Integrand size = 21, antiderivative size = 241 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {5 a b^4 \text {arctanh}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \cos (c+d x) \sqrt {\sec ^2(c+d x)}}{\left (a^2+b^2\right )^{7/2} d}+\frac {b \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sec (c+d x)}{3 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\cos ^3(c+d x) (b+a \tan (c+d x))}{3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\cos (c+d x) \left (b \left (a^2-4 b^2\right )-a \left (2 a^2+7 b^2\right ) \tan (c+d x)\right )}{3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \]
-5*a*b^4*arctanh((b-a*tan(d*x+c))/(a^2+b^2)^(1/2)/(sec(d*x+c)^2)^(1/2))*co s(d*x+c)*(sec(d*x+c)^2)^(1/2)/(a^2+b^2)^(7/2)/d+1/3*b*(2*a^4+9*a^2*b^2-8*b ^4)*sec(d*x+c)/(a^2+b^2)^3/d/(a+b*tan(d*x+c))+1/3*cos(d*x+c)^3*(b+a*tan(d* x+c))/(a^2+b^2)/d/(a+b*tan(d*x+c))-1/3*cos(d*x+c)*(b*(a^2-4*b^2)-a*(2*a^2+ 7*b^2)*tan(d*x+c))/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Time = 1.80 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\sec (c+d x) \left (240 a b^4 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))+\left (a^2+b^2\right ) \left (15 a^4 b+90 a^2 b^3-45 b^5+20 b^3 \left (a^2+b^2\right ) \cos (2 (c+d x))+b \left (a^2+b^2\right )^2 \cos (4 (c+d x))+10 a^5 \sin (2 (c+d x))+40 a^3 b^2 \sin (2 (c+d x))+30 a b^4 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+a b^4 \sin (4 (c+d x))\right )\right )}{24 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))} \]
(Sec[c + d*x]*(240*a*b^4*Sqrt[a^2 + b^2]*ArcTanh[(-b + a*Tan[(c + d*x)/2]) /Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x]) + (a^2 + b^2)*(15*a^4* b + 90*a^2*b^3 - 45*b^5 + 20*b^3*(a^2 + b^2)*Cos[2*(c + d*x)] + b*(a^2 + b ^2)^2*Cos[4*(c + d*x)] + 10*a^5*Sin[2*(c + d*x)] + 40*a^3*b^2*Sin[2*(c + d *x)] + 30*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2*Sin[4* (c + d*x)] + a*b^4*Sin[4*(c + d*x)])))/(24*(a^2 + b^2)^4*d*(a + b*Tan[c + d*x]))
Time = 0.53 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3992, 496, 25, 27, 686, 25, 25, 27, 679, 488, 219}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^3 (a+b \tan (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3992 |
| \(\displaystyle \frac {\sec (c+d x) \int \frac {1}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x)+1\right )^{5/2}}d(b \tan (c+d x))}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}-\frac {b^2 \int -\frac {2 \left (\frac {a^2}{b^2}+2\right ) b^2+3 a \tan (c+d x) b}{b^2 (a+b \tan (c+d x))^2 \left (\tan ^2(c+d x)+1\right )^{3/2}}d(b \tan (c+d x))}{3 \left (a^2+b^2\right )}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {b^2 \int \frac {2 \left (a^2+2 b^2\right )+3 a b \tan (c+d x)}{b^2 (a+b \tan (c+d x))^2 \left (\tan ^2(c+d x)+1\right )^{3/2}}d(b \tan (c+d x))}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\int \frac {2 \left (a^2+2 b^2\right )+3 a b \tan (c+d x)}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x)+1\right )^{3/2}}d(b \tan (c+d x))}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}-\frac {b^4 \int -\frac {2 \left (4-\frac {a^2}{b^2}\right ) b^4+a \left (2 a^2+7 b^2\right ) \tan (c+d x) b}{b^4 (a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {b^4 \int -\frac {2 b^2 \left (a^2-4 b^2\right )-a b \left (2 a^2+7 b^2\right ) \tan (c+d x)}{b^4 (a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}+\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}-\frac {b^4 \int \frac {2 b^2 \left (a^2-4 b^2\right )-a b \left (2 a^2+7 b^2\right ) \tan (c+d x)}{b^4 (a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}-\frac {\int \frac {2 b^2 \left (a^2-4 b^2\right )-a b \left (2 a^2+7 b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2 \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}-\frac {-\frac {15 a b^4 \int \frac {1}{(a+b \tan (c+d x)) \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{a^2+b^2}-\frac {b^2 \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}-\frac {\frac {15 a b^4 \int \frac {1}{\frac {a^2}{b^2}-b^2 \tan ^2(c+d x)+1}d\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\tan ^2(c+d x)+1}}}{a^2+b^2}-\frac {b^2 \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}}{3 \left (a^2+b^2\right )}+\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {a b \tan (c+d x)+b^2}{3 \left (a^2+b^2\right ) \left (\tan ^2(c+d x)+1\right )^{3/2} (a+b \tan (c+d x))}+\frac {\frac {a b \left (2 a^2+7 b^2\right ) \tan (c+d x)+b^4 \left (4-\frac {a^2}{b^2}\right )}{\left (a^2+b^2\right ) \sqrt {\tan ^2(c+d x)+1} (a+b \tan (c+d x))}-\frac {\frac {15 a b^5 \text {arctanh}\left (\frac {b^2 \tan (c+d x)}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {b^2 \left (2 a^4+9 a^2 b^2-8 b^4\right ) \sqrt {\tan ^2(c+d x)+1}}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a^2+b^2}}{3 \left (a^2+b^2\right )}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
(Sec[c + d*x]*((b^2 + a*b*Tan[c + d*x])/(3*(a^2 + b^2)*(a + b*Tan[c + d*x] )*(1 + Tan[c + d*x]^2)^(3/2)) + (((4 - a^2/b^2)*b^4 + a*b*(2*a^2 + 7*b^2)* Tan[c + d*x])/((a^2 + b^2)*(a + b*Tan[c + d*x])*Sqrt[1 + Tan[c + d*x]^2]) - ((15*a*b^5*ArcTanh[(b^2*Tan[c + d*x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2 ) - (b^2*(2*a^4 + 9*a^2*b^2 - 8*b^4)*Sqrt[1 + Tan[c + d*x]^2])/((a^2 + b^2 )*(a + b*Tan[c + d*x])))/(a^2 + b^2))/(3*(a^2 + b^2))))/(b*d*Sqrt[Sec[c + d*x]^2])
3.6.65.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, 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_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
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[Sec[e + f*x]/(b*f*Sqrt[Sec[e + f*x]^2]) 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 - 1)/2]
Time = 8.25 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b -6 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {2}{3} a^{4}-6 a^{2} b^{2}+\frac {8}{3} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 a^{3} b}{3}-\frac {14 a \,b^{3}}{3}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-b}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-\frac {5 a \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{d}\) | \(320\) |
| default | \(\frac {-\frac {2 \left (\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2 a^{3} b -6 a \,b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-\frac {2}{3} a^{4}-6 a^{2} b^{2}+\frac {8}{3} b^{4}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-a^{4}-3 a^{2} b^{2}+2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {2 a^{3} b}{3}-\frac {14 a \,b^{3}}{3}\right )}{\left (a^{2}+b^{2}\right ) \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a}-b}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a}-\frac {5 a \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{d}\) | \(320\) |
| risch | \(-\frac {i {\mathrm e}^{3 i \left (d x +c \right )}}{24 \left (-2 i a b +a^{2}-b^{2}\right ) d}-\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} b}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a}{8 \left (-3 i b \,a^{2}+i b^{3}+a^{3}-3 a \,b^{2}\right ) d}-\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} b}{8 \left (i b +a \right )^{3} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a}{8 \left (i b +a \right )^{3} d}+\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{24 \left (i b +a \right )^{2} d}-\frac {2 i b^{5} {\mathrm e}^{i \left (d x +c \right )}}{\left (-i a +b \right )^{3} d \left (i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )}+\frac {5 b^{4} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {5 b^{4} a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{7}+3 i a^{5} b^{2}+3 i a^{3} b^{4}+i a \,b^{6}-a^{6} b -3 a^{4} b^{3}-3 b^{5} a^{2}-b^{7}}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}\) | \(449\) |
