)*tan(d*x+c))/(a^2+b^2)^5/d

Rubi [A] (verified)

Time = 2.43 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3597, 1661, 1643, 649, 209, 266} \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {a^4 b}{3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3}-\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )}{d \left (a^2+b^2\right )^5 (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 d \left (a^2+b^2\right )^4}+\frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^6}-\frac {a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^5}+\frac {x \left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right )}{8 \left (a^2+b^2\right )^6} \]

[In]

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

[Out]

((3*a^8 - 132*a^6*b^2 + 370*a^4*b^4 - 132*a^2*b^6 + 3*b^8)*x)/(8*(a^2 + b^2)^6) + (4*a*b*(a^2 - b^2)*(a^4 - 8*
a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^6*d) - (a^4*b)/(3*(a^2 + b^2)^3*d*(a + b*Tan
[c + d*x])^3) - (a^3*b*(a^2 - 2*b^2))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])^2) - (3*a^2*b*(a^4 - 5*a^2*b^2 + 2
*b^4))/((a^2 + b^2)^5*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*T
an[c + d*x]))/(4*(a^2 + b^2)^4*d) - (Cos[c + d*x]^2*(16*a*b*(2*a^4 - 5*a^2*b^2 + b^4) + (5*a^6 - 65*a^4*b^2 +
55*a^2*b^4 - 3*b^6)*Tan[c + d*x]))/(8*(a^2 + b^2)^5*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 3597

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {x^4}{(a+x)^4 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^4 \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}-\frac {4 a^3 b^4 \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {2 a^2 b^2 \left (2 a^6+17 a^4 b^2-12 a^2 b^4-3 b^6\right ) x^2}{\left (a^2+b^2\right )^4}-\frac {4 a b^4 \left (3 a^4-14 a^2 b^2-b^4\right ) x^3}{\left (a^2+b^2\right )^4}-\frac {3 b^4 \left (a^4-6 a^2 b^2+b^4\right ) x^4}{\left (a^2+b^2\right )^4}}{(a+x)^4 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d} \\ & = \frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^4 \left (3 a^6-55 a^4 b^2+65 a^2 b^4-5 b^6\right )}{\left (a^2+b^2\right )^5}-\frac {4 a^3 b^4 \left (a^2+5 b^2\right ) \left (5 a^4-10 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^5}-\frac {30 a^2 b^4 \left (a^4-6 a^2 b^2+b^4\right ) x^2}{\left (a^2+b^2\right )^4}-\frac {4 a b^4 \left (5 a^2+b^2\right ) \left (a^4-10 a^2 b^2+5 b^4\right ) x^3}{\left (a^2+b^2\right )^5}-\frac {b^4 \left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) x^4}{\left (a^2+b^2\right )^5}}{(a+x)^4 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d}+\frac {\text {Subst}\left (\int \left (\frac {8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^4}+\frac {16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)^3}+\frac {24 a^2 b^4 \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 (a+x)^2}+\frac {32 a b^4 \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^6 (a+x)}+\frac {b^4 \left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8-32 a \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) x\right )}{\left (a^2+b^2\right )^6 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d} \\ & = \frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^4 b}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^3}-\frac {a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d}+\frac {b \text {Subst}\left (\int \frac {3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8-32 a \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^6 d} \\ & = \frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^4 b}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^3}-\frac {a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d}-\frac {\left (4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^6 d}+\frac {\left (b \left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^6 d} \\ & = \frac {\left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right ) x}{8 \left (a^2+b^2\right )^6}+\frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^6 d}+\frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^4 b}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^3}-\frac {a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.76 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {b \left (-\frac {9 \left (a^2+b^2\right )^2 \left (a^4-6 a^2 b^2+b^4\right ) \arctan (\tan (c+d x))}{b}+\frac {24 a^2 \left (a^2+b^2\right ) \left (a^4-10 a^2 b^2+5 b^4\right ) \arctan (\tan (c+d x))}{b}+48 a \left (a^2+b^2\right ) \left (2 a^4-5 a^2 b^2+b^4\right ) \cos ^2(c+d x)-24 a (a-b) (a+b) \left (a^2+b^2\right )^2 \cos ^4(c+d x)+12 a \left (4 a^6-36 a^4 b^2+36 a^2 b^4-4 b^6+\frac {-a^7+24 a^5 b^2-45 a^3 b^4+10 a b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )-96 a (a-b) (a+b) \left (a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))+12 a \left (4 a^6-36 a^4 b^2+36 a^2 b^4-4 b^6+\frac {a^7-24 a^5 b^2+45 a^3 b^4-10 a b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )-\frac {6 \left (a^2+b^2\right )^2 \left (a^4-6 a^2 b^2+b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{b}-\frac {9 \left (a^2+b^2\right )^2 \left (a^4-6 a^2 b^2+b^4\right ) \sin (2 (c+d x))}{2 b}+\frac {12 a^2 \left (a^2+b^2\right ) \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (2 (c+d x))}{b}+\frac {8 a^4 \left (a^2+b^2\right )^3}{(a+b \tan (c+d x))^3}+\frac {24 a^3 \left (a^2-2 b^2\right ) \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac {72 a^2 \left (a^2+b^2\right ) \left (a^4-5 a^2 b^2+2 b^4\right )}{a+b \tan (c+d x)}\right )}{24 \left (a^2+b^2\right )^6 d} \]

