3.62 \(\int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=217 \[ -\frac{a^4 b}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{4 d \left (a^2+b^2\right )^2}-\frac{\cos ^2(c+d x) \left (\left (-12 a^2 b^2+5 a^4-b^4\right ) \tan (c+d x)+16 a^3 b\right )}{8 d \left (a^2+b^2\right )^3}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-33 a^4 b^2+13 a^2 b^4+3 a^6+b^6\right )}{8 \left (a^2+b^2\right )^4} \]

[Out]

((3*a^6 - 33*a^4*b^2 + 13*a^2*b^4 + b^6)*x)/(8*(a^2 + b^2)^4) + (2*a^3*b*(a^2 - 2*b^2)*Log[a*Cos[c + d*x] + b*
Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^4*b)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(2*a*b + (
a^2 - b^2)*Tan[c + d*x]))/(4*(a^2 + b^2)^2*d) - (Cos[c + d*x]^2*(16*a^3*b + (5*a^4 - 12*a^2*b^2 - b^4)*Tan[c +
d*x]))/(8*(a^2 + b^2)^3*d)

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Rubi [A]  time = 0.561945, antiderivative size = 217, 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} \[ -\frac{a^4 b}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{4 d \left (a^2+b^2\right )^2}-\frac{\cos ^2(c+d x) \left (\left (-12 a^2 b^2+5 a^4-b^4\right ) \tan (c+d x)+16 a^3 b\right )}{8 d \left (a^2+b^2\right )^3}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-33 a^4 b^2+13 a^2 b^4+3 a^6+b^6\right )}{8 \left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]

[Out]

((3*a^6 - 33*a^4*b^2 + 13*a^2*b^4 + b^6)*x)/(8*(a^2 + b^2)^4) + (2*a^3*b*(a^2 - 2*b^2)*Log[a*Cos[c + d*x] + b*
Sin[c + d*x]])/((a^2 + b^2)^4*d) - (a^4*b)/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(2*a*b + (
a^2 - b^2)*Tan[c + d*x]))/(4*(a^2 + b^2)^2*d) - (Cos[c + d*x]^2*(16*a^3*b + (5*a^4 - 12*a^2*b^2 - b^4)*Tan[c +
d*x]))/(8*(a^2 + b^2)^3*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))^2} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^4}{(a+x)^2 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^4 \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac{2 a b^4 \left (3 a^2+b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{b^2 \left (4 a^4+11 a^2 b^2+b^4\right ) x^2}{\left (a^2+b^2\right )^2}}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^4 \left (3 a^4-12 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^3}-\frac{2 a b^4 \left (5 a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{b^4 \left (5 a^4-12 a^2 b^2-b^4\right ) x^2}{\left (a^2+b^2\right )^3}}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^2}+\frac{16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)}+\frac{b^4 \left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6-16 a^3 \left (a^2-2 b^2\right ) x\right )}{\left (a^2+b^2\right )^4 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^6-33 a^4 b^2+13 a^2 b^4+b^6-16 a^3 \left (a^2-2 b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}\\ &=\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}-\frac{\left (2 a^3 b \left (a^2-2 b^2\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 )^4 d}+\frac{\left (b \left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\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 )^4 d}\\ &=\frac{\left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\right ) x}{8 \left (a^2+b^2\right )^4}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}\\ \end{align*}

Mathematica [A]  time = 3.80693, size = 373, normalized size = 1.72 \[ \frac{b \left (\frac{2 \left (a^2+b^2\right ) \left (3 a^2 b^2-2 a^4+b^4\right ) \sin (2 (c+d x))}{b}-16 a^3 \left (a^2+b^2\right ) \cos ^2(c+d x)+4 a \left (a^2+b^2\right )^2 \cos ^4(c+d x)+\frac{4 \left (a^2+b^2\right ) \left (3 a^2 b^2-2 a^4+b^4\right ) \tan ^{-1}(\tan (c+d x))}{b}-\frac{8 a^4 \left (a^2+b^2\right )}{a+b \tan (c+d x)}-4 a^3 \left (\frac{5 a b^2-a^3}{\sqrt{-b^2}}+2 a^2-4 b^2\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+16 a^3 \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))-4 a^3 \left (\frac{a^3-5 a b^2}{\sqrt{-b^2}}+2 a^2-4 b^2\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )+\frac{3 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \left (\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right )}{2 b}+\frac{2 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \sin (c+d x) \cos ^3(c+d x)}{b}\right )}{8 d \left (a^2+b^2\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^2,x]

[Out]

