3.68 \(\int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx\)

Optimal. Leaf size=285 \[ \frac {\cos ^4(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{4 d \left (a^2+b^2\right )^3}-\frac {a^4 b}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 d \left (a^2+b^2\right )^4}+\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}+\frac {3 a x \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right )}{8 \left (a^2+b^2\right )^5} \]

[Out]

3/8*a*(a^6-25*a^4*b^2+35*a^2*b^4-3*b^6)*x/(a^2+b^2)^5+3*a^2*b*(a^4-5*a^2*b^2+2*b^4)*ln(a*cos(d*x+c)+b*sin(d*x+
c))/(a^2+b^2)^5/d-1/2*a^4*b/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^2-2*a^3*b*(a^2-2*b^2)/(a^2+b^2)^4/d/(a+b*tan(d*x+c)
)+1/4*cos(d*x+c)^4*(b*(3*a^2-b^2)+a*(a^2-3*b^2)*tan(d*x+c))/(a^2+b^2)^3/d-1/8*a*cos(d*x+c)^2*(24*a*b*(a^2-b^2)
+(5*a^4-34*a^2*b^2+9*b^4)*tan(d*x+c))/(a^2+b^2)^4/d

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Rubi [A]  time = 0.85, antiderivative size = 285, normalized size of antiderivative = 1.00, 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} \[ -\frac {a^4 b}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac {a \cos ^2(c+d x) \left (\left (-34 a^2 b^2+5 a^4+9 b^4\right ) \tan (c+d x)+24 a b \left (a^2-b^2\right )\right )}{8 d \left (a^2+b^2\right )^4}+\frac {\cos ^4(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{4 d \left (a^2+b^2\right )^3}+\frac {3 a^2 b \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac {3 a x \left (-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right )}{8 \left (a^2+b^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]

[Out]

(3*a*(a^6 - 25*a^4*b^2 + 35*a^2*b^4 - 3*b^6)*x)/(8*(a^2 + b^2)^5) + (3*a^2*b*(a^4 - 5*a^2*b^2 + 2*b^4)*Log[a*C
os[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^4*b)/(2*(a^2 + b^2)^3*d*(a + b*Tan[c + d*x])^2) - (2*a^3
*b*(a^2 - 2*b^2))/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) + (Cos[c + d*x]^4*(b*(3*a^2 - b^2) + a*(a^2 - 3*b^2)*
Tan[c + d*x]))/(4*(a^2 + b^2)^3*d) - (a*Cos[c + d*x]^2*(24*a*b*(a^2 - b^2) + (5*a^4 - 34*a^2*b^2 + 9*b^4)*Tan[
c + d*x]))/(8*(a^2 + b^2)^4*d)

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]

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {x^4}{(a+x)^3 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {\cos ^4(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 )}{4 \left (a^2+b^2\right )^3 d}-\frac {\operatorname {Subst}\left (\int \frac {\frac {a^4 b^4 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}-\frac {a^3 b^4 \left (9 a^2+5 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac {a^2 b^2 \left (4 a^4+21 a^2 b^2-3 b^4\right ) x^2}{\left (a^2+b^2\right )^3}-\frac {3 a b^4 \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 )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d}\\ &=\frac {\cos ^4(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 )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3 a^4 b^4 \left (a^4-10 a^2 b^2+5 b^4\right )}{\left (a^2+b^2\right )^4}-\frac {a^3 b^4 \left (15 a^4+26 a^2 b^2-37 b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac {3 a^2 b^4 \left (5 a^4-18 a^2 b^2-7 b^4\right ) x^2}{\left (a^2+b^2\right )^4}-\frac {a b^4 \left (5 a^4-34 a^2 b^2+9 b^4\right ) x^3}{\left (a^2+b^2\right )^4}}{(a+x)^3 \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 (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac {\operatorname {Subst}\left (\int \left (\frac {8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^3}+\frac {16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)^2}+\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)}+\frac {3 a b^4 \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 )}{8 b^3 d}\\ &=\frac {3 a^2 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^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(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 )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac {(3 a 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 )}{8 \left (a^2+b^2\right )^5 d}\\ &=\frac {3 a^2 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^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(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 )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}-\frac {\left (3 a^2 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 (3 a 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 )}{8 \left (a^2+b^2\right )^5 d}\\ &=\frac {3 a \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac {3 a^2 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 {3 a^2 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^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac {2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac {\cos ^4(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 )}{4 \left (a^2+b^2\right )^3 d}-\frac {a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}\\ \end {align*}

