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

Optimal. Leaf size=264 \[ -\frac{a^2 b}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^3}-\frac{a b \left (a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac{b \left (-8 a^2 b^2+3 a^4+b^4\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (\left (-6 a^2 b^2+a^4+b^4\right ) \tan (c+d x)+4 a b \left (a^2-b^2\right )\right )}{2 d \left (a^2+b^2\right )^4}+\frac{4 a b \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac{x \left (-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right )}{2 \left (a^2+b^2\right )^5} \]

[Out]

((a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(2*(a^2 + b^2)^5) + (4*a*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*Cos[c +
d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^2*b)/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^3) - (a*b*(a^2 -
b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^4 - 8*a^2*b^2 + b^4))/((a^2 + b^2)^4*d*(a + b*Tan[c +
d*x])) - (Cos[c + d*x]^2*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/(2*(a^2 + b^2)^4*d)

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Rubi [A]  time = 0.570884, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{a^2 b}{3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^3}-\frac{a b \left (a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac{b \left (-8 a^2 b^2+3 a^4+b^4\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac{\cos ^2(c+d x) \left (\left (-6 a^2 b^2+a^4+b^4\right ) \tan (c+d x)+4 a b \left (a^2-b^2\right )\right )}{2 d \left (a^2+b^2\right )^4}+\frac{4 a b \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac{x \left (-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right )}{2 \left (a^2+b^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(2*(a^2 + b^2)^5) + (4*a*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*Cos[c +
d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^2*b)/(3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^3) - (a*b*(a^2 -
b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (b*(3*a^4 - 8*a^2*b^2 + b^4))/((a^2 + b^2)^4*d*(a + b*Tan[c +
d*x])) - (Cos[c + d*x]^2*(4*a*b*(a^2 - b^2) + (a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/(2*(a^2 + b^2)^4*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 ^2(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^2}{(a+x)^4 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac{\cos ^2(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 )}{2 \left (a^2+b^2\right )^4 d}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^4 b^2 \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}+\frac{4 a^3 b^2 \left (a^4+4 a^2 b^2-b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac{2 b^2 \left (3 a^4-6 a^2 b^2-b^4\right ) x^2}{\left (a^2+b^2\right )^3}+\frac{4 a b^2 \left (a^4-4 a^2 b^2-b^4\right ) x^3}{\left (a^2+b^2\right )^4}+\frac{b^2 \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 )} \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=-\frac{\cos ^2(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 )}{2 \left (a^2+b^2\right )^4 d}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)^4}+\frac{4 a b^2 \left (-a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+x)^3}-\frac{2 \left (3 a^4 b^2-8 a^2 b^4+b^6\right )}{\left (a^2+b^2\right )^4 (a+x)^2}-\frac{8 a b^2 \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 (a+x)}+\frac{b^2 \left (-a^6+25 a^4 b^2-35 a^2 b^4+3 b^6+8 a \left (a^4-5 a^2 b^2+2 b^4\right ) x\right )}{\left (a^2+b^2\right )^5 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 b d}\\ &=\frac{4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac{a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac{a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac{b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac{\cos ^2(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 )}{2 \left (a^2+b^2\right )^4 d}-\frac{b \operatorname{Subst}\left (\int \frac{-a^6+25 a^4 b^2-35 a^2 b^4+3 b^6+8 a \left (a^4-5 a^2 b^2+2 b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 d}\\ &=\frac{4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac{a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac{a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac{b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac{\cos ^2(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 )}{2 \left (a^2+b^2\right )^4 d}-\frac{\left (4 a b \left (a^4-5 a^2 b^2+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 )^5 d}+\frac{\left (b \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 d}\\ &=\frac{\left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{2 \left (a^2+b^2\right )^5}+\frac{4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac{4 a b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac{a^2 b}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^3}-\frac{a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac{b \left (3 a^4-8 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac{\cos ^2(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 )}{2 \left (a^2+b^2\right )^4 d}\\ \end{align*}

