Integrand size = 21, antiderivative size = 297 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right ) x}{16 \left (a^2+b^2\right )^5}+\frac {2 a^5 b \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^6 b}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d} \]
1/16*(5*a^8-80*a^6*b^2+50*a^4*b^4+8*a^2*b^6+b^8)*x/(a^2+b^2)^5+2*a^5*b*(a^ 2-3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^5/d-a^6*b/(a^2+b^2)^4/d/( a+b*tan(d*x+c))-1/6*cos(d*x+c)^6*(2*a*b+(a^2-b^2)*tan(d*x+c))/(a^2+b^2)^2/ d+1/24*cos(d*x+c)^4*(12*a*b*(3*a^2+b^2)+(13*a^4-18*a^2*b^2-7*b^4)*tan(d*x+ c))/(a^2+b^2)^3/d-1/16*cos(d*x+c)^2*(48*a^5*b+(11*a^6-43*a^4*b^2-7*a^2*b^4 -b^6)*tan(d*x+c))/(a^2+b^2)^4/d
Leaf count is larger than twice the leaf count of optimal. \(664\) vs. \(2(297)=594\).
Time = 6.72 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.24 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {b \left (-\frac {5 \left (a^2-b^2\right ) \arctan (\tan (c+d x))}{16 b \left (a^2+b^2\right )^2}+\frac {3 \left (3 a^4-3 a^2 b^2-2 b^4\right ) \arctan (\tan (c+d x))}{8 b \left (a^2+b^2\right )^3}-\frac {\left (3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right ) \arctan (\tan (c+d x))}{2 b \left (a^2+b^2\right )^4}-\frac {3 a^5 \cos ^2(c+d x)}{\left (a^2+b^2\right )^4}+\frac {a \left (3 a^2+b^2\right ) \cos ^4(c+d x)}{2 \left (a^2+b^2\right )^3}-\frac {a \cos ^6(c+d x)}{3 \left (a^2+b^2\right )^2}-\frac {a^5 \left (2 a^2-6 b^2-\frac {a^3-7 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}+\frac {2 a^5 \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}-\frac {a^5 \left (2 a^2-6 b^2+\frac {a^3-7 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}-\frac {5 (a-b) (a+b) \cos (c+d x) \sin (c+d x)}{16 b \left (a^2+b^2\right )^2}+\frac {3 \left (3 a^4-3 a^2 b^2-2 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b \left (a^2+b^2\right )^3}-\frac {\left (3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right ) \cos (c+d x) \sin (c+d x)}{2 b \left (a^2+b^2\right )^4}-\frac {5 \left (a^2-b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 b \left (a^2+b^2\right )^2}+\frac {\left (3 a^4-3 a^2 b^2-2 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{4 b \left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{6 b \left (a^2+b^2\right )^2}-\frac {a^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}\right )}{d} \]
(b*((-5*(a^2 - b^2)*ArcTan[Tan[c + d*x]])/(16*b*(a^2 + b^2)^2) + (3*(3*a^4 - 3*a^2*b^2 - 2*b^4)*ArcTan[Tan[c + d*x]])/(8*b*(a^2 + b^2)^3) - ((3*a^6 - 6*a^4*b^2 - 4*a^2*b^4 - b^6)*ArcTan[Tan[c + d*x]])/(2*b*(a^2 + b^2)^4) - (3*a^5*Cos[c + d*x]^2)/(a^2 + b^2)^4 + (a*(3*a^2 + b^2)*Cos[c + d*x]^4)/( 2*(a^2 + b^2)^3) - (a*Cos[c + d*x]^6)/(3*(a^2 + b^2)^2) - (a^5*(2*a^2 - 6* b^2 - (a^3 - 7*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(2*(a^ 2 + b^2)^5) + (2*a^5*(a^2 - 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^5 - (a^5*(2*a^2 - 6*b^2 + (a^3 - 7*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan [c + d*x]])/(2*(a^2 + b^2)^5) - (5*(a - b)*(a + b)*Cos[c + d*x]*Sin[c + d* x])/(16*b*(a^2 + b^2)^2) + (3*(3*a^4 - 3*a^2*b^2 - 2*b^4)*Cos[c + d*x]*Sin [c + d*x])/(8*b*(a^2 + b^2)^3) - ((3*a^6 - 6*a^4*b^2 - 4*a^2*b^4 - b^6)*Co s[c + d*x]*Sin[c + d*x])/(2*b*(a^2 + b^2)^4) - (5*(a^2 - b^2)*Cos[c + d*x] ^3*Sin[c + d*x])/(24*b*(a^2 + b^2)^2) + ((3*a^4 - 3*a^2*b^2 - 2*b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*(a^2 + b^2)^3) - ((a^2 - b^2)*Cos[c + d*x]^5* Sin[c + d*x])/(6*b*(a^2 + b^2)^2) - a^6/((a^2 + b^2)^4*(a + b*Tan[c + d*x] ))))/d
Time = 1.52 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3999, 601, 25, 2178, 27, 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.
