Integrand size = 21, antiderivative size = 295 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {3 a \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac {3 b^5 \left (7 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {3 b \left (a^4+5 a^2 b^2-4 b^4\right )}{8 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {3 a b \left (a^4+6 a^2 b^2-27 b^4\right )}{8 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^2(c+d x) \left (2 b \left (a^2-3 b^2\right )-a \left (3 a^2+11 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2} \]
3/8*a*(a^6+7*a^4*b^2+35*a^2*b^4-35*b^6)*x/(a^2+b^2)^5+3*b^5*(7*a^2-b^2)*ln (a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^5/d+3/8*b*(a^4+5*a^2*b^2-4*b^4)/(a^2 +b^2)^3/d/(a+b*tan(d*x+c))^2+1/4*cos(d*x+c)^4*(b+a*tan(d*x+c))/(a^2+b^2)/d /(a+b*tan(d*x+c))^2+3/8*a*b*(a^4+6*a^2*b^2-27*b^4)/(a^2+b^2)^4/d/(a+b*tan( d*x+c))-1/8*cos(d*x+c)^2*(2*b*(a^2-3*b^2)-a*(3*a^2+11*b^2)*tan(d*x+c))/(a^ 2+b^2)^2/d/(a+b*tan(d*x+c))^2
Leaf count is larger than twice the leaf count of optimal. \(596\) vs. \(2(295)=590\).
Time = 6.29 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.02 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b^5 \left (\frac {\cos ^4(c+d x) \left (b^2+a b \tan (c+d x)\right )}{4 b^6 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\frac {\cos ^2(c+d x) \left (5 a^2 b^2-3 b^2 \left (a^2+2 b^2\right )+b \left (-5 a b^2-3 a \left (a^2+2 b^2\right )\right ) \tan (c+d x)\right )}{2 b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (-3 a^2 \left (3 a^2+11 b^2\right )+3 \left (a^4+a^2 b^2+8 b^4\right )\right ) \left (-\frac {\left (3 a^2-b^2-\frac {a^3-3 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}+\frac {\left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {\left (3 a^2-b^2+\frac {a^3-3 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}-\frac {1}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}\right )+3 a \left (3 a^2+11 b^2\right ) \left (-\frac {\left (2 a-\frac {a^2-b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}+\frac {2 a \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac {\left (2 a+\frac {a^2-b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}-\frac {1}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}\right )}{2 b^2 \left (a^2+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}\right )}{d} \]
(b^5*((Cos[c + d*x]^4*(b^2 + a*b*Tan[c + d*x]))/(4*b^6*(a^2 + b^2)*(a + b* Tan[c + d*x])^2) - ((Cos[c + d*x]^2*(5*a^2*b^2 - 3*b^2*(a^2 + 2*b^2) + b*( -5*a*b^2 - 3*a*(a^2 + 2*b^2))*Tan[c + d*x]))/(2*b^4*(a^2 + b^2)*(a + b*Tan [c + d*x])^2) - ((-3*a^2*(3*a^2 + 11*b^2) + 3*(a^4 + a^2*b^2 + 8*b^4))*(-1 /2*((3*a^2 - b^2 - (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(a^2 + b^2)^3 + ((3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2) ^3 - ((3*a^2 - b^2 + (a^3 - 3*a*b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^3) - 1/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2 *a)/((a^2 + b^2)^2*(a + b*Tan[c + d*x]))) + 3*a*(3*a^2 + 11*b^2)*(-1/2*((2 *a - (a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(a^2 + b^2) ^2 + (2*a*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^2 - ((2*a + (a^2 - b^2)/Sqr t[-b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^2) - 1/((a^2 + b ^2)*(a + b*Tan[c + d*x]))))/(2*b^2*(a^2 + b^2)))/(4*b^2*(a^2 + b^2))))/d
Time = 0.60 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.31, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3987, 27, 496, 25, 686, 27, 657, 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.
