Integrand size = 20, antiderivative size = 848 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a+i b) (i a+b)^2 \left (i a-b+(i a+b) e^{2 i e+2 i f x}\right ) f}+\frac {(c+d x)^4}{4 (a-i b)^2 d}+\frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {2 b (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 (c+d x)^3 \log \left (1+\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}+\frac {3 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{\left (a^2+b^2\right )^2 f^3}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4} \]
-3*I*b^2*d^2*(d*x+c)*polylog(3,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b ^2)^2/f^3+2*b^2*(d*x+c)^3/(a+I*b)/(I*a+b)^2/(I*a-b+(I*a+b)*exp(2*I*e+2*I*f *x))/f+1/4*(d*x+c)^4/(a-I*b)^2/d+b*(d*x+c)^4/(I*a-b)/(a-I*b)^2/d-b^2*(d*x+ c)^4/(a^2+b^2)^2/d+3*b^2*d*(d*x+c)^2*ln(1+(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I* b))/(a^2+b^2)^2/f^2+2*b*(d*x+c)^3*ln(1+(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b)) /(a-I*b)^2/(a+I*b)/f-3*I*b^2*d^2*(d*x+c)*polylog(2,-(a-I*b)*exp(2*I*e+2*I* f*x)/(a+I*b))/(a^2+b^2)^2/f^3-2*I*b^2*(d*x+c)^3/(a^2+b^2)^2/f+3*b*d*(d*x+c )^2*polylog(2,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(I*a-b)/(a-I*b)^2/f^2-3 *b^2*d*(d*x+c)^2*polylog(2,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2)^ 2/f^2+3/2*b^2*d^3*polylog(3,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a^2+b^2) ^2/f^4+3*b*d^2*(d*x+c)*polylog(3,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/(a-I *b)^2/(a+I*b)/f^3-2*I*b^2*(d*x+c)^3*ln(1+(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b ))/(a^2+b^2)^2/f-3/2*b*d^3*polylog(4,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a+I*b))/ (I*a-b)/(a-I*b)^2/f^4+3/2*b^2*d^3*polylog(4,-(a-I*b)*exp(2*I*e+2*I*f*x)/(a +I*b))/(a^2+b^2)^2/f^4
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1724\) vs. \(2(848)=1696\).
Time = 9.64 (sec) , antiderivative size = 1724, normalized size of antiderivative = 2.03 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx =\text {Too large to display} \]
(b*((-4*c^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(3*b*d + 2*a *c*f)*x)/(a^2 + b^2) + (4*b*(c + d*x)^3)/(a - I*b) + (2*a*f*(c + d*x)^4)/( (a - I*b)*d) + (12*c*d*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*( b*d + a*c*f)*x*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/((a + I *b)*(I*a + b)*f) + (6*d^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)) )*(b*d + 2*a*c*f)*x^2*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/ ((a + I*b)*(I*a + b)*f) + (4*a*d^3*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^( (2*I)*e)))*x^3*Log[1 + (a + I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/((a + I *b)*(I*a + b)) + (2*c^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))* (3*b*d + 2*a*c*f)*Log[a + I*b + (a - I*b)*E^((2*I)*(e + f*x))])/((a + I*b) *(I*a + b)*f) + (6*c*d*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*( b*d + a*c*f)*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))])/((a^2 + b^2)*f^2) + (3*d^2*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(b *d + 2*a*c*f)*(2*f*x*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x))) ] - I*PolyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2 )*f^3) + (3*a*d^3*((-I)*b*(-1 + E^((2*I)*e)) + a*(1 + E^((2*I)*e)))*(2*f^2 *x^2*PolyLog[2, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))] - (2*I)*f*x*Po lyLog[3, (-a - I*b)/((a - I*b)*E^((2*I)*(e + f*x)))] - PolyLog[4, (-a - I* b)/((a - I*b)*E^((2*I)*(e + f*x)))]))/((a^2 + b^2)*f^3)))/(2*(a - I*b)*(a + I*b)*(b - b*E^((2*I)*e) - I*a*(1 + E^((2*I)*e)))*f) + (3*x^2*(a*c^2*d...
Time = 2.25 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4217, 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.