| \(\displaystyle \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^6}{(a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3999 |
| \(\displaystyle \frac {b \int \frac {b^6 \tan ^6(c+d x)}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^4}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {b \left (-\frac {\int -\frac {-\frac {2 a \left (5 a^2+b^2\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^2}+6 \tan ^4(c+d x) b^6-\frac {\left (6 a^4+17 b^2 a^2+b^4\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^2}+\frac {a^2 \left (a^2-b^2\right ) b^6}{\left (a^2+b^2\right )^2}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b \left (\frac {\int \frac {-\frac {2 a \left (5 a^2+b^2\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^2}+6 \tan ^4(c+d x) b^6-\frac {\left (6 a^4+17 b^2 a^2+b^4\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^2}+\frac {a^2 \left (a^2-b^2\right ) b^6}{\left (a^2+b^2\right )^2}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {\int \frac {3 \left (-\frac {2 a \left (13 a^4+6 b^2 a^2+b^4\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^3}-\frac {\left (8 a^6+37 b^2 a^4+6 b^4 a^2+b^6\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (3 a^4-6 b^2 a^2-b^4\right ) b^6}{\left (a^2+b^2\right )^3}\right )}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \int \frac {-\frac {2 a \left (13 a^4+6 b^2 a^2+b^4\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^3}-\frac {\left (8 a^6+37 b^2 a^4+6 b^4 a^2+b^6\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (3 a^4-6 b^2 a^2-b^4\right ) b^6}{\left (a^2+b^2\right )^3}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
| \(\Big \downarrow \) 2178 |
| \(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \left (\frac {b^4 \left (48 a^5 b^2+b \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {\int \frac {-\frac {\left (11 a^6-43 b^2 a^4-7 b^4 a^2-b^6\right ) \tan ^2(c+d x) b^8}{\left (a^2+b^2\right )^4}-\frac {2 a \left (11 a^4-6 b^2 a^2-b^4\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (5 a^6-37 b^2 a^4+7 b^4 a^2+b^6\right ) b^6}{\left (a^2+b^2\right )^4}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \left (\frac {b^4 \left (48 a^5 b^2+b \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {\int \left (\frac {32 a^5 \left (a^2-3 b^2\right ) b^6}{\left (a^2+b^2\right )^5 (a+b \tan (c+d x))}+\frac {\left (5 a^8-80 b^2 a^6-32 b \left (a^2-3 b^2\right ) \tan (c+d x) a^5+50 b^4 a^4+8 b^6 a^2+b^8\right ) b^6}{\left (a^2+b^2\right )^5 \left (\tan ^2(c+d x) b^2+b^2\right )}+\frac {16 a^6 b^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}\right )d(b \tan (c+d x))}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \left (\frac {b^4 \left (48 a^5 b^2+b \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {-\frac {16 a^6 b^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac {16 a^5 b^6 \left (a^2-3 b^2\right ) \log \left (b^2 \tan ^2(c+d x)+b^2\right )}{\left (a^2+b^2\right )^5}+\frac {32 a^5 b^6 \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}+\frac {b^5 \left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right ) \arctan (\tan (c+d x))}{\left (a^2+b^2\right )^5}}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\) |