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Mathematica [A]  time = 6.49, size = 501, normalized size = 1.76 \[ \frac {b \left (\frac {\left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 \left (a^2+b^2\right )^3}-\frac {3 a^2 (a-b) (a+b) \cos ^2(c+d x)}{\left (a^2+b^2\right )^4}+\frac {a \left (a^2-3 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b \left (a^2+b^2\right )^3}+\frac {3 a \left (a^2-3 b^2\right ) \left (\frac {\tan ^{-1}(\tan (c+d x))}{b}+\frac {\sin (c+d x) \cos (c+d x)}{b}\right )}{8 \left (a^2+b^2\right )^3}-\frac {a^4}{2 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}+\frac {3 a^2 \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}-\frac {a^3 \left (a^2-5 b^2\right ) \tan ^{-1}(\tan (c+d x))}{b \left (a^2+b^2\right )^4}-\frac {2 a^3 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac {a^3 \left (a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{b \left (a^2+b^2\right )^4}-\frac {a^2 \left (3 a^4-15 a^2 b^2-\frac {a^5-13 a^3 b^2+10 a b^4}{\sqrt {-b^2}}+6 b^4\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}-\frac {a^2 \left (3 a^4-15 a^2 b^2+\frac {a^5-13 a^3 b^2+10 a b^4}{\sqrt {-b^2}}+6 b^4\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^4/(a + b*Tan[c + d*x])^3,x]

[Out]

(b*(-((a^3*(a^2 - 5*b^2)*ArcTan[Tan[c + d*x]])/(b*(a^2 + b^2)^4)) - (3*a^2*(a - b)*(a + b)*Cos[c + d*x]^2)/(a^
2 + b^2)^4 + ((3*a^2 - b^2)*Cos[c + d*x]^4)/(4*(a^2 + b^2)^3) - (a^2*(3*a^4 - 15*a^2*b^2 + 6*b^4 - (a^5 - 13*a
^3*b^2 + 10*a*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(2*(a^2 + b^2)^5) + (3*a^2*(a^4 - 5*a^2*b^2 +
2*b^4)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^5 - (a^2*(3*a^4 - 15*a^2*b^2 + 6*b^4 + (a^5 - 13*a^3*b^2 + 10*a*b
^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^5) - (a^3*(a^2 - 5*b^2)*Cos[c + d*x]*Sin[c +
d*x])/(b*(a^2 + b^2)^4) + (a*(a^2 - 3*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*(a^2 + b^2)^3) + (3*a*(a^2 - 3*b^
2)*(ArcTan[Tan[c + d*x]]/b + (Cos[c + d*x]*Sin[c + d*x])/b))/(8*(a^2 + b^2)^3) - a^4/(2*(a^2 + b^2)^3*(a + b*T
an[c + d*x])^2) - (2*a^3*(a^2 - 2*b^2))/((a^2 + b^2)^4*(a + b*Tan[c + d*x]))))/d