\(\displaystyle \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2}dx\) |
\(\Big \downarrow \) 4217 |
\(\displaystyle \int \left (-\frac {4 b^2 (c+d x)^3}{(b+i a)^2 \left (i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}+i a \left (1+\frac {i b}{a}\right )\right )^2}+\frac {4 b (c+d x)^3}{(a-i b)^2 \left (i a \left (1-\frac {i b}{a}\right ) e^{2 i e+2 i f x}+i a \left (1+\frac {i b}{a}\right )\right )}+\frac {(c+d x)^3}{(a-i b)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b (c+d x)^4}{(i a-b) (a-i b)^2 d}+\frac {(c+d x)^4}{4 (a-i b)^2 d}-\frac {b^2 (c+d x)^4}{\left (a^2+b^2\right )^2 d}+\frac {2 b \log \left (\frac {e^{2 i e+2 i f x} (a-i b)}{a+i b}+1\right ) (c+d x)^3}{(a-i b)^2 (a+i b) f}-\frac {2 i b^2 \log \left (\frac {e^{2 i e+2 i f x} (a-i b)}{a+i b}+1\right ) (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {2 b^2 (c+d x)^3}{(a+i b) (i a+b)^2 \left (i a+(i a+b) e^{2 i e+2 i f x}-b\right ) f}-\frac {2 i b^2 (c+d x)^3}{\left (a^2+b^2\right )^2 f}+\frac {3 b^2 d \log \left (\frac {e^{2 i e+2 i f x} (a-i b)}{a+i b}+1\right ) (c+d x)^2}{\left (a^2+b^2\right )^2 f^2}+\frac {3 b d \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)^2}{(i a-b) (a-i b)^2 f^2}-\frac {3 b^2 d \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)^2}{\left (a^2+b^2\right )^2 f^2}-\frac {3 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b d^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)}{(a-i b)^2 (a+i b) f^3}-\frac {3 i b^2 d^2 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right ) (c+d x)}{\left (a^2+b^2\right )^2 f^3}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}-\frac {3 b d^3 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 (i a-b) (a-i b)^2 f^4}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (4,-\frac {(a-i b) e^{2 i e+2 i f x}}{a+i b}\right )}{2 \left (a^2+b^2\right )^2 f^4}\) |
((-2*I)*b^2*(c + d*x)^3)/((a^2 + b^2)^2*f) + (2*b^2*(c + d*x)^3)/((a + I*b )*(I*a + b)^2*(I*a - b + (I*a + b)*E^((2*I)*e + (2*I)*f*x))*f) + (c + d*x) ^4/(4*(a - I*b)^2*d) + (b*(c + d*x)^4)/((I*a - b)*(a - I*b)^2*d) - (b^2*(c + d*x)^4)/((a^2 + b^2)^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + ((a - I*b)*E^( (2*I)*e + (2*I)*f*x))/(a + I*b)])/((a^2 + b^2)^2*f^2) + (2*b*(c + d*x)^3*L og[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b)])/((a - I*b)^2*(a + I *b)*f) - ((2*I)*b^2*(c + d*x)^3*Log[1 + ((a - I*b)*E^((2*I)*e + (2*I)*f*x) )/(a + I*b)])/((a^2 + b^2)^2*f) - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((( a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^3) + (3*b* d*(c + d*x)^2*PolyLog[2, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))] )/((I*a - b)*(a - I*b)^2*f^2) - (3*b^2*d*(c + d*x)^2*PolyLog[2, -(((a - I* b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2)^2*f^2) + (3*b^2*d^3* PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(a^2 + b^ 2)^2*f^4) + (3*b*d^2*(c + d*x)*PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)* f*x))/(a + I*b))])/((a - I*b)^2*(a + I*b)*f^3) - ((3*I)*b^2*d^2*(c + d*x)* PolyLog[3, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/((a^2 + b^2) ^2*f^3) - (3*b*d^3*PolyLog[4, -(((a - I*b)*E^((2*I)*e + (2*I)*f*x))/(a + I *b))])/(2*(I*a - b)*(a - I*b)^2*f^4) + (3*b^2*d^3*PolyLog[4, -(((a - I*b)* E^((2*I)*e + (2*I)*f*x))/(a + I*b))])/(2*(a^2 + b^2)^2*f^4)
3.1.59.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (1/(a - I*b) - 2*I*(b/(a^2 + b^2 + (a - I*b)^2*E^(2*I*(e + f*x)))))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && ILtQ[n, 0] && IGtQ[m, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5958 vs. \(2 (763 ) = 1526\).