[In]

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

[Out]

-1/24*(b*((-9*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/b + (24*a^2*(a^2 + b^2)*(a^4 - 10*a^
2*b^2 + 5*b^4)*ArcTan[Tan[c + d*x]])/b + 48*a*(a^2 + b^2)*(2*a^4 - 5*a^2*b^2 + b^4)*Cos[c + d*x]^2 - 24*a*(a -
b)*(a + b)*(a^2 + b^2)^2*Cos[c + d*x]^4 + 12*a*(4*a^6 - 36*a^4*b^2 + 36*a^2*b^4 - 4*b^6 + (-a^7 + 24*a^5*b^2
- 45*a^3*b^4 + 10*a*b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 96*a*(a - b)*(a + b)*(a^4 - 8*a^2*b^2
+ b^4)*Log[a + b*Tan[c + d*x]] + 12*a*(4*a^6 - 36*a^4*b^2 + 36*a^2*b^4 - 4*b^6 + (a^7 - 24*a^5*b^2 + 45*a^3*b^
4 - 10*a*b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] - (6*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*Cos[c +
d*x]^3*Sin[c + d*x])/b - (9*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*Sin[2*(c + d*x)])/(2*b) + (12*a^2*(a^2 + b^2
)*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[2*(c + d*x)])/b + (8*a^4*(a^2 + b^2)^3)/(a + b*Tan[c + d*x])^3 + (24*a^3*(a^2
- 2*b^2)*(a^2 + b^2)^2)/(a + b*Tan[c + d*x])^2 + (72*a^2*(a^2 + b^2)*(a^4 - 5*a^2*b^2 + 2*b^4))/(a + b*Tan[c
+ d*x])))/((a^2 + b^2)^6*d)

Maple [A] (verified)