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fricas [B]  time = 0.57, size = 705, normalized size = 2.47 \[ \frac {119 \, a^{6} b^{3} - 159 \, a^{4} b^{5} - 51 \, a^{2} b^{7} + 3 \, b^{9} + 8 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} - 8 \, {\left (5 \, a^{8} b + 16 \, a^{6} b^{3} + 18 \, a^{4} b^{5} + 8 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4} + 12 \, {\left (a^{7} b^{2} - 25 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 3 \, a b^{8}\right )} d x - {\left (a^{8} b + 110 \, a^{6} b^{3} - 420 \, a^{4} b^{5} - 78 \, a^{2} b^{7} + 3 \, b^{9} - 12 \, {\left (a^{9} - 26 \, a^{7} b^{2} + 60 \, a^{5} b^{4} - 38 \, a^{3} b^{6} + 3 \, a b^{8}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (a^{6} b^{3} - 5 \, a^{4} b^{5} + 2 \, a^{2} b^{7} + {\left (a^{8} b - 6 \, a^{6} b^{3} + 7 \, a^{4} b^{5} - 2 \, a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{2} - 5 \, a^{5} b^{4} + 2 \, a^{3} 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 (4 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{5} - 2 \, {\left (5 \, a^{9} + 12 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} \cos \left (d x + c\right )^{3} + {\left (77 \, a^{7} b^{2} - 69 \, a^{5} b^{4} + 63 \, a^{3} b^{6} - 15 \, a b^{8} + 12 \, {\left (a^{8} b - 25 \, a^{6} b^{3} + 35 \, a^{4} b^{5} - 3 \, a^{2} b^{7}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{32 \, {\left ({\left (a^{12} + 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} - 4 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b + 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} + 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} + a b^{11}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{10} b^{2} + 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} + 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} + b^{12}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/32*(119*a^6*b^3 - 159*a^4*b^5 - 51*a^2*b^7 + 3*b^9 + 8*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cos
(d*x + c)^6 - 8*(5*a^8*b + 16*a^6*b^3 + 18*a^4*b^5 + 8*a^2*b^7 + b^9)*cos(d*x + c)^4 + 12*(a^7*b^2 - 25*a^5*b^
4 + 35*a^3*b^6 - 3*a*b^8)*d*x - (a^8*b + 110*a^6*b^3 - 420*a^4*b^5 - 78*a^2*b^7 + 3*b^9 - 12*(a^9 - 26*a^7*b^2
+ 60*a^5*b^4 - 38*a^3*b^6 + 3*a*b^8)*d*x)*cos(d*x + c)^2 + 48*(a^6*b^3 - 5*a^4*b^5 + 2*a^2*b^7 + (a^8*b - 6*a
^6*b^3 + 7*a^4*b^5 - 2*a^2*b^7)*cos(d*x + c)^2 + 2*(a^7*b^2 - 5*a^5*b^4 + 2*a^3*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*(4*(a^9 + 4*a^7*b^2 + 6*a^5*b^4
+ 4*a^3*b^6 + a*b^8)*cos(d*x + c)^5 - 2*(5*a^9 + 12*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 - 3*a*b^8)*cos(d*x + c)^3
+ (77*a^7*b^2 - 69*a^5*b^4 + 63*a^3*b^6 - 15*a*b^8 + 12*(a^8*b - 25*a^6*b^3 + 35*a^4*b^5 - 3*a^2*b^7)*d*x)*cos
(d*x + c))*sin(d*x + c))/((a^12 + 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 - 4*a^2*b^10 - b^12)*d*cos(d*x + c)^2 + 2
*(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*d*cos(d*x + c)*sin(d*x + c) + (a^10*b^2 +
5*a^8*b^4 + 10*a^6*b^6 + 10*a^4*b^8 + 5*a^2*b^10 + b^12)*d)

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giac [B]  time = 3.87, size = 588, normalized size = 2.06 \[ \frac {\frac {3 \, {\left (a^{7} - 25 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 3 \, a b^{6}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {12 \, {\left (a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {24 \, {\left (a^{6} b^{2} - 5 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} + 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} + b^{11}} - \frac {21 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} - 66 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 9 \, a b^{6} \tan \left (d x + c\right )^{5} + 30 \, a^{6} b \tan \left (d x + c\right )^{4} - 72 \, a^{4} b^{3} \tan \left (d x + c\right )^{4} - 6 \, a^{2} b^{5} \tan \left (d x + c\right )^{4} + 5 \, a^{7} \tan \left (d x + c\right )^{3} + 49 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} - 133 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 15 \, a b^{6} \tan \left (d x + c\right )^{3} + 70 \, a^{6} b \tan \left (d x + c\right )^{2} - 122 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 2 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 2 \, b^{7} \tan \left (d x + c\right )^{2} + 3 \, a^{7} \tan \left (d x + c\right ) + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) - 73 \, a^{3} b^{4} \tan \left (d x + c\right ) + 4 \, a b^{6} \tan \left (d x + c\right ) + 38 \, a^{6} b - 56 \, a^{4} b^{3} + 2 \, a^{2} b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right )^{3} + a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right ) + a\right )}^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(3*(a^7 - 25*a^5*b^2 + 35*a^3*b^4 - 3*a*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^6*b - 5*a^4*b^3 + 2*a^2*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^6*b^2 - 5*a^4*b^4 + 2*a^2*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) - (21*a^5*b^2*tan(d*x + c)^5 - 66*a^3*b^4*tan(d*x + c)^5 +
9*a*b^6*tan(d*x + c)^5 + 30*a^6*b*tan(d*x + c)^4 - 72*a^4*b^3*tan(d*x + c)^4 - 6*a^2*b^5*tan(d*x + c)^4 + 5*a
^7*tan(d*x + c)^3 + 49*a^5*b^2*tan(d*x + c)^3 - 133*a^3*b^4*tan(d*x + c)^3 + 15*a*b^6*tan(d*x + c)^3 + 70*a^6*
b*tan(d*x + c)^2 - 122*a^4*b^3*tan(d*x + c)^2 + 2*a^2*b^5*tan(d*x + c)^2 + 2*b^7*tan(d*x + c)^2 + 3*a^7*tan(d*
x + c) + 22*a^5*b^2*tan(d*x + c) - 73*a^3*b^4*tan(d*x + c) + 4*a*b^6*tan(d*x + c) + 38*a^6*b - 56*a^4*b^3 + 2*
a^2*b^5)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*(b*tan(d*x + c)^3 + a*tan(d*x + c)^2 + b*tan(d*x + c
) + a)^2))/d