(b*(-1/6*(b^4*(2*a*b^2 + b*(a^2 - b^2)*Tan[c + d*x]))/((a^2 + b^2)^2*(b^2 + b^2*Tan[c + d*x]^2)^3) + ((b^4*(12*a*b^2*(3*a^2 + b^2) + b*(13*a^4 - 18* a^2*b^2 - 7*b^4)*Tan[c + d*x]))/(4*(a^2 + b^2)^3*(b^2 + b^2*Tan[c + d*x]^2 )^2) - (3*((b^4*(48*a^5*b^2 + b*(11*a^6 - 43*a^4*b^2 - 7*a^2*b^4 - b^6)*Ta n[c + d*x]))/(2*(a^2 + b^2)^4*(b^2 + b^2*Tan[c + d*x]^2)) - ((b^5*(5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*ArcTan[Tan[c + d*x]])/(a^2 + b ^2)^5 + (32*a^5*b^6*(a^2 - 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^5 - (16*a^5*b^6*(a^2 - 3*b^2)*Log[b^2 + b^2*Tan[c + d*x]^2])/(a^2 + b^2)^5 - (16*a^6*b^6)/((a^2 + b^2)^4*(a + b*Tan[c + d*x])))/(2*b^2)))/(4*b^2))/(6*b ^2)))/d
3.1.61.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
Time = 33.91 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.29
| method | result | size |
| derivativedivides | \(\frac {-\frac {b \,a^{6}}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \,a^{5} \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (-\frac {11}{16} a^{8}+2 a^{6} b^{2}+\frac {25}{8} a^{4} b^{4}+\frac {1}{2} b^{6} a^{2}+\frac {1}{16} b^{8}\right ) \left (\tan ^{5}\left (d x +c \right )\right )+\left (-3 b \,a^{7}-3 b^{3} a^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )+\left (-\frac {5}{6} a^{8}+\frac {13}{3} a^{6} b^{2}+5 a^{4} b^{4}-\frac {1}{3} b^{6} a^{2}-\frac {1}{6} b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-\frac {9}{2} b \,a^{7}-\frac {5}{2} b^{3} a^{5}+\frac {5}{2} a^{3} b^{5}+\frac {1}{2} a \,b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {5}{16} a^{8}+2 a^{6} b^{2}+\frac {15}{8} a^{4} b^{4}-\frac {1}{2} b^{6} a^{2}-\frac {1}{16} b^{8}\right ) \tan \left (d x +c \right )-\frac {11 b \,a^{7}}{6}-\frac {b^{3} a^{5}}{2}+\frac {3 a^{3} b^{5}}{2}+\frac {a \,b^{7}}{6}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {\left (-32 b \,a^{7}+96 b^{3} a^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{32}+\frac {\left (5 a^{8}-80 a^{6} b^{2}+50 a^{4} b^{4}+8 b^{6} a^{2}+b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{16}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(383\) |
| default | \(\frac {-\frac {b \,a^{6}}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \,a^{5} \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (-\frac {11}{16} a^{8}+2 a^{6} b^{2}+\frac {25}{8} a^{4} b^{4}+\frac {1}{2} b^{6} a^{2}+\frac {1}{16} b^{8}\right ) \left (\tan ^{5}\left (d x +c \right )\right )+\left (-3 b \,a^{7}-3 b^{3} a^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )+\left (-\frac {5}{6} a^{8}+\frac {13}{3} a^{6} b^{2}+5 a^{4} b^{4}-\frac {1}{3} b^{6} a^{2}-\frac {1}{6} b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-\frac {9}{2} b \,a^{7}-\frac {5}{2} b^{3} a^{5}+\frac {5}{2} a^{3} b^{5}+\frac {1}{2} a \,b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {5}{16} a^{8}+2 a^{6} b^{2}+\frac {15}{8} a^{4} b^{4}-\frac {1}{2} b^{6} a^{2}-\frac {1}{16} b^{8}\right ) \tan \left (d x +c \right )-\frac {11 b \,a^{7}}{6}-\frac {b^{3} a^{5}}{2}+\frac {3 a^{3} b^{5}}{2}+\frac {a \,b^{7}}{6}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {\left (-32 b \,a^{7}+96 b^{3} a^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{32}+\frac {\left (5 a^{8}-80 a^{6} b^{2}+50 a^{4} b^{4}+8 b^{6} a^{2}+b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{16}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(383\) |