Mathematica [A]  time = 3.59427, size = 395, normalized size = 1.5 \[ -\frac{b \left (\frac{3 \left (-6 a^2 b^2+a^4+b^4\right ) \left (a^2+b^2\right ) \sin (2 (c+d x))}{2 b}+12 a (a-b) (a+b) \left (a^2+b^2\right ) \cos ^2(c+d x)+\frac{3 \left (-6 a^2 b^2+a^4+b^4\right ) \left (a^2+b^2\right ) \tan ^{-1}(\tan (c+d x))}{b}+\frac{2 a^2 \left (a^2+b^2\right )^3}{(a+b \tan (c+d x))^3}+\frac{6 a (a-b) (a+b) \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac{6 \left (-8 a^2 b^2+3 a^4+b^4\right ) \left (a^2+b^2\right )}{a+b \tan (c+d x)}+3 \left (-20 a^3 b^2+\frac{15 a^4 b^2-15 a^2 b^4-a^6+b^6}{\sqrt{-b^2}}+4 a^5+8 a b^4\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )-24 a \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a+b \tan (c+d x))+3 \left (-20 a^3 b^2+\frac{-15 a^4 b^2+15 a^2 b^4+a^6-b^6}{\sqrt{-b^2}}+4 a^5+8 a b^4\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )\right )}{6 d \left (a^2+b^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*((3*(a^2 + b^2)*(a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/b + 12*a*(a - b)*(a + b)*(a^2 + b^2)*Cos[c +
d*x]^2 + 3*(4*a^5 - 20*a^3*b^2 + 8*a*b^4 + (-a^6 + 15*a^4*b^2 - 15*a^2*b^4 + b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2]
- b*Tan[c + d*x]] - 24*a*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a + b*Tan[c + d*x]] + 3*(4*a^5 - 20*a^3*b^2 + 8*a*b^4 +
(a^6 - 15*a^4*b^2 + 15*a^2*b^4 - b^6)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (3*(a^2 + b^2)*(a^4 - 6*
a^2*b^2 + b^4)*Sin[2*(c + d*x)])/(2*b) + (2*a^2*(a^2 + b^2)^3)/(a + b*Tan[c + d*x])^3 + (6*a*(a - b)*(a + b)*(
a^2 + b^2)^2)/(a + b*Tan[c + d*x])^2 + (6*(a^2 + b^2)*(3*a^4 - 8*a^2*b^2 + b^4))/(a + b*Tan[c + d*x])))/(6*(a^
2 + b^2)^5*d)

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Maple [B]  time = 0.12, size = 668, normalized size = 2.5 \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)^2/(a+b*tan(d*x+c))^4,x)

[Out]

-1/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)*tan(d*x+c)*a^6+5/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)*tan(d*x+c)*a^4*b^2+5/2/d
/(a^2+b^2)^5/(1+tan(d*x+c)^2)*tan(d*x+c)*a^2*b^4-1/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)*tan(d*x+c)*b^6-2/d/(a^2+b^
2)^5/(1+tan(d*x+c)^2)*a^5*b+2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)*a*b^5-2/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a^5*b+10
/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a^3*b^3-4/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a*b^5-25/2/d/(a^2+b^2)^5*arctan(t
an(d*x+c))*a^4*b^2+35/2/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^2*b^4-3/2/d/(a^2+b^2)^5*arctan(tan(d*x+c))*b^6+1/2/
d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^6-1/3*a^2*b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^3-3/d*b/(a^2+b^2)^4/(a+b*tan(d*x
+c))*a^4+8/d*b^3/(a^2+b^2)^4/(a+b*tan(d*x+c))*a^2-1/d*b^5/(a^2+b^2)^4/(a+b*tan(d*x+c))+4/d*a^5*b/(a^2+b^2)^5*l
n(a+b*tan(d*x+c))-20/d*a^3*b^3/(a^2+b^2)^5*ln(a+b*tan(d*x+c))+8/d*a*b^5/(a^2+b^2)^5*ln(a+b*tan(d*x+c))-1/d*b*a
^3/(a^2+b^2)^3/(a+b*tan(d*x+c))^2+1/d*b^3*a/(a^2+b^2)^3/(a+b*tan(d*x+c))^2

