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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.37025, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3516, 1647, 1629, 635, 203, 260} \[ -\frac{a^4 b}{3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3}-\frac{a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}-\frac{3 a^2 b \left (-5 a^2 b^2+a^4+2 b^4\right )}{d \left (a^2+b^2\right )^5 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (\left (-6 a^2 b^2+a^4+b^4\right ) \tan (c+d x)+4 a b \left (a^2-b^2\right )\right )}{4 d \left (a^2+b^2\right )^4}-\frac{\cos ^2(c+d x) \left (\left (-65 a^4 b^2+55 a^2 b^4+5 a^6-3 b^6\right ) \tan (c+d x)+16 a b \left (-5 a^2 b^2+2 a^4+b^4\right )\right )}{8 d \left (a^2+b^2\right )^5}+\frac{4 a b \left (a^2-b^2\right ) \left (-8 a^2 b^2+a^4+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^6}+\frac{x \left (-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 a^8+3 b^8\right )}{8 \left (a^2+b^2\right )^6} \]

Antiderivative was successfully verified.

[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 3516

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]

Rule 1647

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 1629

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 635

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 203

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

Rule 260

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]

Rubi steps

\begin{align*} \int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=\frac{b \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 5.41854, size = 564, normalized size = 1.54 \[ -\frac{b \left (\frac{12 a^2 \left (a^2+b^2\right ) \left (-10 a^2 b^2+a^4+5 b^4\right ) \sin (2 (c+d x))}{b}-24 a (a-b) (a+b) \left (a^2+b^2\right )^2 \cos ^4(c+d x)+48 a \left (a^2+b^2\right ) \left (-5 a^2 b^2+2 a^4+b^4\right ) \cos ^2(c+d x)+\frac{24 a^2 \left (a^2+b^2\right ) \left (-10 a^2 b^2+a^4+5 b^4\right ) \tan ^{-1}(\tan (c+d x))}{b}+\frac{8 a^4 \left (a^2+b^2\right )^3}{(a+b \tan (c+d x))^3}+\frac{24 a^3 \left (a^2-2 b^2\right ) \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac{72 a^2 \left (a^2+b^2\right ) \left (-5 a^2 b^2+a^4+2 b^4\right )}{a+b \tan (c+d x)}+12 a \left (-36 a^4 b^2+36 a^2 b^4+\frac{24 a^5 b^2-45 a^3 b^4-a^7+10 a b^6}{\sqrt{-b^2}}+4 a^6-4 b^6\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )-96 a (a-b) (a+b) \left (-8 a^2 b^2+a^4+b^4\right ) \log (a+b \tan (c+d x))+12 a \left (-36 a^4 b^2+36 a^2 b^4+\frac{-24 a^5 b^2+45 a^3 b^4+a^7-10 a b^6}{\sqrt{-b^2}}+4 a^6-4 b^6\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )-\frac{9 \left (a^2+b^2\right )^2 \left (-6 a^2 b^2+a^4+b^4\right ) \left (\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right )}{2 b}-\frac{6 \left (a^2+b^2\right )^2 \left (-6 a^2 b^2+a^4+b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{b}\right )}{24 d \left (a^2+b^2\right )^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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]] - 9
6*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 + (12*a^2*(a^2 + b^2)*(a^4 - 10*a^2*b^2 + 5*b^4)*S
in[2*(c + d*x)])/b - (9*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*(2*ArcTan[Tan[c + d*x]] + Sin[2*(c + d*x)]))/(2*
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])))/(24*(a^2 + b^2)^6*d)

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Maple [B]  time = 0.123, size = 1215, normalized size = 3.3 \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*tan(d*x+c))^4,x)

[Out]

-3/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*a*b^7+4/d*b*a^7/(a^2+b^2)^6*ln(a+b*tan(d*x+c))-36/d*b^3*a^5/(a^2+b^2)^6*ln
(a+b*tan(d*x+c))+36/d*b^5*a^3/(a^2+b^2)^6*ln(a+b*tan(d*x+c))-4/d*b^7*a/(a^2+b^2)^6*ln(a+b*tan(d*x+c))-3/d*b*a^
6/(a^2+b^2)^5/(a+b*tan(d*x+c))-1/3*a^4*b/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^3-5/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2
*tan(d*x+c)*a^4*b^4-3/8/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^8+15/d*b^3*a^4/(a^2+b^2)^5/(a+b*tan(d*x+
c))-6/d*b^5*a^2/(a^2+b^2)^5/(a+b*tan(d*x+c))-1/d*a^5*b/(a^2+b^2)^4/(a+b*tan(d*x+c))^2+2/d*a^3*b^3/(a^2+b^2)^4/
(a+b*tan(d*x+c))^2-2/d/(a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a^7*b+18/d/(a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a^5*b^3-18/d/(
a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a^3*b^5+2/d/(a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a*b^7-33/2/d/(a^2+b^2)^6*arctan(tan(d
*x+c))*a^6*b^2+185/4/d/(a^2+b^2)^6*arctan(tan(d*x+c))*a^4*b^4-33/2/d/(a^2+b^2)^6*arctan(tan(d*x+c))*a^2*b^6-5/
8/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^8+3/8/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*b^8+5/8/
d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)*b^8-3/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*a^7*b+7/d/(a^2+b^2)^6/(1+ta
n(d*x+c)^2)^2*b^3*a^5+7/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*a^3*b^5+3/8/d/(a^2+b^2)^6*arctan(tan(d*x+c))*b^8+3/8/
d/(a^2+b^2)^6*arctan(tan(d*x+c))*a^8+15/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^6*b^2+5/4/d/(a^2+b^2
)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^4*b^4-13/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^2*b^6-4/d/(a^
2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^7*b+6/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^5*b^3+8/d/(a^
2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^3*b^5-2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a*b^7+13/2/d/
(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^6*b^2-15/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^2*b^6

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Maxima [B]  time = 1.89595, size = 1346, normalized size = 3.68 \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*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

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Fricas [B]  time = 4.00481, size = 2412, normalized size = 6.59 \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*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))

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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*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.34998, size = 1218, normalized size = 3.33 \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*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