Time = 55.09 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-\frac {5}{8} a^{8}+\frac {15}{2} a^{6} b^{2}+\frac {5}{4} a^{4} b^{4}-\frac {13}{2} b^{6} a^{2}+\frac {3}{8} b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-4 b \,a^{7}+6 b^{3} a^{5}+8 a^{3} b^{5}-2 a \,b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{8}+\frac {13}{2} a^{6} b^{2}-\frac {15}{2} b^{6} a^{2}+\frac {5}{8} b^{8}-\frac {5}{4} a^{4} b^{4}\right ) \tan \left (d x +c \right )-3 b \,a^{7}+7 b^{3} a^{5}+7 a^{3} b^{5}-3 a \,b^{7}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (-32 b \,a^{7}+288 b^{3} a^{5}-288 a^{3} b^{5}+32 a \,b^{7}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {\left (3 a^{8}-132 a^{6} b^{2}+370 a^{4} b^{4}-132 b^{6} a^{2}+3 b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{6}}-\frac {a^{4} b}{3 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {3 a^{2} b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right )}{\left (a^{2}+b^{2}\right )^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{3} b \left (a^{2}-2 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b a \left (a^{6}-9 a^{4} b^{2}+9 a^{2} b^{4}-b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{6}}}{d}\)	\(425\)
default	\(\frac {\frac {\frac {\left (-\frac {5}{8} a^{8}+\frac {15}{2} a^{6} b^{2}+\frac {5}{4} a^{4} b^{4}-\frac {13}{2} b^{6} a^{2}+\frac {3}{8} b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-4 b \,a^{7}+6 b^{3} a^{5}+8 a^{3} b^{5}-2 a \,b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {3}{8} a^{8}+\frac {13}{2} a^{6} b^{2}-\frac {15}{2} b^{6} a^{2}+\frac {5}{8} b^{8}-\frac {5}{4} a^{4} b^{4}\right ) \tan \left (d x +c \right )-3 b \,a^{7}+7 b^{3} a^{5}+7 a^{3} b^{5}-3 a \,b^{7}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (-32 b \,a^{7}+288 b^{3} a^{5}-288 a^{3} b^{5}+32 a \,b^{7}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {\left (3 a^{8}-132 a^{6} b^{2}+370 a^{4} b^{4}-132 b^{6} a^{2}+3 b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{6}}-\frac {a^{4} b}{3 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {3 a^{2} b \left (a^{4}-5 a^{2} b^{2}+2 b^{4}\right )}{\left (a^{2}+b^{2}\right )^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{3} b \left (a^{2}-2 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b a \left (a^{6}-9 a^{4} b^{2}+9 a^{2} b^{4}-b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{6}}}{d}\)	\(425\)
risch	\(\text {Expression too large to display}\)	\(1620\)

[In]

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

[Out]

1/d*(1/(a^2+b^2)^6*(((-5/8*a^8+15/2*a^6*b^2+5/4*a^4*b^4-13/2*b^6*a^2+3/8*b^8)*tan(d*x+c)^3+(-4*a^7*b+6*a^5*b^3
+8*a^3*b^5-2*a*b^7)*tan(d*x+c)^2+(-3/8*a^8+13/2*a^6*b^2-15/2*b^6*a^2+5/8*b^8-5/4*a^4*b^4)*tan(d*x+c)-3*b*a^7+7
*b^3*a^5+7*a^3*b^5-3*a*b^7)/(1+tan(d*x+c)^2)^2+1/16*(-32*a^7*b+288*a^5*b^3-288*a^3*b^5+32*a*b^7)*ln(1+tan(d*x+
c)^2)+1/8*(3*a^8-132*a^6*b^2+370*a^4*b^4-132*a^2*b^6+3*b^8)*arctan(tan(d*x+c)))-1/3*a^4*b/(a^2+b^2)^3/(a+b*tan
(d*x+c))^3-3*a^2*b*(a^4-5*a^2*b^2+2*b^4)/(a^2+b^2)^5/(a+b*tan(d*x+c))-a^3*b*(a^2-2*b^2)/(a^2+b^2)^4/(a+b*tan(d
*x+c))^2+4*b*a*(a^6-9*a^4*b^2+9*a^2*b^4-b^6)/(a^2+b^2)^6*ln(a+b*tan(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (358) = 716\).