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Maxima [B]  time = 1.71198, size = 894, normalized size = 3.39 \begin{align*} \frac{\frac{3 \,{\left (a^{6} - 25 \, a^{4} b^{2} + 35 \, a^{2} b^{4} - 3 \, b^{6}\right )}{\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac{24 \,{\left (a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac{12 \,{\left (a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac{38 \, a^{6} b - 56 \, a^{4} b^{3} + 2 \, a^{2} b^{5} + 3 \,{\left (7 \, a^{4} b^{3} - 22 \, a^{2} b^{5} + 3 \, b^{7}\right )} \tan \left (d x + c\right )^{4} + 3 \,{\left (17 \, a^{5} b^{2} - 46 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{3} +{\left (35 \, a^{6} b - 44 \, a^{4} b^{3} - 73 \, a^{2} b^{5} + 6 \, b^{7}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{7} + 20 \, a^{5} b^{2} - 43 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \tan \left (d x + c\right )}{a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8} +{\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{5} + 3 \,{\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{4} +{\left (3 \, a^{10} b + 13 \, a^{8} b^{3} + 22 \, a^{6} b^{5} + 18 \, a^{4} b^{7} + 7 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} +{\left (a^{11} + 7 \, a^{9} b^{2} + 18 \, a^{7} b^{4} + 22 \, a^{5} b^{6} + 13 \, a^{3} b^{8} + 3 \, a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/6*(3*(a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*(d*x + c)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b
^8 + b^10) + 24*(a^5*b - 5*a^3*b^3 + 2*a*b^5)*log(b*tan(d*x + c) + a)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*
b^6 + 5*a^2*b^8 + b^10) - 12*(a^5*b - 5*a^3*b^3 + 2*a*b^5)*log(tan(d*x + c)^2 + 1)/(a^10 + 5*a^8*b^2 + 10*a^6*
b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10) - (38*a^6*b - 56*a^4*b^3 + 2*a^2*b^5 + 3*(7*a^4*b^3 - 22*a^2*b^5 + 3*b^7)
*tan(d*x + c)^4 + 3*(17*a^5*b^2 - 46*a^3*b^4 + a*b^6)*tan(d*x + c)^3 + (35*a^6*b - 44*a^4*b^3 - 73*a^2*b^5 + 6
*b^7)*tan(d*x + c)^2 + 3*(a^7 + 20*a^5*b^2 - 43*a^3*b^4 + 2*a*b^6)*tan(d*x + c))/(a^11 + 4*a^9*b^2 + 6*a^7*b^4
+ 4*a^5*b^6 + a^3*b^8 + (a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*tan(d*x + c)^5 + 3*(a^9*b^2 + 4*
a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*tan(d*x + c)^4 + (3*a^10*b + 13*a^8*b^3 + 22*a^6*b^5 + 18*a^4*b^7 +
7*a^2*b^9 + b^11)*tan(d*x + c)^3 + (a^11 + 7*a^9*b^2 + 18*a^7*b^4 + 22*a^5*b^6 + 13*a^3*b^8 + 3*a*b^10)*tan(d*
x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*tan(d*x + c)))/d