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maple [B]  time = 0.49, size = 882, normalized size = 3.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^4/(a+b*tan(d*x+c))^3,x)

[Out]

3/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^2*b^5+29/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^5
*b^2-3/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)^2*a^6*b+27/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^
5*b^2-9/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a*b^6+25/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)
^3*a^3*b^4-15/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a*b^6-1/4/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*b^7+3/8
/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^7+15/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^3*b^4-2/d*b*a^5/(a^2+
b^2)^4/(a+b*tan(d*x+c))+4/d*b^3*a^3/(a^2+b^2)^4/(a+b*tan(d*x+c))-3/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a^2*b^5+3/
d*a^6*b/(a^2+b^2)^5*ln(a+b*tan(d*x+c))-9/8/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a*b^6-75/8/d/(a^2+b^2)^5*arctan(ta
n(d*x+c))*a^5*b^2+105/8/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^3*b^4+5/4/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*a^4*b^3+
13/4/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*a^2*b^5-3/2/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a^6*b+15/2/d/(a^2+b^2)^5*ln
(1+tan(d*x+c)^2)*a^4*b^3-1/2*a^4*b/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^2+6/d*a^2*b^5/(a^2+b^2)^5*ln(a+b*tan(d*x+c))
-15/d*a^4*b^3/(a^2+b^2)^5*ln(a+b*tan(d*x+c))-5/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)^3*a^7-3/8/d/(a^2+
b^2)^5/(1+tan(d*x+c)^2)^2*tan(d*x+c)*a^7-9/4/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^2*a^6*b

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maxima [B]  time = 0.70, size = 744, normalized size = 2.61 \[ \frac {\frac {3 \, {\left (a^{7} - 25 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 3 \, a 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^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} 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^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} 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^{5} b^{2} - 22 \, a^{3} b^{4} + 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + 6 \, {\left (5 \, a^{6} b - 12 \, a^{4} b^{3} - a^{2} b^{5}\right )} \tan \left (d x + c\right )^{4} + {\left (5 \, a^{7} + 49 \, a^{5} b^{2} - 133 \, a^{3} b^{4} + 15 \, a b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (35 \, a^{6} b - 61 \, a^{4} b^{3} + a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{7} + 22 \, a^{5} b^{2} - 73 \, a^{3} b^{4} + 4 \, a b^{6}\right )} \tan \left (d x + c\right )}{a^{10} + 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} + 4 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{5} + {\left (a^{10} + 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} + 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} + 2 \, b^{10}\right )} \tan \left (d x + c\right )^{4} + 4 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (2 \, a^{10} + 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} + 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} + b^{10}\right )} \tan \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 )} \tan \left (d x + c\right )}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^4/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/8*(3*(a^7 - 25*a^5*b^2 + 35*a^3*b^4 - 3*a*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^6*b - 5*a^4*b^3 + 2*a^2*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^6*b - 5*a^4*b^3 + 2*a^2*b^5)*log(tan(d*x + c)^2 + 1)/(a^10 + 5*a^8*b^2 + 1
0*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^5*b^2 - 22*a^3*b^4 +
3*a*b^6)*tan(d*x + c)^5 + 6*(5*a^6*b - 12*a^4*b^3 - a^2*b^5)*tan(d*x + c)^4 + (5*a^7 + 49*a^5*b^2 - 133*a^3*b^
4 + 15*a*b^6)*tan(d*x + c)^3 + 2*(35*a^6*b - 61*a^4*b^3 + a^2*b^5 + b^7)*tan(d*x + c)^2 + (3*a^7 + 22*a^5*b^2
- 73*a^3*b^4 + 4*a*b^6)*tan(d*x + c))/(a^10 + 4*a^8*b^2 + 6*a^6*b^4 + 4*a^4*b^6 + a^2*b^8 + (a^8*b^2 + 4*a^6*b
^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*tan(d*x + c)^6 + 2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*tan(
d*x + c)^5 + (a^10 + 6*a^8*b^2 + 14*a^6*b^4 + 16*a^4*b^6 + 9*a^2*b^8 + 2*b^10)*tan(d*x + c)^4 + 4*(a^9*b + 4*a
^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*tan(d*x + c)^3 + (2*a^10 + 9*a^8*b^2 + 16*a^6*b^4 + 14*a^4*b^6 + 6*a^2
*b^8 + b^10)*tan(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)*tan(d*x + c)))/d