(b*((4*(a^2 + b^2)*(-2*a^4 + 3*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/b - 16*a^3*(a^2 + b^2)*Cos[c + d*x]^2 + 4*
a*(a^2 + b^2)^2*Cos[c + d*x]^4 - 4*a^3*(2*a^2 - 4*b^2 + (-a^3 + 5*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c
+ d*x]] + 16*a^3*(a^2 - 2*b^2)*Log[a + b*Tan[c + d*x]] - 4*a^3*(2*a^2 - 4*b^2 + (a^3 - 5*a*b^2)/Sqrt[-b^2])*Lo
g[Sqrt[-b^2] + b*Tan[c + d*x]] + (2*(a^2 - b^2)*(a^2 + b^2)^2*Cos[c + d*x]^3*Sin[c + d*x])/b + (2*(a^2 + b^2)*
(-2*a^4 + 3*a^2*b^2 + b^4)*Sin[2*(c + d*x)])/b + (3*(a^2 - b^2)*(a^2 + b^2)^2*(2*ArcTan[Tan[c + d*x]] + Sin[2*
(c + d*x)]))/(2*b) - (8*a^4*(a^2 + b^2))/(a + b*Tan[c + d*x])))/(8*(a^2 + b^2)^4*d)

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Maple [B]  time = 0.099, size = 724, 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))^2,x)

[Out]

-5/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^6+7/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^4*b
^2+13/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^2*b^4+1/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^
3*b^6-2/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^5*b-2/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^
3*b^3-3/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^6+9/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^4*
b^2+11/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^2*b^4-1/8/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*tan(d*x+c)*b
^6-3/2/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*a^5*b-1/d/(a^2+b^2)^4/(1+tan(d*x+c)^2)^2*a^3*b^3+1/2/d/(a^2+b^2)^4/(1+
tan(d*x+c)^2)^2*a*b^5-1/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^5*b+2/d/(a^2+b^2)^4*ln(1+tan(d*x+c)^2)*a^3*b^3+3/8/
d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^6-33/8/d/(a^2+b^2)^4*arctan(tan(d*x+c))*a^4*b^2+13/8/d/(a^2+b^2)^4*arctan(t
an(d*x+c))*a^2*b^4+1/8/d/(a^2+b^2)^4*arctan(tan(d*x+c))*b^6-a^4*b/(a^2+b^2)^3/d/(a+b*tan(d*x+c))+2/d*a^5*b/(a^
2+b^2)^4*ln(a+b*tan(d*x+c))-4/d*a^3*b^3/(a^2+b^2)^4*ln(a+b*tan(d*x+c))

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Maxima [B]  time = 1.53015, size = 684, normalized size = 3.15 \begin{align*} \frac{\frac{{\left (3 \, a^{6} - 33 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + b^{6}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{16 \,{\left (a^{5} b - 2 \, a^{3} b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{8 \,{\left (a^{5} b - 2 \, a^{3} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{20 \, a^{4} b - 4 \, a^{2} b^{3} +{\left (13 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{4} +{\left (5 \, a^{5} + 4 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )^{3} +{\left (35 \, a^{4} b - 12 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{5} - a b^{4}\right )} \tan \left (d x + c\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{5} +{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )}}{8 \, d} \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))^2,x, algorithm="maxima")

[Out]

1/8*((3*a^6 - 33*a^4*b^2 + 13*a^2*b^4 + b^6)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 16*(a
^5*b - 2*a^3*b^3)*log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 8*(a^5*b - 2*a^3*b
^3)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - (20*a^4*b - 4*a^2*b^3 + (13*a^4*
b - 12*a^2*b^3 - b^5)*tan(d*x + c)^4 + (5*a^5 + 4*a^3*b^2 - a*b^4)*tan(d*x + c)^3 + (35*a^4*b - 12*a^2*b^3 + b
^5)*tan(d*x + c)^2 + 3*(a^5 - a*b^4)*tan(d*x + c))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + (a^6*b + 3*a^4*b^3 +
3*a^2*b^5 + b^7)*tan(d*x + c)^5 + (a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*tan(d*x + c)^4 + 2*(a^6*b + 3*a^4*b^3
+ 3*a^2*b^5 + b^7)*tan(d*x + c)^3 + 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*tan(d*x + c)^2 + (a^6*b + 3*a^4*b
^3 + 3*a^2*b^5 + b^7)*tan(d*x + c)))/d