1/d*(-2/(a^2+b^2)/(a^4+2*a^2*b^2+b^4)*((-a^4-3*a^2*b^2+2*b^4)*tan(1/2*d*x+ 1/2*c)^5+(-2*a^3*b-6*a*b^3)*tan(1/2*d*x+1/2*c)^4+(-2/3*a^4-6*a^2*b^2+8/3*b ^4)*tan(1/2*d*x+1/2*c)^3-8*a*b^3*tan(1/2*d*x+1/2*c)^2+(-a^4-3*a^2*b^2+2*b^ 4)*tan(1/2*d*x+1/2*c)-2/3*a^3*b-14/3*a*b^3)/(1+tan(1/2*d*x+1/2*c)^2)^3-2*b ^4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-b^2/a*tan(1/2*d*x+1/2*c)-b)/(tan(1/2*d *x+1/2*c)^2*a-2*b*tan(1/2*d*x+1/2*c)-a)-5*a/(a^2+b^2)^(1/2)*arctanh(1/2*(2 *a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2))))
Time = 0.30 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {4 \, a^{6} b + 22 \, a^{4} b^{3} + 2 \, a^{2} b^{5} - 16 \, b^{7} + 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 2 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (a^{2} b^{4} \cos \left (d x + c\right ) + a b^{5} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (2 \, a^{7} + 11 \, a^{5} b^{2} + 16 \, a^{3} b^{4} + 7 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} d \sin \left (d x + c\right )\right )}} \]
1/6*(4*a^6*b + 22*a^4*b^3 + 2*a^2*b^5 - 16*b^7 + 2*(a^6*b + 3*a^4*b^3 + 3* a^2*b^5 + b^7)*cos(d*x + c)^4 - 2*(a^6*b - 2*a^4*b^3 - 7*a^2*b^5 - 4*b^7)* cos(d*x + c)^2 + 15*(a^2*b^4*cos(d*x + c) + a*b^5*sin(d*x + c))*sqrt(a^2 + b^2)*log(-(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b *cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) + 2*((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(d*x + c)^3 + (2*a^7 + 11*a^5*b^2 + 16 *a^3*b^4 + 7*a*b^6)*cos(d*x + c))*sin(d*x + c))/((a^9 + 4*a^7*b^2 + 6*a^5* b^4 + 4*a^3*b^6 + a*b^8)*d*cos(d*x + c) + (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*d*sin(d*x + c))
Timed out. \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 772 vs. \(2 (229) = 458\).
Time = 0.46 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.20 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {15 \, a b^{4} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (2 \, a^{5} b + 14 \, a^{3} b^{3} - 3 \, a b^{5} - \frac {15 \, a b^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (3 \, a^{6} + 13 \, a^{4} b^{2} + 22 \, a^{2} b^{4} - 3 \, b^{6}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {{\left (4 \, a^{5} b + 28 \, a^{3} b^{3} - 21 \, a b^{5}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (a^{6} - 9 \, a^{4} b^{2} - 46 \, a^{2} b^{4} + 9 \, b^{6}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, {\left (2 \, a^{5} b + 6 \, a^{3} b^{3} - 5 \, a b^{5}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 38 \, a^{2} b^{4} - 9 \, b^{6}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3 \, {\left (a^{6} + 3 \, a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + \frac {2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {2 \, {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2 \, {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}}{3 \, d} \]
-1/3*(15*a*b^4*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2 ))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - sqrt(a^2 + b^2)))/((a^6 + 3*a^ 4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 2*(2*a^5*b + 14*a^3*b^3 - 3*a* b^5 - 15*a*b^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (3*a^6 + 13*a^4*b^2 + 22*a^2*b^4 - 3*b^6)*sin(d*x + c)/(cos(d*x + c) + 1) + (4*a^5*b + 28*a^3*b ^3 - 21*a*b^5)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - (a^6 - 9*a^4*b^2 - 46 *a^2*b^4 + 9*b^6)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 5*(2*a^5*b + 6*a^3 *b^3 - 5*a*b^5)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + (a^6 + 3*a^4*b^2 + 3 8*a^2*b^4 - 9*b^6)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3*(a^6 + 3*a^4*b^ 2 - 2*a^2*b^4 + b^6)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(d*x + c)/(cos(d*x + c) + 1) + 2*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*sin(d *x + c)^2/(cos(d*x + c) + 1)^2 + 6*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7) *sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 6*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 2*(a^8 + 3*a^6*b^2 + 3*a^4*b^ 4 + a^2*b^6)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - (a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8))/d