| risch | \(\text {Expression too large to display}\) | \(1139\) |
1/d*(-b*a^6/(a^2+b^2)^4/(a+b*tan(d*x+c))+2*b*a^5*(a^2-3*b^2)/(a^2+b^2)^5*l n(a+b*tan(d*x+c))+1/(a^2+b^2)^5*(((-11/16*a^8+2*a^6*b^2+25/8*a^4*b^4+1/2*b ^6*a^2+1/16*b^8)*tan(d*x+c)^5+(-3*a^7*b-3*a^5*b^3)*tan(d*x+c)^4+(-5/6*a^8+ 13/3*a^6*b^2+5*a^4*b^4-1/3*b^6*a^2-1/6*b^8)*tan(d*x+c)^3+(-9/2*b*a^7-5/2*b ^3*a^5+5/2*a^3*b^5+1/2*a*b^7)*tan(d*x+c)^2+(-5/16*a^8+2*a^6*b^2+15/8*a^4*b ^4-1/2*b^6*a^2-1/16*b^8)*tan(d*x+c)-11/6*b*a^7-1/2*b^3*a^5+3/2*a^3*b^5+1/6 *a*b^7)/(1+tan(d*x+c)^2)^3+1/32*(-32*a^7*b+96*a^5*b^3)*ln(1+tan(d*x+c)^2)+ 1/16*(5*a^8-80*a^6*b^2+50*a^4*b^4+8*a^2*b^6+b^8)*arctan(tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (289) = 578\).
Time = 0.34 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.08 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {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 )^{7} - 2 \, {\left (19 \, a^{8} b + 68 \, a^{6} b^{3} + 90 \, a^{4} b^{5} + 52 \, a^{2} b^{7} + 11 \, b^{9}\right )} \cos \left (d x + c\right )^{5} + {\left (85 \, a^{8} b + 224 \, a^{6} b^{3} + 210 \, a^{4} b^{5} + 88 \, a^{2} b^{7} + 17 \, b^{9}\right )} \cos \left (d x + c\right )^{3} - {\left (17 \, a^{8} b + 72 \, a^{6} b^{3} + 120 \, a^{4} b^{5} + 20 \, a^{2} b^{7} + 3 \, b^{9} + 3 \, {\left (5 \, a^{9} - 80 \, a^{7} b^{2} + 50 \, a^{5} b^{4} + 8 \, a^{3} b^{6} + a b^{8}\right )} d x\right )} \cos \left (d x + c\right ) - 48 \, {\left ({\left (a^{8} b - 3 \, a^{6} b^{3}\right )} \cos \left (d x + c\right ) + {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (98 \, a^{7} b^{2} + 24 \, a^{5} b^{4} - 30 \, a^{3} b^{6} - 4 \, a b^{8} - 8 \, {\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 )^{6} + 2 \, {\left (13 \, a^{9} + 44 \, a^{7} b^{2} + 54 \, a^{5} b^{4} + 28 \, a^{3} b^{6} + 5 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{8} b - 80 \, a^{6} b^{3} + 50 \, a^{4} b^{5} + 8 \, a^{2} b^{7} + b^{9}\right )} d x - 3 \, {\left (11 \, a^{9} + 16 \, a^{7} b^{2} - 2 \, a^{5} b^{4} - 8 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, {\left ({\left (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}\right )} d \cos \left (d x + c\right ) + {\left (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}\right )} d \sin \left (d x + c\right )\right )}} \]
-1/48*(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)^7 - 2*(19*a^8*b + 68*a^6*b^3 + 90*a^4*b^5 + 52*a^2*b^7 + 11*b^9)*cos(d*x + c )^5 + (85*a^8*b + 224*a^6*b^3 + 210*a^4*b^5 + 88*a^2*b^7 + 17*b^9)*cos(d*x + c)^3 - (17*a^8*b + 72*a^6*b^3 + 120*a^4*b^5 + 20*a^2*b^7 + 3*b^9 + 3*(5 *a^9 - 80*a^7*b^2 + 50*a^5*b^4 + 8*a^3*b^6 + a*b^8)*d*x)*cos(d*x + c) - 48 *((a^8*b - 3*a^6*b^3)*cos(d*x + c) + (a^7*b^2 - 3*a^5*b^4)*sin(d*x + c))*l og(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - ( 98*a^7*b^2 + 24*a^5*b^4 - 30*a^3*b^6 - 4*a*b^8 - 8*(a^9 + 4*a^7*b^2 + 6*a^ 5*b^4 + 4*a^3*b^6 + a*b^8)*cos(d*x + c)^6 + 2*(13*a^9 + 44*a^7*b^2 + 54*a^ 5*b^4 + 28*a^3*b^6 + 5*a*b^8)*cos(d*x + c)^4 + 3*(5*a^8*b - 80*a^6*b^3 + 5 0*a^4*b^5 + 8*a^2*b^7 + b^9)*d*x - 3*(11*a^9 + 16*a^7*b^2 - 2*a^5*b^4 - 8* a^3*b^6 - a*b^8)*cos(d*x + c)^2)*sin(d*x + c))/((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)*d*cos(d*x + c) + (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)*d*sin(d*x + c))
