\(\int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [73]

Optimal result

Integrand size = 21, antiderivative size = 366 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {\left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right ) x}{8 \left (a^2+b^2\right )^6}+\frac {4 a b \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^4 b}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^3}-\frac {a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (4 a b \left (a^2-b^2\right )+\left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 d}-\frac {\cos ^2(c+d x) \left (16 a b \left (2 a^4-5 a^2 b^2+b^4\right )+\left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d} \] Output:

1/8*(3*a^8-132*a^6*b^2+370*a^4*b^4-132*a^2*b^6+3*b^8)*x/(a^2+b^2)^6+4*a*b* 
(a^2-b^2)*(a^4-8*a^2*b^2+b^4)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^6/d- 
1/3*a^4*b/(a^2+b^2)^3/d/(a+b*tan(d*x+c))^3-a^3*b*(a^2-2*b^2)/(a^2+b^2)^4/d 
/(a+b*tan(d*x+c))^2-3*a^2*b*(a^4-5*a^2*b^2+2*b^4)/(a^2+b^2)^5/d/(a+b*tan(d 
*x+c))+1/4*cos(d*x+c)^4*(4*a*b*(a^2-b^2)+(a^4-6*a^2*b^2+b^4)*tan(d*x+c))/( 
a^2+b^2)^4/d-1/8*cos(d*x+c)^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(d*x+c))/(a^2+b^2)^5/d
 

Mathematica [A] (verified)

Time = 3.58 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=-\frac {b \left (-\frac {9 \left (a^2+b^2\right )^2 \left (a^4-6 a^2 b^2+b^4\right ) \arctan (\tan (c+d x))}{b}+\frac {24 a^2 \left (a^2+b^2\right ) \left (a^4-10 a^2 b^2+5 b^4\right ) \arctan (\tan (c+d x))}{b}+48 a \left (a^2+b^2\right ) \left (2 a^4-5 a^2 b^2+b^4\right ) \cos ^2(c+d x)-24 a (a-b) (a+b) \left (a^2+b^2\right )^2 \cos ^4(c+d x)+12 a \left (4 a^6-36 a^4 b^2+36 a^2 b^4-4 b^6+\frac {-a^7+24 a^5 b^2-45 a^3 b^4+10 a b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )-96 a (a-b) (a+b) \left (a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))+12 a \left (4 a^6-36 a^4 b^2+36 a^2 b^4-4 b^6+\frac {a^7-24 a^5 b^2+45 a^3 b^4-10 a b^6}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )-\frac {6 \left (a^2+b^2\right )^2 \left (a^4-6 a^2 b^2+b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{b}-\frac {9 \left (a^2+b^2\right )^2 \left (a^4-6 a^2 b^2+b^4\right ) \sin (2 (c+d x))}{2 b}+\frac {12 a^2 \left (a^2+b^2\right ) \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (2 (c+d x))}{b}+\frac {8 a^4 \left (a^2+b^2\right )^3}{(a+b \tan (c+d x))^3}+\frac {24 a^3 \left (a^2-2 b^2\right ) \left (a^2+b^2\right )^2}{(a+b \tan (c+d x))^2}+\frac {72 a^2 \left (a^2+b^2\right ) \left (a^4-5 a^2 b^2+2 b^4\right )}{a+b \tan (c+d x)}\right )}{24 \left (a^2+b^2\right )^6 d} \] Input:

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

Output:

-1/24*(b*((-9*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*ArcTan[Tan[c + d*x]])/ 
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]] - 96*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 - (9*(a^2 + b^2)^2*(a^4 - 6*a^2*b^2 + b^4)*Sin[2*(c + d*x 
)])/(2*b) + (12*a^2*(a^2 + b^2)*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[2*(c + d*x) 
])/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])))/((a^2 + b^2)^6*d)
 

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3999, 601, 2178, 2160, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3999

\(\displaystyle \frac {b \int \frac {b^4 \tan ^4(c+d x)}{(a+b \tan (c+d x))^4 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 601