Time = 0.37 (sec) , antiderivative size = 1053, normalized size of antiderivative = 2.88 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/24*(6*(a^10*b + 5*a^8*b^3 + 10*a^6*b^5 + 10*a^4*b^7 + 5*a^2*b^9 + b^11)*cos(d*x + c)^7 - 3*(11*a^10*b + 45*a
^8*b^3 + 70*a^6*b^5 + 50*a^4*b^7 + 15*a^2*b^9 + b^11)*cos(d*x + c)^5 - (6*a^10*b + 342*a^8*b^3 - 1830*a^6*b^5
+ 614*a^4*b^7 - 216*a^2*b^9 + 12*b^11 - 3*(3*a^11 - 141*a^9*b^2 + 766*a^7*b^4 - 1242*a^5*b^6 + 399*a^3*b^8 - 9
*a*b^10)*d*x)*cos(d*x + c)^3 + 3*(114*a^8*b^3 - 381*a^6*b^5 + 187*a^4*b^7 - 67*a^2*b^9 + 3*b^11 + 3*(3*a^9*b^2
- 132*a^7*b^4 + 370*a^5*b^6 - 132*a^3*b^8 + 3*a*b^10)*d*x)*cos(d*x + c) + 48*((a^10*b - 12*a^8*b^3 + 36*a^6*b
^5 - 28*a^4*b^7 + 3*a^2*b^9)*cos(d*x + c)^3 + 3*(a^8*b^3 - 9*a^6*b^5 + 9*a^4*b^7 - a^2*b^9)*cos(d*x + c) + (a^
7*b^4 - 9*a^5*b^6 + 9*a^3*b^8 - a*b^10 + (3*a^9*b^2 - 28*a^7*b^4 + 36*a^5*b^6 - 12*a^3*b^8 + a*b^10)*cos(d*x +
c)^2)*sin(d*x + c))*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) + (143*a^7*b^4 -
537*a^5*b^6 + 105*a^3*b^8 + 33*a*b^10 + 6*(a^11 + 5*a^9*b^2 + 10*a^7*b^4 + 10*a^5*b^6 + 5*a^3*b^8 + a*b^10)*co
s(d*x + c)^6 - 15*(a^11 + 3*a^9*b^2 + 2*a^7*b^4 - 2*a^5*b^6 - 3*a^3*b^8 - a*b^10)*cos(d*x + c)^4 + 3*(3*a^8*b^
3 - 132*a^6*b^5 + 370*a^4*b^7 - 132*a^2*b^9 + 3*b^11)*d*x + (216*a^9*b^2 - 734*a^7*b^4 + 1590*a^5*b^6 - 522*a^
3*b^8 - 54*a*b^10 + 3*(9*a^10*b - 399*a^8*b^3 + 1242*a^6*b^5 - 766*a^4*b^7 + 141*a^2*b^9 - 3*b^11)*d*x)*cos(d*
x + c)^2)*sin(d*x + c))/((a^15 + 3*a^13*b^2 - 3*a^11*b^4 - 25*a^9*b^6 - 45*a^7*b^8 - 39*a^5*b^10 - 17*a^3*b^12
- 3*a*b^14)*d*cos(d*x + c)^3 + 3*(a^13*b^2 + 6*a^11*b^4 + 15*a^9*b^6 + 20*a^7*b^8 + 15*a^5*b^10 + 6*a^3*b^12
+ a*b^14)*d*cos(d*x + c) + ((3*a^14*b + 17*a^12*b^3 + 39*a^10*b^5 + 45*a^8*b^7 + 25*a^6*b^9 + 3*a^4*b^11 - 3*a
^2*b^13 - b^15)*d*cos(d*x + c)^2 + (a^12*b^3 + 6*a^10*b^5 + 15*a^8*b^7 + 20*a^6*b^9 + 15*a^4*b^11 + 6*a^2*b^13
+ b^15)*d)*sin(d*x + c))

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (358) = 716\).