Timed out. \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (289) = 578\).
Time = 0.32 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.69 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (5 \, a^{8} - 80 \, a^{6} b^{2} + 50 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\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 {96 \, {\left (a^{7} b - 3 \, a^{5} b^{3}\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 {48 \, {\left (a^{7} b - 3 \, a^{5} b^{3}\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 {136 \, a^{6} b - 64 \, a^{4} b^{3} - 8 \, a^{2} b^{5} + 3 \, {\left (27 \, a^{6} b - 43 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - b^{7}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (11 \, a^{7} + 5 \, a^{5} b^{2} - 7 \, a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (41 \, a^{6} b - 31 \, a^{4} b^{3} + a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{4} + 8 \, {\left (5 \, a^{7} - 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (125 \, a^{6} b - 69 \, a^{4} b^{3} - a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (15 \, a^{7} - 23 \, a^{5} b^{2} - 43 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \tan \left (d x + c\right )}{a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8} + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{7} + {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{5} + 3 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )}}{48 \, d} \]
1/48*(3*(5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*(d*x + c)/(a^1 0 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10) + 96*(a^7*b - 3*a^5*b^3)*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) - 48*(a^7*b - 3*a^5*b^3)*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) - (136*a^6 *b - 64*a^4*b^3 - 8*a^2*b^5 + 3*(27*a^6*b - 43*a^4*b^3 - 7*a^2*b^5 - b^7)* tan(d*x + c)^6 + 3*(11*a^7 + 5*a^5*b^2 - 7*a^3*b^4 - a*b^6)*tan(d*x + c)^5 + 8*(41*a^6*b - 31*a^4*b^3 + a^2*b^5 + b^7)*tan(d*x + c)^4 + 8*(5*a^7 - 4 *a^5*b^2 - 11*a^3*b^4 - 2*a*b^6)*tan(d*x + c)^3 + 3*(125*a^6*b - 69*a^4*b^ 3 - a^2*b^5 + b^7)*tan(d*x + c)^2 + (15*a^7 - 23*a^5*b^2 - 43*a^3*b^4 - 5* a*b^6)*tan(d*x + c))/(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8 + (a ^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*tan(d*x + c)^7 + (a^9 + 4* a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*tan(d*x + c)^6 + 3*(a^8*b + 4*a^6 *b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*tan(d*x + c)^5 + 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*tan(d*x + c)^4 + 3*(a^8*b + 4*a^6*b^3 + 6*a ^4*b^5 + 4*a^2*b^7 + b^9)*tan(d*x + c)^3 + 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*tan(d*x + c)^2 + (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a ^2*b^7 + b^9)*tan(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (289) = 578\).