\(\displaystyle \frac {b \left (\frac {b^2 \left (4 a b^2 \left (a^2-b^2\right )+b \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {\int \frac {-\frac {3 \left (a^4-6 b^2 a^2+b^4\right ) \tan ^4(c+d x) b^8}{\left (a^2+b^2\right )^4}-\frac {4 a \left (3 a^4-14 b^2 a^2-b^4\right ) \tan ^3(c+d x) b^7}{\left (a^2+b^2\right )^4}-\frac {4 a^3 \left (3 a^2-b^2\right ) \tan (c+d x) b^5}{\left (a^2+b^2\right )^3}-\frac {2 a^2 \left (2 a^6+17 b^2 a^4-12 b^4 a^2-3 b^6\right ) \tan ^2(c+d x) b^4}{\left (a^2+b^2\right )^4}+\frac {a^4 \left (a^4-6 b^2 a^2+b^4\right ) b^4}{\left (a^2+b^2\right )^4}}{(a+b \tan (c+d x))^4 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2}\right )}{d}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {b \left (\frac {b^2 \left (4 a b^2 \left (a^2-b^2\right )+b \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {\frac {b^2 \left (16 a b^2 \left (2 a^4-5 a^2 b^2+b^4\right )+b \left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {\int \frac {-\frac {\left (5 a^6-65 b^2 a^4+55 b^4 a^2-3 b^6\right ) \tan ^4(c+d x) b^8}{\left (a^2+b^2\right )^5}-\frac {4 a \left (5 a^2+b^2\right ) \left (a^4-10 b^2 a^2+5 b^4\right ) \tan ^3(c+d x) b^7}{\left (a^2+b^2\right )^5}-\frac {30 a^2 \left (a^4-6 b^2 a^2+b^4\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^4}-\frac {4 a^3 \left (a^2+5 b^2\right ) \left (5 a^4-10 b^2 a^2+b^4\right ) \tan (c+d x) b^5}{\left (a^2+b^2\right )^5}+\frac {a^4 \left (3 a^6-55 b^2 a^4+65 b^4 a^2-5 b^6\right ) b^4}{\left (a^2+b^2\right )^5}}{(a+b \tan (c+d x))^4 \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{2 b^2}}{4 b^2}\right )}{d}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {b \left (\frac {b^2 \left (4 a b^2 \left (a^2-b^2\right )+b \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {\frac {b^2 \left (16 a b^2 \left (2 a^4-5 a^2 b^2+b^4\right )+b \left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {\int \left (\frac {32 a \left (a^2-b^2\right ) \left (a^4-8 b^2 a^2+b^4\right ) b^4}{\left (a^2+b^2\right )^6 (a+b \tan (c+d x))}+\frac {\left (3 a^8-132 b^2 a^6+370 b^4 a^4-132 b^6 a^2-32 b \left (a^2-b^2\right ) \left (a^4-8 b^2 a^2+b^4\right ) \tan (c+d x) a+3 b^8\right ) b^4}{\left (a^2+b^2\right )^6 \left (\tan ^2(c+d x) b^2+b^2\right )}+\frac {24 a^2 \left (a^4-5 b^2 a^2+2 b^4\right ) b^4}{\left (a^2+b^2\right )^5 (a+b \tan (c+d x))^2}+\frac {16 a^3 \left (a^2-2 b^2\right ) b^4}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))^3}+\frac {8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))^4}\right )d(b \tan (c+d x))}{2 b^2}}{4 b^2}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {b^2 \left (4 a b^2 \left (a^2-b^2\right )+b \left (a^4-6 a^2 b^2+b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {\frac {b^2 \left (16 a b^2 \left (2 a^4-5 a^2 b^2+b^4\right )+b \left (5 a^6-65 a^4 b^2+55 a^2 b^4-3 b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {-\frac {24 a^2 b^4 \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 (a+b \tan (c+d x))}-\frac {8 a^4 b^4}{3 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3}-\frac {16 a b^4 \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log \left (b^2 \tan ^2(c+d x)+b^2\right )}{\left (a^2+b^2\right )^6}+\frac {32 a b^4 \left (a^2-b^2\right ) \left (a^4-8 a^2 b^2+b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6}-\frac {8 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}+\frac {b^3 \left (3 a^8-132 a^6 b^2+370 a^4 b^4-132 a^2 b^6+3 b^8\right ) \arctan (\tan (c+d x))}{\left (a^2+b^2\right )^6}}{2 b^2}}{4 b^2}\right )}{d}\)