Time = 0.42 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.72 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (3 \, a^{8} - 132 \, a^{6} b^{2} + 370 \, a^{4} b^{4} - 132 \, a^{2} b^{6} + 3 \, b^{8}\right )} {\left (d x + c\right )}}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} + \frac {96 \, {\left (a^{7} b - 9 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - a b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} - \frac {48 \, {\left (a^{7} b - 9 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - a b^{7}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} - \frac {176 \, a^{8} b - 608 \, a^{6} b^{3} + 176 \, a^{4} b^{5} + 3 \, {\left (29 \, a^{6} b^{3} - 185 \, a^{4} b^{5} + 103 \, a^{2} b^{7} - 3 \, b^{9}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (71 \, a^{7} b^{2} - 411 \, a^{5} b^{4} + 165 \, a^{3} b^{6} + 7 \, a b^{8}\right )} \tan \left (d x + c\right )^{5} + {\left (149 \, a^{8} b - 512 \, a^{6} b^{3} - 1006 \, a^{4} b^{5} + 600 \, a^{2} b^{7} - 15 \, b^{9}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{9} + 152 \, a^{7} b^{2} - 822 \, a^{5} b^{4} + 320 \, a^{3} b^{6} + 9 \, a b^{8}\right )} \tan \left (d x + c\right )^{3} + {\left (331 \, a^{8} b - 1183 \, a^{6} b^{3} - 239 \, a^{4} b^{5} + 315 \, a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (3 \, a^{9} + 73 \, a^{7} b^{2} - 423 \, a^{5} b^{4} + 147 \, a^{3} b^{6}\right )} \tan \left (d x + c\right )}{a^{13} + 5 \, a^{11} b^{2} + 10 \, a^{9} b^{4} + 10 \, a^{7} b^{6} + 5 \, a^{5} b^{8} + a^{3} b^{10} + {\left (a^{10} b^{3} + 5 \, a^{8} b^{5} + 10 \, a^{6} b^{7} + 10 \, a^{4} b^{9} + 5 \, a^{2} b^{11} + b^{13}\right )} \tan \left (d x + c\right )^{7} + 3 \, {\left (a^{11} b^{2} + 5 \, a^{9} b^{4} + 10 \, a^{7} b^{6} + 10 \, a^{5} b^{8} + 5 \, a^{3} b^{10} + a b^{12}\right )} \tan \left (d x + c\right )^{6} + {\left (3 \, a^{12} b + 17 \, a^{10} b^{3} + 40 \, a^{8} b^{5} + 50 \, a^{6} b^{7} + 35 \, a^{4} b^{9} + 13 \, a^{2} b^{11} + 2 \, b^{13}\right )} \tan \left (d x + c\right )^{5} + {\left (a^{13} + 11 \, a^{11} b^{2} + 40 \, a^{9} b^{4} + 70 \, a^{7} b^{6} + 65 \, a^{5} b^{8} + 31 \, a^{3} b^{10} + 6 \, a b^{12}\right )} \tan \left (d x + c\right )^{4} + {\left (6 \, a^{12} b + 31 \, a^{10} b^{3} + 65 \, a^{8} b^{5} + 70 \, a^{6} b^{7} + 40 \, a^{4} b^{9} + 11 \, a^{2} b^{11} + b^{13}\right )} \tan \left (d x + c\right )^{3} + {\left (2 \, a^{13} + 13 \, a^{11} b^{2} + 35 \, a^{9} b^{4} + 50 \, a^{7} b^{6} + 40 \, a^{5} b^{8} + 17 \, a^{3} b^{10} + 3 \, a b^{12}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{12} b + 5 \, a^{10} b^{3} + 10 \, a^{8} b^{5} + 10 \, a^{6} b^{7} + 5 \, a^{4} b^{9} + a^{2} b^{11}\right )} \tan \left (d x + c\right )}}{24 \, d} \]

[In]

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

[Out]

1/24*(3*(3*a^8 - 132*a^6*b^2 + 370*a^4*b^4 - 132*a^2*b^6 + 3*b^8)*(d*x + c)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) + 96*(a^7*b - 9*a^5*b^3 + 9*a^3*b^5 - a*b^7)*log(b*tan(d*x + c) +
a)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) - 48*(a^7*b - 9*a^5*b^3 + 9
*a^3*b^5 - a*b^7)*log(tan(d*x + c)^2 + 1)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^
10 + b^12) - (176*a^8*b - 608*a^6*b^3 + 176*a^4*b^5 + 3*(29*a^6*b^3 - 185*a^4*b^5 + 103*a^2*b^7 - 3*b^9)*tan(d
*x + c)^6 + 3*(71*a^7*b^2 - 411*a^5*b^4 + 165*a^3*b^6 + 7*a*b^8)*tan(d*x + c)^5 + (149*a^8*b - 512*a^6*b^3 - 1
006*a^4*b^5 + 600*a^2*b^7 - 15*b^9)*tan(d*x + c)^4 + 3*(5*a^9 + 152*a^7*b^2 - 822*a^5*b^4 + 320*a^3*b^6 + 9*a*
b^8)*tan(d*x + c)^3 + (331*a^8*b - 1183*a^6*b^3 - 239*a^4*b^5 + 315*a^2*b^7)*tan(d*x + c)^2 + 3*(3*a^9 + 73*a^
7*b^2 - 423*a^5*b^4 + 147*a^3*b^6)*tan(d*x + c))/(a^13 + 5*a^11*b^2 + 10*a^9*b^4 + 10*a^7*b^6 + 5*a^5*b^8 + a^
3*b^10 + (a^10*b^3 + 5*a^8*b^5 + 10*a^6*b^7 + 10*a^4*b^9 + 5*a^2*b^11 + b^13)*tan(d*x + c)^7 + 3*(a^11*b^2 + 5
*a^9*b^4 + 10*a^7*b^6 + 10*a^5*b^8 + 5*a^3*b^10 + a*b^12)*tan(d*x + c)^6 + (3*a^12*b + 17*a^10*b^3 + 40*a^8*b^
5 + 50*a^6*b^7 + 35*a^4*b^9 + 13*a^2*b^11 + 2*b^13)*tan(d*x + c)^5 + (a^13 + 11*a^11*b^2 + 40*a^9*b^4 + 70*a^7
*b^6 + 65*a^5*b^8 + 31*a^3*b^10 + 6*a*b^12)*tan(d*x + c)^4 + (6*a^12*b + 31*a^10*b^3 + 65*a^8*b^5 + 70*a^6*b^7
+ 40*a^4*b^9 + 11*a^2*b^11 + b^13)*tan(d*x + c)^3 + (2*a^13 + 13*a^11*b^2 + 35*a^9*b^4 + 50*a^7*b^6 + 40*a^5*
b^8 + 17*a^3*b^10 + 3*a*b^12)*tan(d*x + c)^2 + 3*(a^12*b + 5*a^10*b^3 + 10*a^8*b^5 + 10*a^6*b^7 + 5*a^4*b^9 +
a^2*b^11)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 902 vs. \(2 (358) = 716\).

