Integrand size = 21, antiderivative size = 95 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {\sec (c+d x) (b-a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2} \] Output:
-1/2*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/(a^2+b^2)^(3/2)/ d-1/2*sec(d*x+c)*(b-a*tan(d*x+c))/(a^2+b^2)/d/(a+b*tan(d*x+c))^2
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.39 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\left (a^2+b^2\right ) (-b \cos (c+d x)+a \sin (c+d x))+2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{2 (a-i b)^2 (a+i b)^2 d (a \cos (c+d x)+b \sin (c+d x))^2} \] Input:
Integrate[Sec[c + d*x]^3/(a + b*Tan[c + d*x])^3,x]
Output:
((a^2 + b^2)*(-(b*Cos[c + d*x]) + a*Sin[c + d*x]) + 2*Sqrt[a^2 + b^2]*ArcT anh[(-b + a*Tan[(c + d*x)/2])/Sqrt[a^2 + b^2]]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(2*(a - I*b)^2*(a + I*b)^2*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^2 )
Time = 0.32 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3992, 486, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^3}{(a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3992 |
| \(\displaystyle \frac {\sec (c+d x) \int \frac {\sqrt {\tan ^2(c+d x)+1}}{(a+b \tan (c+d x))^3}d(b \tan (c+d x))}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {\sec (c+d x) \left (\frac {\int \frac {1}{(a+b \tan (c+d x)) \sqrt {\tan ^2(c+d x)+1}}d(b \tan (c+d x))}{2 \left (a^2+b^2\right )}-\frac {\sqrt {\tan ^2(c+d x)+1} \left (b^2-a b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {\sec (c+d x) \left (-\frac {\int \frac {1}{\frac {a^2}{b^2}-b^2 \tan ^2(c+d x)+1}d\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\tan ^2(c+d x)+1}}}{2 \left (a^2+b^2\right )}-\frac {\sqrt {\tan ^2(c+d x)+1} \left (b^2-a b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sec (c+d x) \left (-\frac {b \text {arctanh}\left (\frac {b^2 \tan (c+d x)}{\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2}}-\frac {\sqrt {\tan ^2(c+d x)+1} \left (b^2-a b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\right )}{b d \sqrt {\sec ^2(c+d x)}}\) |
Input:
Int[Sec[c + d*x]^3/(a + b*Tan[c + d*x])^3,x]
Output:
(Sec[c + d*x]*(-1/2*(b*ArcTanh[(b^2*Tan[c + d*x])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - ((b^2 - a*b*Tan[c + d*x])*Sqrt[1 + Tan[c + d*x]^2])/(2*(a^2 + b^2)*(a + b*Tan[c + d*x])^2)))/(b*d*Sqrt[Sec[c + d*x]^2])
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && GtQ[p, 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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(190\) vs. \(2(87)=174\).
Time = 16.75 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.01
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a \left (a^{2}+b^{2}\right )}-\frac {b \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \left (a^{2}+b^{2}\right ) a^{2}}-\frac {\left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+b^{2}\right ) a}+\frac {b}{2 a^{2}+2 b^{2}}\right )}{\left (a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d}\) | \(191\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (a^{2}+2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a \left (a^{2}+b^{2}\right )}-\frac {b \left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \left (a^{2}+b^{2}\right ) a^{2}}-\frac {\left (a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}+b^{2}\right ) a}+\frac {b}{2 a^{2}+2 b^{2}}\right )}{\left (a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}}{d}\) | \(191\) |
| risch | \(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (i a \,{\mathrm e}^{2 i \left (d x +c \right )}+b \,{\mathrm e}^{2 i \left (d x +c \right )}-i a +b \right )}{\left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{2} \left (-i a +b \right ) d \left (i a +b \right )}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{3}+i a \,b^{2}-a^{2} b -b^{3}}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}\) | \(224\) |
Input:
int(sec(d*x+c)^3/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2*(-1/2*(a^2+2*b^2)/a/(a^2+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*b*(a^2-2*b^ 2)/(a^2+b^2)/a^2*tan(1/2*d*x+1/2*c)^2-1/2*(a^2-2*b^2)/(a^2+b^2)/a*tan(1/2* d*x+1/2*c)+1/2*b/(a^2+b^2))/(a*tan(1/2*d*x+1/2*c)^2-2*b*tan(1/2*d*x+1/2*c) -a)^2+1/(a^2+b^2)^(3/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2) ^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (88) = 176\).
Time = 0.10 (sec) , antiderivative size = 294, normalized size of antiderivative = 3.09 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {{\left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 2 \, {\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right ) + 2 \, {\left (a^{3} + a b^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{6} + a^{4} b^{2} - a^{2} b^{4} - b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} d\right )}} \] Input:
integrate(sec(d*x+c)^3/(a+b*tan(d*x+c))^3,x, algorithm="fricas")
Output:
1/4*((2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)* 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^2*b + b^3)*cos(d*x + c) + 2*(a^3 + a*b^2)*sin(d*x + c))/((a^6 + a^4 *b^2 - a^2*b^4 - b^6)*d*cos(d*x + c)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d*c os(d*x + c)*sin(d*x + c) + (a^4*b^2 + 2*a^2*b^4 + b^6)*d)
\[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(sec(d*x+c)**3/(a+b*tan(d*x+c))**3,x)
Output:
Integral(sec(c + d*x)**3/(a + b*tan(c + d*x))**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (88) = 176\).