Input:

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

Output:

(b*((b^2*(4*a*b^2*(a^2 - b^2) + b*(a^4 - 6*a^2*b^2 + b^4)*Tan[c + d*x]))/( 
4*(a^2 + b^2)^4*(b^2 + b^2*Tan[c + d*x]^2)^2) - ((b^2*(16*a*b^2*(2*a^4 - 5 
*a^2*b^2 + b^4) + b*(5*a^6 - 65*a^4*b^2 + 55*a^2*b^4 - 3*b^6)*Tan[c + d*x] 
))/(2*(a^2 + b^2)^5*(b^2 + b^2*Tan[c + d*x]^2)) - ((b^3*(3*a^8 - 132*a^6*b 
^2 + 370*a^4*b^4 - 132*a^2*b^6 + 3*b^8)*ArcTan[Tan[c + d*x]])/(a^2 + b^2)^ 
6 + (32*a*b^4*(a^2 - b^2)*(a^4 - 8*a^2*b^2 + b^4)*Log[a + b*Tan[c + d*x]]) 
/(a^2 + b^2)^6 - (16*a*b^4*(a^2 - b^2)*(a^4 - 8*a^2*b^2 + b^4)*Log[b^2 + b 
^2*Tan[c + d*x]^2])/(a^2 + b^2)^6 - (8*a^4*b^4)/(3*(a^2 + b^2)^3*(a + b*Ta 
n[c + d*x])^3) - (8*a^3*b^4*(a^2 - 2*b^2))/((a^2 + b^2)^4*(a + b*Tan[c + d 
*x])^2) - (24*a^2*b^4*(a^4 - 5*a^2*b^2 + 2*b^4))/((a^2 + b^2)^5*(a + b*Tan 
[c + d*x])))/(2*b^2))/(4*b^2)))/d
 

Defintions of rubi rules used

rule 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2160

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] 
:> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, 
 d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
 

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3999

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ 
), x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 89.27 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {-\frac {a^{4} b}{3 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {3 a^{2} b \left (a^{4}-5 b^{2} a^{2}+2 b^{4}\right )}{\left (a^{2}+b^{2}\right )^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{3} b \left (a^{2}-2 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b a \left (a^{6}-9 a^{4} b^{2}+9 a^{2} b^{4}-b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{6}}+\frac {\frac {\left (-\frac {5}{8} a^{8}+\frac {15}{2} a^{6} b^{2}+\frac {5}{4} a^{4} b^{4}-\frac {13}{2} a^{2} b^{6}+\frac {3}{8} b^{8}\right ) \tan \left (d x +c \right )^{3}+\left (-4 a^{7} b +6 b^{3} a^{5}+8 a^{3} b^{5}-2 a \,b^{7}\right ) \tan \left (d x +c \right )^{2}+\left (-\frac {3}{8} a^{8}+\frac {13}{2} a^{6} b^{2}-\frac {15}{2} a^{2} b^{6}+\frac {5}{8} b^{8}-\frac {5}{4} a^{4} b^{4}\right ) \tan \left (d x +c \right )-3 a^{7} b +7 b^{3} a^{5}+7 a^{3} b^{5}-3 a \,b^{7}}{\left (1+\tan \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (-32 a^{7} b +288 b^{3} a^{5}-288 a^{3} b^{5}+32 a \,b^{7}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{16}+\frac {\left (3 a^{8}-132 a^{6} b^{2}+370 a^{4} b^{4}-132 a^{2} b^{6}+3 b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{6}}}{d}\)	\(425\)
default	\(\frac {-\frac {a^{4} b}{3 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{3}}-\frac {3 a^{2} b \left (a^{4}-5 b^{2} a^{2}+2 b^{4}\right )}{\left (a^{2}+b^{2}\right )^{5} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{3} b \left (a^{2}-2 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {4 b a \left (a^{6}-9 a^{4} b^{2}+9 a^{2} b^{4}-b^{6}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{6}}+\frac {\frac {\left (-\frac {5}{8} a^{8}+\frac {15}{2} a^{6} b^{2}+\frac {5}{4} a^{4} b^{4}-\frac {13}{2} a^{2} b^{6}+\frac {3}{8} b^{8}\right ) \tan \left (d x +c \right )^{3}+\left (-4 a^{7} b +6 b^{3} a^{5}+8 a^{3} b^{5}-2 a \,b^{7}\right ) \tan \left (d x +c \right )^{2}+\left (-\frac {3}{8} a^{8}+\frac {13}{2} a^{6} b^{2}-\frac {15}{2} a^{2} b^{6}+\frac {5}{8} b^{8}-\frac {5}{4} a^{4} b^{4}\right ) \tan \left (d x +c \right )-3 a^{7} b +7 b^{3} a^{5}+7 a^{3} b^{5}-3 a \,b^{7}}{\left (1+\tan \left (d x +c \right )^{2}\right )^{2}}+\frac {\left (-32 a^{7} b +288 b^{3} a^{5}-288 a^{3} b^{5}+32 a \,b^{7}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{16}+\frac {\left (3 a^{8}-132 a^{6} b^{2}+370 a^{4} b^{4}-132 a^{2} b^{6}+3 b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{6}}}{d}\)	\(425\)
risch	\(\text {Expression too large to display}\)	\(1620\)