Time = 0.91 (sec) , antiderivative size = 902, normalized size of antiderivative = 2.46 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (3 \, a^{8} - 132 \, a^{6} b^{2} + 370 \, a^{4} b^{4} - 132 \, a^{2} b^{6} + 3 \, b^{8}\right )} {\left (d x + c\right )}}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} - \frac {48 \, {\left (a^{7} b - 9 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - a b^{7}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} + \frac {96 \, {\left (a^{7} b^{2} - 9 \, a^{5} b^{4} + 9 \, a^{3} b^{6} - a b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b + 6 \, a^{10} b^{3} + 15 \, a^{8} b^{5} + 20 \, a^{6} b^{7} + 15 \, a^{4} b^{9} + 6 \, a^{2} b^{11} + b^{13}} + \frac {3 \, {\left (24 \, a^{7} b \tan \left (d x + c\right )^{4} - 216 \, a^{5} b^{3} \tan \left (d x + c\right )^{4} + 216 \, a^{3} b^{5} \tan \left (d x + c\right )^{4} - 24 \, a b^{7} \tan \left (d x + c\right )^{4} - 5 \, a^{8} \tan \left (d x + c\right )^{3} + 60 \, a^{6} b^{2} \tan \left (d x + c\right )^{3} + 10 \, a^{4} b^{4} \tan \left (d x + c\right )^{3} - 52 \, a^{2} b^{6} \tan \left (d x + c\right )^{3} + 3 \, b^{8} \tan \left (d x + c\right )^{3} + 16 \, a^{7} b \tan \left (d x + c\right )^{2} - 384 \, a^{5} b^{3} \tan \left (d x + c\right )^{2} + 496 \, a^{3} b^{5} \tan \left (d x + c\right )^{2} - 64 \, a b^{7} \tan \left (d x + c\right )^{2} - 3 \, a^{8} \tan \left (d x + c\right ) + 52 \, a^{6} b^{2} \tan \left (d x + c\right ) - 10 \, a^{4} b^{4} \tan \left (d x + c\right ) - 60 \, a^{2} b^{6} \tan \left (d x + c\right ) + 5 \, b^{8} \tan \left (d x + c\right ) - 160 \, a^{5} b^{3} + 272 \, a^{3} b^{5} - 48 \, a b^{7}\right )}}{{\left (a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}} - \frac {8 \, {\left (22 \, a^{7} b^{4} \tan \left (d x + c\right )^{3} - 198 \, a^{5} b^{6} \tan \left (d x + c\right )^{3} + 198 \, a^{3} b^{8} \tan \left (d x + c\right )^{3} - 22 \, a b^{10} \tan \left (d x + c\right )^{3} + 75 \, a^{8} b^{3} \tan \left (d x + c\right )^{2} - 630 \, a^{6} b^{5} \tan \left (d x + c\right )^{2} + 567 \, a^{4} b^{7} \tan \left (d x + c\right )^{2} - 48 \, a^{2} b^{9} \tan \left (d x + c\right )^{2} + 87 \, a^{9} b^{2} \tan \left (d x + c\right ) - 666 \, a^{7} b^{4} \tan \left (d x + c\right ) + 531 \, a^{5} b^{6} \tan \left (d x + c\right ) - 36 \, a^{3} b^{8} \tan \left (d x + c\right ) + 35 \, a^{10} b - 231 \, a^{8} b^{3} + 165 \, a^{6} b^{5} - 9 \, a^{4} b^{7}\right )}}{{\left (a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{24 \, d} \]