Time = 0.13 (sec) , antiderivative size = 326, normalized size of antiderivative = 3.43 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {2 \, {\left (a^{2} b - \frac {{\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{6} + a^{4} b^{2} + \frac {4 \, {\left (a^{5} b + a^{3} b^{3}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, {\left (a^{6} - a^{4} b^{2} - 2 \, a^{2} b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {4 \, {\left (a^{5} b + a^{3} b^{3}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {{\left (a^{6} + a^{4} b^{2}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}}}{2 \, d} \] Input:
integrate(sec(d*x+c)^3/(a+b*tan(d*x+c))^3,x, algorithm="maxima")
Output:
-1/2*(2*(a^2*b - (a^3 - 2*a*b^2)*sin(d*x + c)/(cos(d*x + c) + 1) - (a^2*b - 2*b^3)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - (a^3 + 2*a*b^2)*sin(d*x + c )^3/(cos(d*x + c) + 1)^3)/(a^6 + a^4*b^2 + 4*(a^5*b + a^3*b^3)*sin(d*x + c )/(cos(d*x + c) + 1) - 2*(a^6 - a^4*b^2 - 2*a^2*b^4)*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 4*(a^5*b + a^3*b^3)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + (a^6 + a^4*b^2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^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^2 + b^2)^(3/2))/d
Leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (88) = 176\).
Time = 0.35 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.33 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\frac {\log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2} b\right )}}{{\left (a^{4} + a^{2} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )}^{2}}}{2 \, d} \] Input:
integrate(sec(d*x+c)^3/(a+b*tan(d*x+c))^3,x, algorithm="giac")
Output:
-1/2*(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^2 + b^2)^(3/2) - 2*(a^ 3*tan(1/2*d*x + 1/2*c)^3 + 2*a*b^2*tan(1/2*d*x + 1/2*c)^3 + a^2*b*tan(1/2* d*x + 1/2*c)^2 - 2*b^3*tan(1/2*d*x + 1/2*c)^2 + a^3*tan(1/2*d*x + 1/2*c) - 2*a*b^2*tan(1/2*d*x + 1/2*c) - a^2*b)/((a^4 + a^2*b^2)*(a*tan(1/2*d*x + 1 /2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)^2))/d
Time = 3.03 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.74 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^2-2\,b^2\right )}{a\,\left (a^2+b^2\right )}-\frac {b}{a^2+b^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^2+2\,b^2\right )}{a\,\left (a^2+b^2\right )}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^2-2\,b^2\right )}{a^2\,\left (a^2+b^2\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {\mathrm {atanh}\left (\frac {\left (2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {2\,a^2\,b+2\,b^3}{a^2+b^2}\right )\,\left (\frac {a^2}{2}+\frac {b^2}{2}\right )}{{\left (a^2+b^2\right )}^{3/2}}\right )}{d\,{\left (a^2+b^2\right )}^{3/2}} \] Input:
int(1/(cos(c + d*x)^3*(a + b*tan(c + d*x))^3),x)
Output:
((tan(c/2 + (d*x)/2)*(a^2 - 2*b^2))/(a*(a^2 + b^2)) - b/(a^2 + b^2) + (tan (c/2 + (d*x)/2)^3*(a^2 + 2*b^2))/(a*(a^2 + b^2)) + (b*tan(c/2 + (d*x)/2)^2 *(a^2 - 2*b^2))/(a^2*(a^2 + b^2)))/(d*(a^2*tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)^2*(2*a^2 - 4*b^2) + a^2 - 4*a*b*tan(c/2 + (d*x)/2)^3 + 4*a*b*ta n(c/2 + (d*x)/2))) + atanh(((2*a*tan(c/2 + (d*x)/2) - (2*a^2*b + 2*b^3)/(a ^2 + b^2))*(a^2/2 + b^2/2))/(a^2 + b^2)^(3/2))/(d*(a^2 + b^2)^(3/2))
Time = 0.18 (sec) , antiderivative size = 2842, normalized size of antiderivative = 29.92 \[ \int \frac {\sec ^3(c+d x)}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^3/(a+b*tan(d*x+c))^3,x)
Output:
( - 4*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2 ))*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)**2*a**2*b**3*i + 4*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)* sin(c + d*x)**2*tan(c + d*x)**2*b**5*i - 8*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)*a**3*b**2*i + 8*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i )/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*tan(c + d*x)*a*b**4*i - 4*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*c os(c + d*x)*sin(c + d*x)**2*a**4*b*i + 4*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b* *3*i + 4*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b **2))*cos(c + d*x)*tan(c + d*x)**2*a**2*b**3*i + 8*sqrt(a**2 + b**2)*atan( (tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*cos(c + d*x)*tan(c + d*x)* a**3*b**2*i + 4*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a **2 + b**2))*cos(c + d*x)*a**4*b*i - 8*sqrt(a**2 + b**2)*atan((tan((c + d* x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**3*tan(c + d*x)**2*a*b**4 *i - 16*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b* *2))*sin(c + d*x)**3*tan(c + d*x)*a**2*b**3*i - 8*sqrt(a**2 + b**2)*atan(( tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b**2))*sin(c + d*x)**3*a**3*b**2*i + 8*sqrt(a**2 + b**2)*atan((tan((c + d*x)/2)*a*i - b*i)/sqrt(a**2 + b*...