Input:

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

Output:

1/d*(-1/3*a^4*b/(a^2+b^2)^3/(a+b*tan(d*x+c))^3-3*a^2*b*(a^4-5*a^2*b^2+2*b^ 
4)/(a^2+b^2)^5/(a+b*tan(d*x+c))-a^3*b*(a^2-2*b^2)/(a^2+b^2)^4/(a+b*tan(d*x 
+c))^2+4*b*a*(a^6-9*a^4*b^2+9*a^2*b^4-b^6)/(a^2+b^2)^6*ln(a+b*tan(d*x+c))+ 
1/(a^2+b^2)^6*(((-5/8*a^8+15/2*a^6*b^2+5/4*a^4*b^4-13/2*a^2*b^6+3/8*b^8)*t 
an(d*x+c)^3+(-4*a^7*b+6*a^5*b^3+8*a^3*b^5-2*a*b^7)*tan(d*x+c)^2+(-3/8*a^8+ 
13/2*a^6*b^2-15/2*a^2*b^6+5/8*b^8-5/4*a^4*b^4)*tan(d*x+c)-3*a^7*b+7*b^3*a^ 
5+7*a^3*b^5-3*a*b^7)/(1+tan(d*x+c)^2)^2+1/16*(-32*a^7*b+288*a^5*b^3-288*a^ 
3*b^5+32*a*b^7)*ln(1+tan(d*x+c)^2)+1/8*(3*a^8-132*a^6*b^2+370*a^4*b^4-132* 
a^2*b^6+3*b^8)*arctan(tan(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (358) = 716\).

Time = 0.28 (sec) , antiderivative size = 1053, normalized size of antiderivative = 2.88 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Too large to display} \] Input:

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

Output:

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 - 1 
32*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)*cos(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)*si 
n(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 - 3 
9*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...
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\text {Exception raised: AttributeError} \] Input:

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

Output:

Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr 
imitive'
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (358) = 716\).