[In]

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

[Out]

1/24*(3*(3*a^8 - 132*a^6*b^2 + 370*a^4*b^4 - 132*a^2*b^6 + 3*b^8)*(d*x + c)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) - 48*(a^7*b - 9*a^5*b^3 + 9*a^3*b^5 - a*b^7)*log(tan(d*x + c)^2 +
1)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) + 96*(a^7*b^2 - 9*a^5*b^4 +
9*a^3*b^6 - a*b^8)*log(abs(b*tan(d*x + c) + a))/(a^12*b + 6*a^10*b^3 + 15*a^8*b^5 + 20*a^6*b^7 + 15*a^4*b^9 +
6*a^2*b^11 + b^13) + 3*(24*a^7*b*tan(d*x + c)^4 - 216*a^5*b^3*tan(d*x + c)^4 + 216*a^3*b^5*tan(d*x + c)^4 - 2
4*a*b^7*tan(d*x + c)^4 - 5*a^8*tan(d*x + c)^3 + 60*a^6*b^2*tan(d*x + c)^3 + 10*a^4*b^4*tan(d*x + c)^3 - 52*a^2
*b^6*tan(d*x + c)^3 + 3*b^8*tan(d*x + c)^3 + 16*a^7*b*tan(d*x + c)^2 - 384*a^5*b^3*tan(d*x + c)^2 + 496*a^3*b^
5*tan(d*x + c)^2 - 64*a*b^7*tan(d*x + c)^2 - 3*a^8*tan(d*x + c) + 52*a^6*b^2*tan(d*x + c) - 10*a^4*b^4*tan(d*x
+ c) - 60*a^2*b^6*tan(d*x + c) + 5*b^8*tan(d*x + c) - 160*a^5*b^3 + 272*a^3*b^5 - 48*a*b^7)/((a^12 + 6*a^10*b
^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*(tan(d*x + c)^2 + 1)^2) - 8*(22*a^7*b^4*tan(d*x
+ c)^3 - 198*a^5*b^6*tan(d*x + c)^3 + 198*a^3*b^8*tan(d*x + c)^3 - 22*a*b^10*tan(d*x + c)^3 + 75*a^8*b^3*tan(
d*x + c)^2 - 630*a^6*b^5*tan(d*x + c)^2 + 567*a^4*b^7*tan(d*x + c)^2 - 48*a^2*b^9*tan(d*x + c)^2 + 87*a^9*b^2*
tan(d*x + c) - 666*a^7*b^4*tan(d*x + c) + 531*a^5*b^6*tan(d*x + c) - 36*a^3*b^8*tan(d*x + c) + 35*a^10*b - 231
*a^8*b^3 + 165*a^6*b^5 - 9*a^4*b^7)/((a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 +
b^12)*(b*tan(d*x + c) + a)^3))/d

Mupad [B] (verification not implemented)