| \(\displaystyle \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^4 (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3987 |
| \(\displaystyle \frac {\int \frac {b^6}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \int \frac {1}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {b^5 \left (\frac {a b \tan (c+d x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {\int -\frac {3 \left (a^2+2 b^2\right )+5 a b \tan (c+d x)}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2 \left (a^2+b^2\right )}\right )}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^5 \left (\frac {\int \frac {3 \left (a^2+2 b^2\right )+5 a b \tan (c+d x)}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right )^2 (a+b \tan (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {b^5 \left (\frac {-\frac {\int -\frac {3 \left (a^4+b^2 a^2+b \left (3 a^2+11 b^2\right ) \tan (c+d x) a+8 b^4\right )}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{2 b^2 \left (a^2+b^2\right )}-\frac {2 b^2 \left (a^2-3 b^2\right )-a b \left (3 a^2+11 b^2\right ) \tan (c+d x)}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right ) (a+b \tan (c+d x))^2}}{4 b^2 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right )^2 (a+b \tan (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \int \frac {a^4+b^2 a^2+b \left (3 a^2+11 b^2\right ) \tan (c+d x) a+8 b^4}{(a+b \tan (c+d x))^3 \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{2 b^2 \left (a^2+b^2\right )}-\frac {2 b^2 \left (a^2-3 b^2\right )-a b \left (3 a^2+11 b^2\right ) \tan (c+d x)}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right ) (a+b \tan (c+d x))^2}}{4 b^2 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right )^2 (a+b \tan (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {b^5 \left (\frac {\frac {3 \int \left (-\frac {a \left (a^4+6 b^2 a^2-27 b^4\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {8 \left (7 a^2 b^4-b^6\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {a \left (a^6+7 b^2 a^4+35 b^4 a^2-35 b^6\right )-8 b^5 \left (7 a^2-b^2\right ) \tan (c+d x)}{\left (a^2+b^2\right )^3 \left (\tan ^2(c+d x) b^2+b^2\right )}-\frac {2 \left (a^4+5 b^2 a^2-4 b^4\right )}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}\right )d(b \tan (c+d x))}{2 b^2 \left (a^2+b^2\right )}-\frac {2 b^2 \left (a^2-3 b^2\right )-a b \left (3 a^2+11 b^2\right ) \tan (c+d x)}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right ) (a+b \tan (c+d x))^2}}{4 b^2 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right )^2 (a+b \tan (c+d x))^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^5 \left (\frac {a b \tan (c+d x)+b^2}{4 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {\frac {3 \left (-\frac {4 b^4 \left (7 a^2-b^2\right ) \log \left (b^2 \tan ^2(c+d x)+b^2\right )}{\left (a^2+b^2\right )^3}+\frac {8 b^4 \left (7 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac {a \left (a^4+6 a^2 b^2-27 b^4\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {a^4+5 a^2 b^2-4 b^4}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (a^6+7 a^4 b^2+35 a^2 b^4-35 b^6\right ) \arctan (\tan (c+d x))}{b \left (a^2+b^2\right )^3}\right )}{2 b^2 \left (a^2+b^2\right )}-\frac {2 b^2 \left (a^2-3 b^2\right )-a b \left (3 a^2+11 b^2\right ) \tan (c+d x)}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \tan ^2(c+d x)+b^2\right ) (a+b \tan (c+d x))^2}}{4 b^2 \left (a^2+b^2\right )}\right )}{d}\) |