Time = 0.15 (sec) , antiderivative size = 997, normalized size of antiderivative = 2.72 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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 - 1006*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 - 82 
2*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...
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.83 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx=\frac {{\left (3 \, a^{8} - 132 \, a^{6} b^{2} + 370 \, a^{4} b^{4} - 132 \, a^{2} b^{6} + 3 \, b^{8}\right )} {\left (d x + c\right )}}{8 \, {\left (a^{12} d + 6 \, a^{10} b^{2} d + 15 \, a^{8} b^{4} d + 20 \, a^{6} b^{6} d + 15 \, a^{4} b^{8} d + 6 \, a^{2} b^{10} d + b^{12} d\right )}} - \frac {2 \, {\left (a^{7} b - 9 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - a b^{7}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{12} d + 6 \, a^{10} b^{2} d + 15 \, a^{8} b^{4} d + 20 \, a^{6} b^{6} d + 15 \, a^{4} b^{8} d + 6 \, a^{2} b^{10} d + b^{12} d} + \frac {4 \, {\left (a^{7} b^{2} - 9 \, a^{5} b^{4} + 9 \, a^{3} b^{6} - a b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b d + 6 \, a^{10} b^{3} d + 15 \, a^{8} b^{5} d + 20 \, a^{6} b^{7} d + 15 \, a^{4} b^{9} d + 6 \, a^{2} b^{11} d + b^{13} d} - \frac {176 \, a^{10} b - 432 \, a^{8} b^{3} - 432 \, a^{6} b^{5} + 176 \, a^{4} b^{7} + 3 \, {\left (29 \, a^{8} b^{3} - 156 \, a^{6} b^{5} - 82 \, a^{4} b^{7} + 100 \, a^{2} b^{9} - 3 \, b^{11}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (71 \, a^{9} b^{2} - 340 \, a^{7} b^{4} - 246 \, a^{5} b^{6} + 172 \, a^{3} b^{8} + 7 \, a b^{10}\right )} \tan \left (d x + c\right )^{5} + {\left (149 \, a^{10} b - 363 \, a^{8} b^{3} - 1518 \, a^{6} b^{5} - 406 \, a^{4} b^{7} + 585 \, a^{2} b^{9} - 15 \, b^{11}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{11} + 157 \, a^{9} b^{2} - 670 \, a^{7} b^{4} - 502 \, a^{5} b^{6} + 329 \, a^{3} b^{8} + 9 \, a b^{10}\right )} \tan \left (d x + c\right )^{3} + {\left (331 \, a^{10} b - 852 \, a^{8} b^{3} - 1422 \, a^{6} b^{5} + 76 \, a^{4} b^{7} + 315 \, a^{2} b^{9}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (3 \, a^{11} + 76 \, a^{9} b^{2} - 350 \, a^{7} b^{4} - 276 \, a^{5} b^{6} + 147 \, a^{3} b^{8}\right )} \tan \left (d x + c\right )}{24 \, {\left (a^{2} + b^{2}\right )}^{6} {\left (b \tan \left (d x + c\right ) + a\right )}^{3} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} d} \] Input:

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

Output:

1/8*(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*d + 6*a^10*b^2*d + 15*a^8*b^4*d + 20*a^6*b^6*d + 15*a^4*b^8*d + 6*a^2* 
b^10*d + b^12*d) - 2*(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*d + 6*a^10*b^2*d + 15*a^8*b^4*d + 20*a^6*b^6*d + 15*a^4*b 
^8*d + 6*a^2*b^10*d + b^12*d) + 4*(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*d + 6*a^10*b^3*d + 15*a^8*b^5*d + 2 
0*a^6*b^7*d + 15*a^4*b^9*d + 6*a^2*b^11*d + b^13*d) - 1/24*(176*a^10*b - 4 
32*a^8*b^3 - 432*a^6*b^5 + 176*a^4*b^7 + 3*(29*a^8*b^3 - 156*a^6*b^5 - 82* 
a^4*b^7 + 100*a^2*b^9 - 3*b^11)*tan(d*x + c)^6 + 3*(71*a^9*b^2 - 340*a^7*b 
^4 - 246*a^5*b^6 + 172*a^3*b^8 + 7*a*b^10)*tan(d*x + c)^5 + (149*a^10*b - 
363*a^8*b^3 - 1518*a^6*b^5 - 406*a^4*b^7 + 585*a^2*b^9 - 15*b^11)*tan(d*x 
+ c)^4 + 3*(5*a^11 + 157*a^9*b^2 - 670*a^7*b^4 - 502*a^5*b^6 + 329*a^3*b^8 
 + 9*a*b^10)*tan(d*x + c)^3 + (331*a^10*b - 852*a^8*b^3 - 1422*a^6*b^5 + 7 
6*a^4*b^7 + 315*a^2*b^9)*tan(d*x + c)^2 + 3*(3*a^11 + 76*a^9*b^2 - 350*a^7 
*b^4 - 276*a^5*b^6 + 147*a^3*b^8)*tan(d*x + c))/((a^2 + b^2)^6*(b*tan(d*x 
+ c) + a)^3*(tan(d*x + c)^2 + 1)^2*d)
 