Time = 7.61 (sec) , antiderivative size = 962, normalized size of antiderivative = 2.63 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {4\,a\,b}{{\left (a^2+b^2\right )}^3}-\frac {48\,a\,b^3}{{\left (a^2+b^2\right )}^4}+\frac {120\,a\,b^5}{{\left (a^2+b^2\right )}^5}-\frac {80\,a\,b^7}{{\left (a^2+b^2\right )}^6}\right )}{d}-\frac {\frac {2\,\left (11\,a^8\,b-38\,a^6\,b^3+11\,a^4\,b^5\right )}{3\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (-29\,a^6\,b^3+185\,a^4\,b^5-103\,a^2\,b^7+3\,b^9\right )}{8\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (71\,a^7\,b^2-411\,a^5\,b^4+165\,a^3\,b^6+7\,a\,b^8\right )}{8\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (331\,a^8\,b-1183\,a^6\,b^3-239\,a^4\,b^5+315\,a^2\,b^7\right )}{24\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (5\,a^9+152\,a^7\,b^2-822\,a^5\,b^4+320\,a^3\,b^6+9\,a\,b^8\right )}{8\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (-149\,a^8\,b+512\,a^6\,b^3+1006\,a^4\,b^5-600\,a^2\,b^7+15\,b^9\right )}{24\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^8+73\,a^6\,b^2-423\,a^4\,b^4+147\,a^2\,b^6\right )}{8\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^3+3\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (3\,a^2\,b+2\,b^3\right )+a^3+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (6\,a^2\,b+b^3\right )+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^7+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^6+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,14{}\mathrm {i}+3\,b^2\right )}{16\,d\,\left (-a^6\,1{}\mathrm {i}+6\,a^5\,b+a^4\,b^2\,15{}\mathrm {i}-20\,a^3\,b^3-a^2\,b^4\,15{}\mathrm {i}+6\,a\,b^5+b^6\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (3\,a^2+a\,b\,14{}\mathrm {i}-3\,b^2\right )}{16\,d\,\left (a^6\,1{}\mathrm {i}+6\,a^5\,b-a^4\,b^2\,15{}\mathrm {i}-20\,a^3\,b^3+a^2\,b^4\,15{}\mathrm {i}+6\,a\,b^5-b^6\,1{}\mathrm {i}\right )} \]

[In]

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

[Out]

(log(a + b*tan(c + d*x))*((4*a*b)/(a^2 + b^2)^3 - (48*a*b^3)/(a^2 + b^2)^4 + (120*a*b^5)/(a^2 + b^2)^5 - (80*a
*b^7)/(a^2 + b^2)^6))/d - ((2*(11*a^8*b + 11*a^4*b^5 - 38*a^6*b^3))/(3*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 +
10*a^6*b^4 + 5*a^8*b^2)) - (tan(c + d*x)^6*(3*b^9 - 103*a^2*b^7 + 185*a^4*b^5 - 29*a^6*b^3))/(8*(a^10 + b^10
+ 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^5*(7*a*b^8 + 165*a^3*b^6 - 411*a^5*b^4 + 7
1*a^7*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^2*(331*a^8*b +
315*a^2*b^7 - 239*a^4*b^5 - 1183*a^6*b^3))/(24*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2
)) + (tan(c + d*x)^3*(9*a*b^8 + 5*a^9 + 320*a^3*b^6 - 822*a^5*b^4 + 152*a^7*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8
+ 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) - (tan(c + d*x)^4*(15*b^9 - 149*a^8*b - 600*a^2*b^7 + 1006*a^4*b^5 + 5
12*a^6*b^3))/(24*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (a*tan(c + d*x)*(3*a^8 + 1
47*a^2*b^6 - 423*a^4*b^4 + 73*a^6*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)))/(
d*(tan(c + d*x)^2*(3*a*b^2 + 2*a^3) + tan(c + d*x)^5*(3*a^2*b + 2*b^3) + a^3 + tan(c + d*x)^4*(6*a*b^2 + a^3)
+ tan(c + d*x)^3*(6*a^2*b + b^3) + b^3*tan(c + d*x)^7 + 3*a*b^2*tan(c + d*x)^6 + 3*a^2*b*tan(c + d*x))) + (log
(tan(c + d*x) - 1i)*(a*b*14i - 3*a^2 + 3*b^2))/(16*d*(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a
^3*b^3 + a^4*b^2*15i)) - (log(tan(c + d*x) + 1i)*(a*b*14i + 3*a^2 - 3*b^2))/(16*d*(6*a*b^5 + 6*a^5*b + a^6*1i
- b^6*1i + a^2*b^4*15i - 20*a^3*b^3 - a^4*b^2*15i))