(b^5*((b^2 + a*b*Tan[c + d*x])/(4*b^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2*( b^2 + b^2*Tan[c + d*x]^2)^2) + (-1/2*(2*b^2*(a^2 - 3*b^2) - a*b*(3*a^2 + 1 1*b^2)*Tan[c + d*x])/(b^2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2*(b^2 + b^2*Ta n[c + d*x]^2)) + (3*((a*(a^6 + 7*a^4*b^2 + 35*a^2*b^4 - 35*b^6)*ArcTan[Tan [c + d*x]])/(b*(a^2 + b^2)^3) + (8*b^4*(7*a^2 - b^2)*Log[a + b*Tan[c + d*x ]])/(a^2 + b^2)^3 - (4*b^4*(7*a^2 - b^2)*Log[b^2 + b^2*Tan[c + d*x]^2])/(a ^2 + b^2)^3 + (a^4 + 5*a^2*b^2 - 4*b^4)/((a^2 + b^2)*(a + b*Tan[c + d*x])^ 2) + (a*(a^4 + 6*a^2*b^2 - 27*b^4))/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))) /(2*b^2*(a^2 + b^2)))/(4*b^2*(a^2 + b^2))))/d
3.6.71.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(b*f) Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
Time = 43.90 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.06
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{5}}{2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {6 b^{5} a}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{5} \left (7 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (\frac {3}{8} a^{7}+\frac {21}{8} a^{5} b^{2}-\frac {15}{8} a^{3} b^{4}-\frac {33}{8} a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (5 a^{4} b^{3}+4 b^{5} a^{2}-b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (\frac {19}{8} a^{5} b^{2}-\frac {39}{8} a \,b^{6}+\frac {5}{8} a^{7}-\frac {25}{8} a^{3} b^{4}\right ) \tan \left (d x +c \right )+\frac {3 a^{6} b}{4}+\frac {25 a^{4} b^{3}}{4}+\frac {17 b^{5} a^{2}}{4}-\frac {5 b^{7}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 \left (-56 b^{5} a^{2}+8 b^{7}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {3 \left (a^{7}+7 a^{5} b^{2}+35 a^{3} b^{4}-35 a \,b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(312\) |
| default | \(\frac {-\frac {b^{5}}{2 \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {6 b^{5} a}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{5} \left (7 a^{2}-b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (\frac {3}{8} a^{7}+\frac {21}{8} a^{5} b^{2}-\frac {15}{8} a^{3} b^{4}-\frac {33}{8} a \,b^{6}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (5 a^{4} b^{3}+4 b^{5} a^{2}-b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (\frac {19}{8} a^{5} b^{2}-\frac {39}{8} a \,b^{6}+\frac {5}{8} a^{7}-\frac {25}{8} a^{3} b^{4}\right ) \tan \left (d x +c \right )+\frac {3 a^{6} b}{4}+\frac {25 a^{4} b^{3}}{4}+\frac {17 b^{5} a^{2}}{4}-\frac {5 b^{7}}{4}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 \left (-56 b^{5} a^{2}+8 b^{7}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{16}+\frac {3 \left (a^{7}+7 a^{5} b^{2}+35 a^{3} b^{4}-35 a \,b^{6}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\) | \(312\) |
| risch | \(\frac {15 x a b}{8 i a^{5}-80 i a^{3} b^{2}+40 i a \,b^{4}+40 a^{4} b -80 a^{2} b^{3}+8 b^{5}}-\frac {24 i x \,b^{2}}{8 i a^{5}-80 i a^{3} b^{2}+40 i a \,b^{4}+40 a^{4} b -80 a^{2} b^{3}+8 b^{5}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a}{8 \left (-4 i a^{3} b +4 i a \,b^{3}+a^{4}-6 a^{2} b^{2}+b^{4}\right ) d}+\frac {6 i b^{7} x}{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 {5 \,{\mathrm e}^{2 i \left (d x +c \right )} b}{16 \left (-4 i a^{3} b +4 i a \,b^{3}+a^{4}-6 a^{2} b^{2}+b^{4}\right ) d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} \left (2 i a b +a^{2}-b^{2}\right ) d}-\frac {5 \,{\mathrm e}^{-2 i \left (d x +c \right )} b}{16 \left (i b +a \right )^{2} \left (2 i a b +a^{2}-b^{2}\right ) d}-\frac {42 i b^{5} a^{2} c}{d \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 )}+\frac {6 i b^{7} c}{d \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 )}-\frac {42 i