Mupad [B] (verification not implemented)

Time = 2.87 (sec) , antiderivative size = 962, normalized size of antiderivative = 2.63 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(log(a + b*tan(c + d*x))*((4*a*b)/(a^2 + b^2)^3 - (48*a*b^3)/(a^2 + b^2)^4 
 + (120*a*b^5)/(a^2 + b^2)^5 - (80*a*b^7)/(a^2 + b^2)^6))/d - ((2*(11*a^8* 
b + 11*a^4*b^5 - 38*a^6*b^3))/(3*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 1 
0*a^6*b^4 + 5*a^8*b^2)) - (tan(c + d*x)^6*(3*b^9 - 103*a^2*b^7 + 185*a^4*b 
^5 - 29*a^6*b^3))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 
5*a^8*b^2)) + (tan(c + d*x)^5*(7*a*b^8 + 165*a^3*b^6 - 411*a^5*b^4 + 71*a^ 
7*b^2))/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2) 
) + (tan(c + d*x)^2*(331*a^8*b + 315*a^2*b^7 - 239*a^4*b^5 - 1183*a^6*b^3) 
)/(24*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + ( 
tan(c + d*x)^3*(9*a*b^8 + 5*a^9 + 320*a^3*b^6 - 822*a^5*b^4 + 152*a^7*b^2) 
)/(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) - (t 
an(c + d*x)^4*(15*b^9 - 149*a^8*b - 600*a^2*b^7 + 1006*a^4*b^5 + 512*a^6*b 
^3))/(24*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) 
+ (a*tan(c + d*x)*(3*a^8 + 147*a^2*b^6 - 423*a^4*b^4 + 73*a^6*b^2))/(8*(a^ 
10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)))/(d*(tan(c + 
 d*x)^2*(3*a*b^2 + 2*a^3) + tan(c + d*x)^5*(3*a^2*b + 2*b^3) + a^3 + tan(c 
 + d*x)^4*(6*a*b^2 + a^3) + tan(c + d*x)^3*(6*a^2*b + b^3) + b^3*tan(c + d 
*x)^7 + 3*a*b^2*tan(c + d*x)^6 + 3*a^2*b*tan(c + d*x))) + (log(tan(c + d*x 
) - 1i)*(a*b*14i - 3*a^2 + 3*b^2))/(16*d*(6*a*b^5 + 6*a^5*b - a^6*1i + b^6 
*1i - a^2*b^4*15i - 20*a^3*b^3 + a^4*b^2*15i)) - (log(tan(c + d*x) + 1i...
 

Reduce [B] (verification not implemented)

Time = 6.49 (sec) , antiderivative size = 3021, normalized size of antiderivative = 8.25 \[ \int \frac {\sin ^4(c+d x)}{(a+b \tan (c+d x))^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 96*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**10*b** 
2 + 1152*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**8*b* 
*4 - 3456*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**6*b 
**6 + 2688*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**4* 
b**8 - 288*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**2*a**2* 
b**10 + 96*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**10*b**2 - 864*cos( 
c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**8*b**4 + 864*cos(c + d*x)*log(tan 
((c + d*x)/2)**2 + 1)*a**6*b**6 - 96*cos(c + d*x)*log(tan((c + d*x)/2)**2 
+ 1)*a**4*b**8 + 96*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d* 
x)/2)*b - a)*sin(c + d*x)**2*a**10*b**2 - 1152*cos(c + d*x)*log(tan((c + d 
*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2*a**8*b**4 + 3456*c 
os(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + 
d*x)**2*a**6*b**6 - 2688*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c 
 + d*x)/2)*b - a)*sin(c + d*x)**2*a**4*b**8 + 288*cos(c + d*x)*log(tan((c 
+ d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)**2*a**2*b**10 - 96 
*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**10* 
b**2 + 864*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - 
 a)*a**8*b**4 - 864*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d* 
x)/2)*b - a)*a**6*b**6 + 96*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan 
((c + d*x)/2)*b - a)*a**4*b**8 + 6*cos(c + d*x)*sin(c + d*x)**6*a**10*b...