Time = 0.54 (sec) , antiderivative size = 735, normalized size of antiderivative = 2.47 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (5 \, a^{8} - 80 \, a^{6} b^{2} + 50 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\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 {48 \, {\left (a^{7} b - 3 \, a^{5} b^{3}\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 {96 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\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 {48 \, {\left (2 \, a^{7} b^{2} \tan \left (d x + c\right ) - 6 \, a^{5} b^{4} \tan \left (d x + c\right ) + 3 \, a^{8} b - 5 \, a^{6} b^{3}\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 )}} + \frac {88 \, a^{7} b \tan \left (d x + c\right )^{6} - 264 \, a^{5} b^{3} \tan \left (d x + c\right )^{6} - 33 \, a^{8} \tan \left (d x + c\right )^{5} + 96 \, a^{6} b^{2} \tan \left (d x + c\right )^{5} + 150 \, a^{4} b^{4} \tan \left (d x + c\right )^{5} + 24 \, a^{2} b^{6} \tan \left (d x + c\right )^{5} + 3 \, b^{8} \tan \left (d x + c\right )^{5} + 120 \, a^{7} b \tan \left (d x + c\right )^{4} - 936 \, a^{5} b^{3} \tan \left (d x + c\right )^{4} - 40 \, a^{8} \tan \left (d x + c\right )^{3} + 208 \, a^{6} b^{2} \tan \left (d x + c\right )^{3} + 240 \, a^{4} b^{4} \tan \left (d x + c\right )^{3} - 16 \, a^{2} b^{6} \tan \left (d x + c\right )^{3} - 8 \, b^{8} \tan \left (d x + c\right )^{3} + 48 \, a^{7} b \tan \left (d x + c\right )^{2} - 912 \, a^{5} b^{3} \tan \left (d x + c\right )^{2} + 120 \, a^{3} b^{5} \tan \left (d x + c\right )^{2} + 24 \, a b^{7} \tan \left (d x + c\right )^{2} - 15 \, a^{8} \tan \left (d x + c\right ) + 96 \, a^{6} b^{2} \tan \left (d x + c\right ) + 90 \, a^{4} b^{4} \tan \left (d x + c\right ) - 24 \, a^{2} b^{6} \tan \left (d x + c\right ) - 3 \, b^{8} \tan \left (d x + c\right ) - 288 \, a^{5} b^{3} + 72 \, a^{3} b^{5} + 8 \, a b^{7}}{{\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 )}^{3}}}{48 \, d} \]
1/48*(3*(5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*(d*x + c)/(a^1 0 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10) - 48*(a^7*b - 3*a^5*b^3)*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) + 96*(a^7*b^2 - 3*a^5*b^4)*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) - 48*(2*a^7*b^2*tan(d*x + c) - 6*a^5*b^4*tan(d*x + c) + 3*a^8*b - 5*a^6*b^3 )/((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)) + (88*a^7*b*tan(d*x + c)^6 - 264*a^5*b^3*tan(d*x + c)^6 - 3 3*a^8*tan(d*x + c)^5 + 96*a^6*b^2*tan(d*x + c)^5 + 150*a^4*b^4*tan(d*x + c )^5 + 24*a^2*b^6*tan(d*x + c)^5 + 3*b^8*tan(d*x + c)^5 + 120*a^7*b*tan(d*x + c)^4 - 936*a^5*b^3*tan(d*x + c)^4 - 40*a^8*tan(d*x + c)^3 + 208*a^6*b^2 *tan(d*x + c)^3 + 240*a^4*b^4*tan(d*x + c)^3 - 16*a^2*b^6*tan(d*x + c)^3 - 8*b^8*tan(d*x + c)^3 + 48*a^7*b*tan(d*x + c)^2 - 912*a^5*b^3*tan(d*x + c) ^2 + 120*a^3*b^5*tan(d*x + c)^2 + 24*a*b^7*tan(d*x + c)^2 - 15*a^8*tan(d*x + c) + 96*a^6*b^2*tan(d*x + c) + 90*a^4*b^4*tan(d*x + c) - 24*a^2*b^6*tan (d*x + c) - 3*b^8*tan(d*x + c) - 288*a^5*b^3 + 72*a^3*b^5 + 8*a*b^7)/((a^1 0 + 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)^3))/d