b^{5} a^{2} x}{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 {3 i x \,a^{2}}{8 i a^{5}-80 i a^{3} b^{2}+40 i a \,b^{4}+40 a^{4} b -80 a^{2} b^{3}+8 b^{5}}-\frac {i {\mathrm e}^{4 i \left (d x +c \right )}}{64 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}+\frac {i {\mathrm e}^{-4 i \left (d x +c \right )}}{64 \left (2 i a b +a^{2}-b^{2}\right ) \left (i b +a \right ) d}+\frac {2 b^{6} \left (-6 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}+7 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+7 i a b +7 a^{2}\right )}{\left (-i a +b \right )^{4} \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} d \left (i a +b \right )^{5}}+\frac {21 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) a^{2}}{d \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 )}-\frac {3 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \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 )}\) | \(877\) |
1/d*(-1/2*b^5/(a^2+b^2)^3/(a+b*tan(d*x+c))^2-6*b^5/(a^2+b^2)^4*a/(a+b*tan( d*x+c))+3*b^5*(7*a^2-b^2)/(a^2+b^2)^5*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^5*((( 3/8*a^7+21/8*a^5*b^2-15/8*a^3*b^4-33/8*a*b^6)*tan(d*x+c)^3+(5*a^4*b^3+4*a^ 2*b^5-b^7)*tan(d*x+c)^2+(19/8*a^5*b^2-39/8*a*b^6+5/8*a^7-25/8*a^3*b^4)*tan (d*x+c)+3/4*a^6*b+25/4*a^4*b^3+17/4*b^5*a^2-5/4*b^7)/(1+tan(d*x+c)^2)^2+3/ 16*(-56*a^2*b^5+8*b^7)*ln(1+tan(d*x+c)^2)+3/8*(a^7+7*a^5*b^2+35*a^3*b^4-35 *a*b^6)*arctan(tan(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 671 vs. \(2 (284) = 568\).
Time = 0.33 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.27 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {9 \, a^{6} b^{3} + 95 \, a^{4} b^{5} - 141 \, 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 (a^{8} b - 6 \, a^{4} b^{5} - 8 \, a^{2} b^{7} - 3 \, b^{9}\right )} \cos \left (d x + c\right )^{4} - 12 \, {\left (a^{7} b^{2} + 7 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 35 \, a b^{8}\right )} d x - {\left (15 \, a^{8} b + 82 \, a^{6} b^{3} + 68 \, a^{4} b^{5} - 498 \, a^{2} b^{7} - 51 \, b^{9} + 12 \, {\left (a^{9} + 6 \, a^{7} b^{2} + 28 \, a^{5} b^{4} - 70 \, a^{3} b^{6} + 35 \, a b^{8}\right )} d x\right )} \cos \left (d x + c\right )^{2} - 48 \, {\left (7 \, a^{2} b^{7} - b^{9} + {\left (7 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (7 \, a^{3} b^{6} - a b^{8}\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 (3 \, a^{9} + 20 \, a^{7} b^{2} + 42 \, a^{5} b^{4} + 36 \, a^{3} b^{6} + 11 \, a b^{8}\right )} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{7} b^{2} + 53 \, a^{5} b^{4} - 15 \, a^{3} b^{6} + 159 \, a b^{8} - 12 \, {\left (a^{8} b + 7 \, a^{6} b^{3} + 35 \, a^{4} b^{5} - 35 \, 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 )}} \]
-1/32*(9*a^6*b^3 + 95*a^4*b^5 - 141*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*(a^8*b - 6*a^4*b^5 - 8* a^2*b^7 - 3*b^9)*cos(d*x + c)^4 - 12*(a^7*b^2 + 7*a^5*b^4 + 35*a^3*b^6 - 3 5*a*b^8)*d*x - (15*a^8*b + 82*a^6*b^3 + 68*a^4*b^5 - 498*a^2*b^7 - 51*b^9 + 12*(a^9 + 6*a^7*b^2 + 28*a^5*b^4 - 70*a^3*b^6 + 35*a*b^8)*d*x)*cos(d*x + c)^2 - 48*(7*a^2*b^7 - b^9 + (7*a^4*b^5 - 8*a^2*b^7 + b^9)*cos(d*x + c)^2 + 2*(7*a^3*b^6 - a*b^8)*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*(3*a^9 + 20*a^7*b^2 + 42*a^5*b^4 + 36*a^3*b^6 + 11*a*b^8)*cos(d*x + c)^3 - (3*a^7*b^2 + 53*a^5*b ^4 - 15*a^3*b^6 + 159*a*b^8 - 12*(a^8*b + 7*a^6*b^3 + 35*a^4*b^5 - 35*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)
Exception generated. \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]
Leaf count of result is larger than twice the leaf count of optimal. 738 vs. \(2 (284) = 568\).
