\(\int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [69]

Optimal result

Integrand size = 21, antiderivative size = 206 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {a \left (a^4-14 a^2 b^2+9 b^4\right ) x}{2 \left (a^2+b^2\right )^4}+\frac {b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d} \]

[Out]

1/2*a*(a^4-14*a^2*b^2+9*b^4)*x/(a^2+b^2)^4+b*(3*a^4-8*a^2*b^2+b^4)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^4/d
-1/2*a^2*b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-2*a*b*(a^2-b^2)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))-1/2*cos(d*x+c)^2*(b
*(3*a^2-b^2)+a*(a^2-3*b^2)*tan(d*x+c))/(a^2+b^2)^3/d

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3597, 1661, 1643, 649, 209, 266} \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {a^2 b}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {2 a b \left (a^2-b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{2 d \left (a^2+b^2\right )^3}+\frac {b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {a x \left (a^4-14 a^2 b^2+9 b^4\right )}{2 \left (a^2+b^2\right )^4} \]

[In]

Int[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]

[Out]

(a*(a^4 - 14*a^2*b^2 + 9*b^4)*x)/(2*(a^2 + b^2)^4) + (b*(3*a^4 - 8*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c
+ d*x]])/((a^2 + b^2)^4*d) - (a^2*b)/(2*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x])^2) - (2*a*b*(a^2 - b^2))/((a^2 +
b^2)^3*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^2*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*Tan[c + d*x]))/(2*(a^2 +
b^2)^3*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^2}{(a+x)^3 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\text {Subst}\left (\int \frac {-\frac {a^4 b^2 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac {a^3 b^2 \left (3 a^2+7 b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {b^2 \left (3 a^4-3 a^2 b^2-2 b^4\right ) x^2}{\left (a^2+b^2\right )^3}+\frac {a b^2 \left (a^2-3 b^2\right ) x^3}{\left (a^2+b^2\right )^3}}{(a+x)^3 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{2 b d} \\ & = -\frac {\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\text {Subst}\left (\int \left (-\frac {2 a^2 b^2}{\left (a^2+b^2\right )^2 (a+x)^3}+\frac {4 a b^2 \left (-a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+x)^2}-\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)}+\frac {b^2 \left (-a \left (a^4-14 a^2 b^2+9 b^4\right )+2 \left (3 a^4-8 a^2 b^2+b^4\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 )}{2 b d} \\ & = \frac {b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {b \text {Subst}\left (\int \frac {-a \left (a^4-14 a^2 b^2+9 b^4\right )+2 \left (3 a^4-8 a^2 b^2+b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d} \\ & = \frac {b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d}-\frac {\left (b \left (3 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 )^4 d}+\frac {\left (a b \left (a^4-14 a^2 b^2+9 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d} \\ & = \frac {a \left (a^4-14 a^2 b^2+9 b^4\right ) x}{2 \left (a^2+b^2\right )^4}+\frac {b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {b \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 b}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {2 a b \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.21 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.53 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {b \left (\frac {a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \arctan (\tan (c+d x))}{b}+\left (3 a^2-b^2\right ) \left (a^2+b^2\right ) \cos ^2(c+d x)+\left (3 a^4-8 a^2 b^2+b^4-\frac {a^5-8 a^3 b^2+3 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )-2 \left (3 a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))+\left (3 a^4-8 a^2 b^2+b^4+\frac {a^5-8 a^3 b^2+3 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+\frac {a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \sin (2 (c+d x))}{2 b}+\frac {a^2 \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac {4 \left (a^5-a b^4\right )}{a+b \tan (c+d x)}\right )}{2 \left (a^2+b^2\right )^4 d} \]

[In]

Integrate[Sin[c + d*x]^2/(a + b*Tan[c + d*x])^3,x]

[Out]

-1/2*(b*((a*(a^2 - 3*b^2)*(a^2 + b^2)*ArcTan[Tan[c + d*x]])/b + (3*a^2 - b^2)*(a^2 + b^2)*Cos[c + d*x]^2 + (3*
a^4 - 8*a^2*b^2 + b^4 - (a^5 - 8*a^3*b^2 + 3*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]] - 2*(3*a^4 -
8*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]] + (3*a^4 - 8*a^2*b^2 + b^4 + (a^5 - 8*a^3*b^2 + 3*a*b^4)/Sqrt[-b^2])*
Log[Sqrt[-b^2] + b*Tan[c + d*x]] + (a*(a^2 - 3*b^2)*(a^2 + b^2)*Sin[2*(c + d*x)])/(2*b) + (a^2*(a^2 + b^2)^2)/
(a + b*Tan[c + d*x])^2 + (4*(a^5 - a*b^4))/(a + b*Tan[c + d*x])))/((a^2 + b^2)^4*d)

Maple [A] (verified)