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mupad [B]  time = 5.34, size = 717, normalized size = 2.52 \[ \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 {24\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {45\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {24\,b^7}{{\left (a^2+b^2\right )}^5}\right )}{d}-\frac {\frac {19\,a^6\,b-28\,a^4\,b^3+a^2\,b^5}{4\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (35\,a^6\,b-61\,a^4\,b^3+a^2\,b^5+b^7\right )}{4\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}-\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (-5\,a^6\,b+12\,a^4\,b^3+a^2\,b^5\right )}{4\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {3\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (7\,a^5\,b^2-22\,a^3\,b^4+3\,a\,b^6\right )}{8\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (5\,a^7+49\,a^5\,b^2-133\,a^3\,b^4+15\,a\,b^6\right )}{8\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (3\,a^6+22\,a^4\,b^2-73\,a^2\,b^4+4\,b^6\right )}{8\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2+b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^2+2\,b^2\right )+a^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^6+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+4\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a^2\,1{}\mathrm {i}+3\,b\,a\right )}{16\,d\,\left (a^5+a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4+b^5\,1{}\mathrm {i}\right )}-\frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (3\,a\,b-a^2\,1{}\mathrm {i}\right )}{16\,d\,\left (a^5-a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4-b^5\,1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^4/(a + b*tan(c + d*x))^3,x)

[Out]

(log(a + b*tan(c + d*x))*((3*b)/(a^2 + b^2)^2 - (24*b^3)/(a^2 + b^2)^3 + (45*b^5)/(a^2 + b^2)^4 - (24*b^7)/(a^
2 + b^2)^5))/d - ((19*a^6*b + a^2*b^5 - 28*a^4*b^3)/(4*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan
(c + d*x)^2*(35*a^6*b + b^7 + a^2*b^5 - 61*a^4*b^3))/(4*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) - (3*
tan(c + d*x)^4*(a^2*b^5 - 5*a^6*b + 12*a^4*b^3))/(4*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (3*tan(
c + d*x)^5*(3*a*b^6 - 22*a^3*b^4 + 7*a^5*b^2))/(8*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c +
d*x)^3*(15*a*b^6 + 5*a^7 - 133*a^3*b^4 + 49*a^5*b^2))/(8*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (a
*tan(c + d*x)*(3*a^6 + 4*b^6 - 73*a^2*b^4 + 22*a^4*b^2))/(8*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)))/
(d*(tan(c + d*x)^2*(2*a^2 + b^2) + tan(c + d*x)^4*(a^2 + 2*b^2) + a^2 + b^2*tan(c + d*x)^6 + 2*a*b*tan(c + d*x
) + 4*a*b*tan(c + d*x)^3 + 2*a*b*tan(c + d*x)^5)) - (3*log(tan(c + d*x) - 1i)*(3*a*b + a^2*1i))/(16*d*(5*a*b^4
+ a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2)) - (3*log(tan(c + d*x) + 1i)*(3*a*b - a^2*1i))/(16*d*(5
*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2))

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**4/(a+b*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError

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