Time = 0.53 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.50 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{7} + 7 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 35 \, 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 (7 \, a^{2} b^{5} - b^{7}\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 (7 \, a^{2} b^{5} - b^{7}\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 {6 \, a^{6} b + 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} - 4 \, b^{7} + 3 \, {\left (a^{5} b^{2} + 6 \, a^{3} b^{4} - 27 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + 6 \, {\left (a^{6} b + 6 \, a^{4} b^{3} - 13 \, a^{2} b^{5} - 2 \, b^{7}\right )} \tan \left (d x + c\right )^{4} + {\left (3 \, a^{7} + 23 \, a^{5} b^{2} + 61 \, a^{3} b^{4} - 151 \, a b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (5 \, a^{6} b + 37 \, a^{4} b^{3} - 73 \, a^{2} b^{5} - 9 \, b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (5 \, a^{7} + 26 \, a^{5} b^{2} + 49 \, a^{3} b^{4} - 68 \, 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} \]
1/8*(3*(a^7 + 7*a^5*b^2 + 35*a^3*b^4 - 35*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*(7*a^2*b^5 - b^7)*lo g(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*(7*a^2*b^5 - b^7)*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) + (6*a^6*b + 44*a^4*b^3 - 62*a^2*b^5 - 4*b^7 + 3*(a^5*b^2 + 6*a^3*b^4 - 27*a*b^6)*tan(d*x + c)^5 + 6*(a^6*b + 6*a^4*b^3 - 13*a^2*b^5 - 2*b^7)*tan(d*x + c)^4 + (3*a^7 + 23*a^ 5*b^2 + 61*a^3*b^4 - 151*a*b^6)*tan(d*x + c)^3 + 2*(5*a^6*b + 37*a^4*b^3 - 73*a^2*b^5 - 9*b^7)*tan(d*x + c)^2 + (5*a^7 + 26*a^5*b^2 + 49*a^3*b^4 - 6 8*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^1 0 + 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
Leaf count of result is larger than twice the leaf count of optimal. 587 vs. \(2 (284) = 568\).