Time = 5.73 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {-\frac {a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {2 a b \left (a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-\frac {1}{2} a^{5}+a^{3} b^{2}+\frac {3}{2} a \,b^{4}\right ) \tan \left (d x +c \right )-\frac {3 a^{4} b}{2}-a^{2} b^{3}+\frac {b^{5}}{2}}{1+\tan ^{2}\left (d x +c \right )}+\frac {\left (-6 a^{4} b +16 a^{2} b^{3}-2 b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{4}+\frac {\left (a^{5}-14 a^{3} b^{2}+9 a \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(236\)
default	\(\frac {-\frac {a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {2 a b \left (a^{2}-b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-\frac {1}{2} a^{5}+a^{3} b^{2}+\frac {3}{2} a \,b^{4}\right ) \tan \left (d x +c \right )-\frac {3 a^{4} b}{2}-a^{2} b^{3}+\frac {b^{5}}{2}}{1+\tan ^{2}\left (d x +c \right )}+\frac {\left (-6 a^{4} b +16 a^{2} b^{3}-2 b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{4}+\frac {\left (a^{5}-14 a^{3} b^{2}+9 a \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{4}}}{d}\)	\(236\)
risch	\(-\frac {i x b}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {x a}{2 \left (4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}\right )}+\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 \left (3 i b \,a^{2}-i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {6 i b \,a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}+\frac {16 i b^{3} a^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {2 i b^{5} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}}-\frac {6 i b \,a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {16 i b^{3} a^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {2 i b^{5} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {2 i a \,b^{2} \left (-3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-2 a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-3 i a^{3}+2 i a \,b^{2}+3 a^{2} b -2 b^{3}\right )}{\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} \left (-i a +b \right )^{3} d \left (i a +b \right )^{4}}+\frac {3 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}-\frac {8 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{2}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 b^{6} a^{2}+b^{8}\right )}\)	\(776\)

[In]

int(sin(d*x+c)^2/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*a^2*b/(a^2+b^2)^2/(a+b*tan(d*x+c))^2+b*(3*a^4-8*a^2*b^2+b^4)/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-2*a*b*(a
^2-b^2)/(a^2+b^2)^3/(a+b*tan(d*x+c))+1/(a^2+b^2)^4*(((-1/2*a^5+a^3*b^2+3/2*a*b^4)*tan(d*x+c)-3/2*a^4*b-a^2*b^3
+1/2*b^5)/(1+tan(d*x+c)^2)+1/4*(-6*a^4*b+16*a^2*b^3-2*b^5)*ln(1+tan(d*x+c)^2)+1/2*(a^5-14*a^3*b^2+9*a*b^4)*arc
tan(tan(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (200) = 400\).

Time = 0.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.55 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {13 \, a^{4} b^{3} - 8 \, a^{2} b^{5} - b^{7} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (a^{5} b^{2} - 14 \, a^{3} b^{4} + 9 \, a b^{6}\right )} d x - {\left (a^{6} b + 23 \, a^{4} b^{3} - 21 \, a^{2} b^{5} - 3 \, b^{7} - 2 \, {\left (a^{7} - 15 \, a^{5} b^{2} + 23 \, a^{3} b^{4} - 9 \, a b^{6}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{4} b^{3} - 8 \, a^{2} b^{5} + b^{7} + {\left (3 \, a^{6} b - 11 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{5} b^{2} - 8 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (4 \, a^{5} b^{2} - 3 \, a^{3} b^{4} + 3 \, a b^{6} + {\left (a^{6} b - 14 \, a^{4} b^{3} + 9 \, a^{2} b^{5}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \]

[In]

integrate(sin(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/4*(13*a^4*b^3 - 8*a^2*b^5 - b^7 - 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)^4 + 2*(a^5*b^2 - 14*a
^3*b^4 + 9*a*b^6)*d*x - (a^6*b + 23*a^4*b^3 - 21*a^2*b^5 - 3*b^7 - 2*(a^7 - 15*a^5*b^2 + 23*a^3*b^4 - 9*a*b^6)
*d*x)*cos(d*x + c)^2 + 2*(3*a^4*b^3 - 8*a^2*b^5 + b^7 + (3*a^6*b - 11*a^4*b^3 + 9*a^2*b^5 - b^7)*cos(d*x + c)^
2 + 2*(3*a^5*b^2 - 8*a^3*b^4 + a*b^6)*cos(d*x + c)*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) - 2*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(d*x + c)^3 - 2*(4*a^5*b^2 - 3*a^3*b^
4 + 3*a*b^6 + (a^6*b - 14*a^4*b^3 + 9*a^2*b^5)*d*x)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^8*b^2 + 2*a^6*b^4
- 2*a^4*b^6 - 3*a^2*b^8 - b^10)*d*cos(d*x + c)^2 + 2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*d*co
s(d*x + c)*sin(d*x + c) + (a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*d)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(sin(d*x+c)**2/(a+b*tan(d*x+c))**3,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. 463 vs. \(2 (200) = 400\).