Time = 0.51 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.82 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\frac {15 \, a b^{4} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} - \frac {6 \, {\left (b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a b^{5}\right )}}{{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}} - \frac {2 \, {\left (3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{3} b + 14 \, a b^{3}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \]
-1/3*(15*a*b^4*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2)) /abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^ 2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 6*(b^6*tan(1/2*d*x + 1/2*c) + a*b^ 5)/((a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - 2*b* tan(1/2*d*x + 1/2*c) - a)) - 2*(3*a^4*tan(1/2*d*x + 1/2*c)^5 + 9*a^2*b^2*t an(1/2*d*x + 1/2*c)^5 - 6*b^4*tan(1/2*d*x + 1/2*c)^5 + 6*a^3*b*tan(1/2*d*x + 1/2*c)^4 + 18*a*b^3*tan(1/2*d*x + 1/2*c)^4 + 2*a^4*tan(1/2*d*x + 1/2*c) ^3 + 18*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 - 8*b^4*tan(1/2*d*x + 1/2*c)^3 + 24 *a*b^3*tan(1/2*d*x + 1/2*c)^2 + 3*a^4*tan(1/2*d*x + 1/2*c) + 9*a^2*b^2*tan (1/2*d*x + 1/2*c) - 6*b^4*tan(1/2*d*x + 1/2*c) + 2*a^3*b + 14*a*b^3)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(1/2*d*x + 1/2*c)^2 + 1)^3))/d
Time = 8.30 (sec) , antiderivative size = 674, normalized size of antiderivative = 2.80 \[ \int \frac {\cos ^3(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {2\,\left (2\,a^4\,b+14\,a^2\,b^3-3\,b^5\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {10\,b^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {10\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (2\,a^4+6\,a^2\,b^2-5\,b^4\right )}{3\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^4\,b+28\,a^2\,b^3-21\,b^5\right )}{3\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (a^6+3\,a^4\,b^2-2\,a^2\,b^4+b^6\right )}{a\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^6+13\,a^4\,b^2+22\,a^2\,b^4-3\,b^6\right )}{3\,a\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (a^6+3\,a^4\,b^2+38\,a^2\,b^4-9\,b^6\right )}{3\,a\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^6-9\,a^4\,b^2-46\,a^2\,b^4+9\,b^6\right )}{3\,a\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+6\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {10\,a\,b^4\,\mathrm {atanh}\left (\frac {2\,a^6\,b+2\,b^7+6\,a^2\,b^5+6\,a^4\,b^3-2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{2\,{\left (a^2+b^2\right )}^{7/2}}\right )}{d\,{\left (a^2+b^2\right )}^{7/2}} \]
((2*(2*a^4*b - 3*b^5 + 14*a^2*b^3))/(3*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2) ) - (10*b^5*tan(c/2 + (d*x)/2)^6)/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (1 0*b*tan(c/2 + (d*x)/2)^4*(2*a^4 - 5*b^4 + 6*a^2*b^2))/(3*(a^6 + b^6 + 3*a^ 2*b^4 + 3*a^4*b^2)) + (2*tan(c/2 + (d*x)/2)^2*(4*a^4*b - 21*b^5 + 28*a^2*b ^3))/(3*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) - (2*tan(c/2 + (d*x)/2)^7*(a^ 6 + b^6 - 2*a^2*b^4 + 3*a^4*b^2))/(a*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*tan(c/2 + (d*x)/2)*(3*a^6 - 3*b^6 + 22*a^2*b^4 + 13*a^4*b^2))/(3*a*(a ^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) + (2*tan(c/2 + (d*x)/2)^5*(a^6 - 9*b^6 + 38*a^2*b^4 + 3*a^4*b^2))/(3*a*(a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)) - (2* tan(c/2 + (d*x)/2)^3*(a^6 + 9*b^6 - 46*a^2*b^4 - 9*a^4*b^2))/(3*a*(a^2 + b ^2)*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a + 2*b*tan(c/2 + (d*x)/2) + 2*a*tan(c/2 + (d*x)/2)^2 - 2*a*tan(c/2 + (d*x)/2)^6 - a*tan(c/2 + (d*x)/2)^8 + 6*b*ta n(c/2 + (d*x)/2)^3 + 6*b*tan(c/2 + (d*x)/2)^5 + 2*b*tan(c/2 + (d*x)/2)^7)) - (10*a*b^4*atanh((2*a^6*b + 2*b^7 + 6*a^2*b^5 + 6*a^4*b^3 - 2*a*tan(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(2*(a^2 + b^2)^(7/2))))/(d *(a^2 + b^2)^(7/2))