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Fricas [B]  time = 2.63226, size = 986, normalized size = 4.54 \begin{align*} \frac{4 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{5} - 6 \,{\left (3 \, a^{6} b + 7 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{6} b + 8 \, a^{4} b^{3} + 23 \, a^{2} b^{5} + 2 \, b^{7} + 2 \,{\left (3 \, a^{7} - 33 \, a^{5} b^{2} + 13 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \cos \left (d x + c\right ) + 16 \,{\left ({\left (a^{6} b - 2 \, a^{4} b^{3}\right )} \cos \left (d x + c\right ) +{\left (a^{5} b^{2} - 2 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) +{\left (29 \, a^{5} b^{2} + 10 \, a^{3} b^{4} - 3 \, a b^{6} + 4 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (3 \, a^{6} b - 33 \, a^{4} b^{3} + 13 \, a^{2} b^{5} + b^{7}\right )} d x - 2 \,{\left (5 \, a^{7} + 9 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) +{\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} d \sin \left (d x + c\right )\right )}} \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))^2,x, algorithm="fricas")

[Out]

1/16*(4*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^5 - 6*(3*a^6*b + 7*a^4*b^3 + 5*a^2*b^5 + b^7)*cos(d
*x + c)^3 + (3*a^6*b + 8*a^4*b^3 + 23*a^2*b^5 + 2*b^7 + 2*(3*a^7 - 33*a^5*b^2 + 13*a^3*b^4 + a*b^6)*d*x)*cos(d
*x + c) + 16*((a^6*b - 2*a^4*b^3)*cos(d*x + c) + (a^5*b^2 - 2*a^3*b^4)*sin(d*x + c))*log(2*a*b*cos(d*x + c)*si
n(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) + (29*a^5*b^2 + 10*a^3*b^4 - 3*a*b^6 + 4*(a^7 + 3*a^5*b^2 + 3*a
^3*b^4 + a*b^6)*cos(d*x + c)^4 + 2*(3*a^6*b - 33*a^4*b^3 + 13*a^2*b^5 + b^7)*d*x - 2*(5*a^7 + 9*a^5*b^2 + 3*a^
3*b^4 - a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)
+ (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*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))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.21571, size = 693, normalized size = 3.19 \begin{align*} \frac{\frac{{\left (3 \, a^{6} - 33 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + b^{6}\right )}{\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac{8 \,{\left (a^{5} b - 2 \, a^{3} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac{16 \,{\left (a^{5} b^{2} - 2 \, a^{3} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} - \frac{8 \,{\left (2 \, a^{5} b^{2} \tan \left (d x + c\right ) - 4 \, a^{3} b^{4} \tan \left (d x + c\right ) + 3 \, a^{6} b - 3 \, a^{4} b^{3}\right )}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}} + \frac{12 \, a^{5} b \tan \left (d x + c\right )^{4} - 24 \, a^{3} b^{3} \tan \left (d x + c\right )^{4} - 5 \, a^{6} \tan \left (d x + c\right )^{3} + 7 \, a^{4} b^{2} \tan \left (d x + c\right )^{3} + 13 \, a^{2} b^{4} \tan \left (d x + c\right )^{3} + b^{6} \tan \left (d x + c\right )^{3} + 8 \, a^{5} b \tan \left (d x + c\right )^{2} - 64 \, a^{3} b^{3} \tan \left (d x + c\right )^{2} - 3 \, a^{6} \tan \left (d x + c\right ) + 9 \, a^{4} b^{2} \tan \left (d x + c\right ) + 11 \, a^{2} b^{4} \tan \left (d x + c\right ) - b^{6} \tan \left (d x + c\right ) - 32 \, a^{3} b^{3} + 4 \, a b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \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))^2,x, algorithm="giac")

[Out]

1/8*((3*a^6 - 33*a^4*b^2 + 13*a^2*b^4 + b^6)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 8*(a^
5*b - 2*a^3*b^3)*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 16*(a^5*b^2 - 2*a^3
*b^4)*log(abs(b*tan(d*x + c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) - 8*(2*a^5*b^2*tan(d*x +
c) - 4*a^3*b^4*tan(d*x + c) + 3*a^6*b - 3*a^4*b^3)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(d*x
+ c) + a)) + (12*a^5*b*tan(d*x + c)^4 - 24*a^3*b^3*tan(d*x + c)^4 - 5*a^6*tan(d*x + c)^3 + 7*a^4*b^2*tan(d*x
+ c)^3 + 13*a^2*b^4*tan(d*x + c)^3 + b^6*tan(d*x + c)^3 + 8*a^5*b*tan(d*x + c)^2 - 64*a^3*b^3*tan(d*x + c)^2 -
3*a^6*tan(d*x + c) + 9*a^4*b^2*tan(d*x + c) + 11*a^2*b^4*tan(d*x + c) - b^6*tan(d*x + c) - 32*a^3*b^3 + 4*a*b
^5)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(tan(d*x + c)^2 + 1)^2))/d