Time = 0.45 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.25 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (a^{5} - 14 \, a^{3} b^{2} + 9 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {2 \, {\left (3 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (3 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {8 \, a^{4} b - 4 \, a^{2} b^{3} + {\left (5 \, a^{3} b^{2} - 7 \, a b^{4}\right )} \tan \left (d x + c\right )^{3} + {\left (7 \, a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} + 7 \, a^{3} b^{2} - 6 \, a b^{4}\right )} \tan \left (d x + c\right )}{a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )}}{2 \, d} \]

[In]

integrate(sin(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*((a^5 - 14*a^3*b^2 + 9*a*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 2*(3*a^4*b - 8*a
^2*b^3 + b^5)*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) - (3*a^4*b - 8*a^2*b^3 +
b^5)*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) - (8*a^4*b - 4*a^2*b^3 + (5*a^3*
b^2 - 7*a*b^4)*tan(d*x + c)^3 + (7*a^4*b - 6*a^2*b^3 - b^5)*tan(d*x + c)^2 + (a^5 + 7*a^3*b^2 - 6*a*b^4)*tan(d
*x + c))/(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6 + (a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*tan(d*x + c)^4 + 2*(
a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*tan(d*x + c)^3 + (a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*tan(d*
x + c)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 482 vs. \(2 (200) = 400\).

Time = 0.64 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.34 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\left (a^{5} - 14 \, a^{3} b^{2} + 9 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (3 \, a^{4} b - 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {2 \, {\left (3 \, a^{4} b^{2} - 8 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {3 \, a^{4} b \tan \left (d x + c\right )^{2} - 8 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + b^{5} \tan \left (d x + c\right )^{2} - a^{5} \tan \left (d x + c\right ) + 2 \, a^{3} b^{2} \tan \left (d x + c\right ) + 3 \, a b^{4} \tan \left (d x + c\right ) - 10 \, a^{2} b^{3} + 2 \, b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}} - \frac {9 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} - 24 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 3 \, b^{7} \tan \left (d x + c\right )^{2} + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) - 48 \, a^{3} b^{4} \tan \left (d x + c\right ) + 2 \, a b^{6} \tan \left (d x + c\right ) + 14 \, a^{6} b - 22 \, a^{4} b^{3}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]

[In]

integrate(sin(d*x+c)^2/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/2*((a^5 - 14*a^3*b^2 + 9*a*b^4)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - (3*a^4*b - 8*a^2
*b^3 + b^5)*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) + 2*(3*a^4*b^2 - 8*a^2*b^4
+ b^6)*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) + (3*a^4*b*tan(d*x + c)
^2 - 8*a^2*b^3*tan(d*x + c)^2 + b^5*tan(d*x + c)^2 - a^5*tan(d*x + c) + 2*a^3*b^2*tan(d*x + c) + 3*a*b^4*tan(d
*x + c) - 10*a^2*b^3 + 2*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)) - (9*a^4*
b^3*tan(d*x + c)^2 - 24*a^2*b^5*tan(d*x + c)^2 + 3*b^7*tan(d*x + c)^2 + 22*a^5*b^2*tan(d*x + c) - 48*a^3*b^4*t
an(d*x + c) + 2*a*b^6*tan(d*x + c) + 14*a^6*b - 22*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)^2))/d

Mupad [B] (verification not implemented)

Time = 5.52 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.10 \[ \int \frac {\sin ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-7\,a^4\,b+6\,a^2\,b^3+b^5\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (7\,a\,b^4-5\,a^3\,b^2\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {2\,a^2\,\left (2\,a^2\,b-b^3\right )}{\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^4+7\,a^2\,b^2-6\,b^4\right )}{2\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^2+b^2\right )+a^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {3\,b}{{\left (a^2+b^2\right )}^2}-\frac {14\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {12\,b^5}{{\left (a^2+b^2\right )}^4}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-2\,b+a\,1{}\mathrm {i}\right )}{4\,d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a-b\,2{}\mathrm {i}\right )}{4\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]

[In]

int(sin(c + d*x)^2/(a + b*tan(c + d*x))^3,x)

[Out]

((tan(c + d*x)^2*(b^5 - 7*a^4*b + 6*a^2*b^3))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^3*(7*a*b
^4 - 5*a^3*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (2*a^2*(2*a^2*b - b^3))/((a^2 + b^2)*(a^4 + b^4 + 2
*a^2*b^2)) - (a*tan(c + d*x)*(a^4 - 6*b^4 + 7*a^2*b^2))/(2*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)))/(d*(tan(c + d
*x)^2*(a^2 + b^2) + a^2 + b^2*tan(c + d*x)^4 + 2*a*b*tan(c + d*x) + 2*a*b*tan(c + d*x)^3)) + (log(a + b*tan(c
+ d*x))*((3*b)/(a^2 + b^2)^2 - (14*b^3)/(a^2 + b^2)^3 + (12*b^5)/(a^2 + b^2)^4))/d + (log(tan(c + d*x) + 1i)*(
a*1i - 2*b))/(4*d*(a*b^3*4i - a^3*b*4i + a^4 + b^4 - 6*a^2*b^2)) + (log(tan(c + d*x) - 1i)*(a - b*2i))/(4*d*(4
*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i))