Time = 6.11 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.55 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {2\,a\,b}{{\left (a^2+b^2\right )}^2}-\frac {12\,a\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {18\,a\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {8\,a\,b^7}{{\left (a^2+b^2\right )}^5}\right )}{d}+\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-5\,a^5+9\,a^3\,b^2+2\,a\,b^4\right )}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (-11\,a^5+6\,a^3\,b^2+a\,b^4\right )}{16\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (-27\,a^6\,b+43\,a^4\,b^3+7\,a^2\,b^5+b^7\right )}{16\,\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 )\,\left (-15\,a^5+38\,a^3\,b^2+5\,a\,b^4\right )}{48\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {a\,\left (-17\,a^5\,b+8\,a^3\,b^3+a\,b^5\right )}{6\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (41\,a^6\,b-31\,a^4\,b^3+a^2\,b^5+b^7\right )}{6\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (125\,a^6\,b-69\,a^4\,b^3-a^2\,b^5+b^7\right )}{16\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^7+a\,{\mathrm {tan}\left (c+d\,x\right )}^6+3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5+3\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+3\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2+b\,\mathrm {tan}\left (c+d\,x\right )+a\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (a^3\,5{}\mathrm {i}-7\,a^2\,b+a\,b^2\,5{}\mathrm {i}+b^3\right )}{32\,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 {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a^3\,5{}\mathrm {i}+7\,a^2\,b+a\,b^2\,5{}\mathrm {i}-b^3\right )}{32\,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 )} \]
(log(a + b*tan(c + d*x))*((2*a*b)/(a^2 + b^2)^2 - (12*a*b^3)/(a^2 + b^2)^3 + (18*a*b^5)/(a^2 + b^2)^4 - (8*a*b^7)/(a^2 + b^2)^5))/d + ((tan(c + d*x) ^3*(2*a*b^4 - 5*a^5 + 9*a^3*b^2))/(6*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^5*(a*b^4 - 11*a^5 + 6*a^3*b^2))/(16*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^6*(b^7 - 27*a^6*b + 7*a^2*b^5 + 43*a^4*b^3) )/(16*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)*(5* a*b^4 - 15*a^5 + 38*a^3*b^2))/(48*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + ( a*(a*b^5 - 17*a^5*b + 8*a^3*b^3))/(6*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^4*(41*a^6*b + b^7 + a^2*b^5 - 31*a^4*b^3))/(6* (a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^2*(125*a^ 6*b + b^7 - a^2*b^5 - 69*a^4*b^3))/(16*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a + b*tan(c + d*x) + 3*a*tan(c + d*x)^2 + 3*a*tan(c + d *x)^4 + a*tan(c + d*x)^6 + 3*b*tan(c + d*x)^3 + 3*b*tan(c + d*x)^5 + b*tan (c + d*x)^7)) + (log(tan(c + d*x) + 1i)*(a*b^2*5i - 7*a^2*b + a^3*5i + b^3 ))/(32*d*(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a^3*b^2)) - (log(tan(c + d*x) - 1i)*(a*b^2*5i + 7*a^2*b + a^3*5i - b^3))/(32*d*(5*a*b ^4 + a^4*b*5i + a^5 + b^5*1i - a^2*b^3*10i - 10*a^3*b^2))