Time = 0.61 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.99 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (a^{7} + 7 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 35 \, a b^{6}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {12 \, {\left (7 \, a^{2} b^{5} - b^{7}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {24 \, {\left (7 \, a^{2} b^{6} - b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} + 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} + b^{11}} + \frac {3 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} + 18 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} - 81 \, a b^{6} \tan \left (d x + c\right )^{5} + 6 \, a^{6} b \tan \left (d x + c\right )^{4} + 36 \, a^{4} b^{3} \tan \left (d x + c\right )^{4} - 78 \, a^{2} b^{5} \tan \left (d x + c\right )^{4} - 12 \, b^{7} \tan \left (d x + c\right )^{4} + 3 \, a^{7} \tan \left (d x + c\right )^{3} + 23 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 61 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} - 151 \, a b^{6} \tan \left (d x + c\right )^{3} + 10 \, a^{6} b \tan \left (d x + c\right )^{2} + 74 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} - 146 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} - 18 \, b^{7} \tan \left (d x + c\right )^{2} + 5 \, a^{7} \tan \left (d x + c\right ) + 26 \, a^{5} b^{2} \tan \left (d x + c\right ) + 49 \, a^{3} b^{4} \tan \left (d x + c\right ) - 68 \, a b^{6} \tan \left (d x + c\right ) + 6 \, a^{6} b + 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} - 4 \, b^{7}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right )^{3} + a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right ) + a\right )}^{2}}}{8 \, d} \]
1/8*(3*(a^7 + 7*a^5*b^2 + 35*a^3*b^4 - 35*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*(7*a^2*b^5 - b^7)*lo g(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*(7*a^2*b^6 - b^8)*log(abs(b*tan(d*x + c) + a))/(a^10*b + 5*a^8*b^3 + 10*a^6*b^5 + 10*a^4*b^7 + 5*a^2*b^9 + b^11) + (3*a^5*b^2*tan(d *x + c)^5 + 18*a^3*b^4*tan(d*x + c)^5 - 81*a*b^6*tan(d*x + c)^5 + 6*a^6*b* tan(d*x + c)^4 + 36*a^4*b^3*tan(d*x + c)^4 - 78*a^2*b^5*tan(d*x + c)^4 - 1 2*b^7*tan(d*x + c)^4 + 3*a^7*tan(d*x + c)^3 + 23*a^5*b^2*tan(d*x + c)^3 + 61*a^3*b^4*tan(d*x + c)^3 - 151*a*b^6*tan(d*x + c)^3 + 10*a^6*b*tan(d*x + c)^2 + 74*a^4*b^3*tan(d*x + c)^2 - 146*a^2*b^5*tan(d*x + c)^2 - 18*b^7*tan (d*x + c)^2 + 5*a^7*tan(d*x + c) + 26*a^5*b^2*tan(d*x + c) + 49*a^3*b^4*ta n(d*x + c) - 68*a*b^6*tan(d*x + c) + 6*a^6*b + 44*a^4*b^3 - 62*a^2*b^5 - 4 *b^7)/((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
Time = 6.49 (sec) , antiderivative size = 715, normalized size of antiderivative = 2.42 \[ \int \frac {\cos ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3\,a^6\,b+22\,a^4\,b^3-31\,a^2\,b^5-2\,b^7}{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 )\,\left (5\,a^7+26\,a^5\,b^2+49\,a^3\,b^4-68\,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 {3\,{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^5\,b^2+6\,a^3\,b^4-27\,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 (3\,a^7+23\,a^5\,b^2+61\,a^3\,b^4-151\,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 {3\,{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (a^6\,b+6\,a^4\,b^3-13\,a^2\,b^5-2\,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 {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (5\,a^6\,b+37\,a^4\,b^3-73\,a^2\,b^5-9\,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 )}}{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 {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {21\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {24\,b^7}{{\left (a^2+b^2\right )}^5}\right )}{d}+\frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-a^2\,1{}\mathrm {i}+5\,a\,b+b^2\,8{}\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 )}+\frac {3\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (a^2\,1{}\mathrm {i}+5\,a\,b-b^2\,8{}\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 )} \]
((3*a^6*b - 2*b^7 - 31*a^2*b^5 + 22*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)*(5*a^7 - 68*a*b^6 + 49*a^3*b^4 + 26 *a^5*b^2))/(8*(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*(6*a^3*b^4 - 27*a*b^6 + 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*(3*a^7 - 151*a*b^6 + 61*a^3*b^4 + 2 3*a^5*b^2))/(8*(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^6*b - 2*b^7 - 13*a^2*b^5 + 6*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*(5*a^6*b - 9*b^7 - 73*a^2* b^5 + 37*a^4*b^3))/(4*(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)) + (log(a + b*tan(c + d*x))*((21*b^5)/(a^2 + b^2)^4 - (24*b^7)/(a ^2 + b^2)^5))/d + (3*log(tan(c + d*x) - 1i)*(5*a*b - a^2*1i + b^2*8i))/(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*lo g(tan(c + d*x) + 1i)*(5*a*b + a^2*1i - b^2*8i))/(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))