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Fricas [B]  time = 3.00205, size = 1789, normalized size = 6.78 \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)^2/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/6*(3*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cos(d*x + c)^5 + (3*a^8*b + 111*a^6*b^3 - 231*a^4*b^
5 + 65*a^2*b^7 - 12*b^9 - 3*(a^9 - 28*a^7*b^2 + 110*a^5*b^4 - 108*a^3*b^6 + 9*a*b^8)*d*x)*cos(d*x + c)^3 - 3*(
25*a^6*b^3 - 51*a^4*b^5 + 25*a^2*b^7 - 3*b^9 + 3*(a^7*b^2 - 25*a^5*b^4 + 35*a^3*b^6 - 3*a*b^8)*d*x)*cos(d*x +
c) - 12*((a^8*b - 8*a^6*b^3 + 17*a^4*b^5 - 6*a^2*b^7)*cos(d*x + c)^3 + 3*(a^6*b^3 - 5*a^4*b^5 + 2*a^2*b^7)*cos
(d*x + c) + (a^5*b^4 - 5*a^3*b^6 + 2*a*b^8 + (3*a^7*b^2 - 16*a^5*b^4 + 11*a^3*b^6 - 2*a*b^8)*cos(d*x + c)^2)*s
in(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) - (32*a^5*b^4 - 66*a^3*b^
6 + 6*a*b^8 - 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cos(d*x + c)^4 + 3*(a^6*b^3 - 25*a^4*b^5 + 3
5*a^2*b^7 - 3*b^9)*d*x + (45*a^7*b^2 - 143*a^5*b^4 + 219*a^3*b^6 - 9*a*b^8 + 3*(3*a^8*b - 76*a^6*b^3 + 130*a^4
*b^5 - 44*a^2*b^7 + 3*b^9)*d*x)*cos(d*x + c)^2)*sin(d*x + c))/((a^13 + 2*a^11*b^2 - 5*a^9*b^4 - 20*a^7*b^6 - 2
5*a^5*b^8 - 14*a^3*b^10 - 3*a*b^12)*d*cos(d*x + c)^3 + 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)*d*cos(d*x + c) + ((3*a^12*b + 14*a^10*b^3 + 25*a^8*b^5 + 20*a^6*b^7 + 5*a^4*b^9 - 2*a^2*b^11
- b^13)*d*cos(d*x + c)^2 + (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)*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)**2/(a+b*tan(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.32387, size = 867, normalized size = 3.28 \begin{align*} \frac{\frac{3 \,{\left (a^{6} - 25 \, a^{4} b^{2} + 35 \, a^{2} b^{4} - 3 \, b^{6}\right )}{\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac{12 \,{\left (a^{5} b - 5 \, a^{3} b^{3} + 2 \, a b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac{24 \,{\left (a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{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}} + \frac{3 \,{\left (4 \, a^{5} b \tan \left (d x + c\right )^{2} - 20 \, a^{3} b^{3} \tan \left (d x + c\right )^{2} + 8 \, a b^{5} \tan \left (d x + c\right )^{2} - a^{6} \tan \left (d x + c\right ) + 5 \, a^{4} b^{2} \tan \left (d x + c\right ) + 5 \, a^{2} b^{4} \tan \left (d x + c\right ) - b^{6} \tan \left (d x + c\right ) - 20 \, a^{3} b^{3} + 12 \, a b^{5}\right )}}{{\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 )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}} - \frac{2 \,{\left (22 \, a^{5} b^{4} \tan \left (d x + c\right )^{3} - 110 \, a^{3} b^{6} \tan \left (d x + c\right )^{3} + 44 \, a b^{8} \tan \left (d x + c\right )^{3} + 75 \, a^{6} b^{3} \tan \left (d x + c\right )^{2} - 345 \, a^{4} b^{5} \tan \left (d x + c\right )^{2} + 111 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} + 3 \, b^{9} \tan \left (d x + c\right )^{2} + 87 \, a^{7} b^{2} \tan \left (d x + c\right ) - 357 \, a^{5} b^{4} \tan \left (d x + c\right ) + 87 \, a^{3} b^{6} \tan \left (d x + c\right ) + 3 \, a b^{8} \tan \left (d x + c\right ) + 35 \, a^{8} b - 119 \, a^{6} b^{3} + 23 \, a^{4} b^{5} + a^{2} b^{7}\right )}}{{\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 )}{\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(3*(a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*(d*x + c)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b
^8 + b^10) - 12*(a^5*b - 5*a^3*b^3 + 2*a*b^5)*log(tan(d*x + c)^2 + 1)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*
b^6 + 5*a^2*b^8 + b^10) + 24*(a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*log(abs(b*tan(d*x + c) + a))/(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) + 3*(4*a^5*b*tan(d*x + c)^2 - 20*a^3*b^3*tan(d*x + c)^2 + 8*a*b^
5*tan(d*x + c)^2 - a^6*tan(d*x + c) + 5*a^4*b^2*tan(d*x + c) + 5*a^2*b^4*tan(d*x + c) - b^6*tan(d*x + c) - 20*
a^3*b^3 + 12*a*b^5)/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*(tan(d*x + c)^2 + 1)) - 2
*(22*a^5*b^4*tan(d*x + c)^3 - 110*a^3*b^6*tan(d*x + c)^3 + 44*a*b^8*tan(d*x + c)^3 + 75*a^6*b^3*tan(d*x + c)^2
- 345*a^4*b^5*tan(d*x + c)^2 + 111*a^2*b^7*tan(d*x + c)^2 + 3*b^9*tan(d*x + c)^2 + 87*a^7*b^2*tan(d*x + c) -
357*a^5*b^4*tan(d*x + c) + 87*a^3*b^6*tan(d*x + c) + 3*a*b^8*tan(d*x + c) + 35*a^8*b - 119*a^6*b^3 + 23*a^4*b^
5 + a^2*b^7)/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*(b*tan(d*x + c) + a)^3))/d