Time = 0.97 (sec) , antiderivative size = 5959, normalized size of antiderivative = 7.03
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2537 vs. \(2 (695) = 1390\).
Time = 0.33 (sec) , antiderivative size = 2537, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \]
1/4*((a^3 - a*b^2)*d^3*f^4*x^4 - 4*b^3*c^3*f^3 - 4*(b^3*d^3*f^3 - (a^3 - a *b^2)*c*d^2*f^4)*x^3 - 6*(2*b^3*c*d^2*f^3 - (a^3 - a*b^2)*c^2*d*f^4)*x^2 - 4*(3*b^3*c^2*d*f^3 - (a^3 - a*b^2)*c^3*f^4)*x - 6*(-I*a^2*b*d^3*f^2*x^2 - I*a^2*b*c^2*d*f^2 - I*a*b^2*c*d^2*f - I*(2*a^2*b*c*d^2*f^2 + a*b^2*d^3*f) *x + (-I*a*b^2*d^3*f^2*x^2 - I*a*b^2*c^2*d*f^2 - I*b^3*c*d^2*f - I*(2*a*b^ 2*c*d^2*f^2 + b^3*d^3*f)*x)*tan(f*x + e))*dilog(2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*t an(f*x + e)^2 + a^2 + b^2) + 1) - 6*(I*a^2*b*d^3*f^2*x^2 + I*a^2*b*c^2*d*f ^2 + I*a*b^2*c*d^2*f + I*(2*a^2*b*c*d^2*f^2 + a*b^2*d^3*f)*x + (I*a*b^2*d^ 3*f^2*x^2 + I*a*b^2*c^2*d*f^2 + I*b^3*c*d^2*f + I*(2*a*b^2*c*d^2*f^2 + b^3 *d^3*f)*x)*tan(f*x + e))*dilog(2*((-I*a*b - b^2)*tan(f*x + e)^2 - a^2 + I* a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2) + 1) + 2*(2*a^2*b*d^3*f^3*x^3 + 2*a^2*b*d^3*e^3 + 6*a^2*b*c^2* d*e*f^2 - 3*a*b^2*d^3*e^2 + 3*(2*a^2*b*c*d^2*f^3 + a*b^2*d^3*f^2)*x^2 - 6* (a^2*b*c*d^2*e^2 - a*b^2*c*d^2*e)*f + 6*(a^2*b*c^2*d*f^3 + a*b^2*c*d^2*f^2 )*x + (2*a*b^2*d^3*f^3*x^3 + 2*a*b^2*d^3*e^3 + 6*a*b^2*c^2*d*e*f^2 - 3*b^3 *d^3*e^2 + 3*(2*a*b^2*c*d^2*f^3 + b^3*d^3*f^2)*x^2 - 6*(a*b^2*c*d^2*e^2 - b^3*c*d^2*e)*f + 6*(a*b^2*c^2*d*f^3 + b^3*c*d^2*f^2)*x)*tan(f*x + e))*log( -2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*t an(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 2*(2*a^2*b*d^3...
\[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4631 vs. \(2 (695) = 1390\).
Time = 2.41 (sec) , antiderivative size = 4631, normalized size of antiderivative = 5.46 \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \]
-1/12*(36*c^2*d*e*(2*a*b*log(b*tan(f*x + e) + a)/((a^4 + 2*a^2*b^2 + b^4)* f) - a*b*log(tan(f*x + e)^2 + 1)/((a^4 + 2*a^2*b^2 + b^4)*f) - b/((a^2*b + b^3)*f*tan(f*x + e) + (a^3 + a*b^2)*f) + (a^2 - b^2)*(f*x + e)/((a^4 + 2* a^2*b^2 + b^4)*f)) - 12*(2*a*b*log(b*tan(f*x + e) + a)/(a^4 + 2*a^2*b^2 + b^4) - a*b*log(tan(f*x + e)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (a^2 - b^2)*( f*x + e)/(a^4 + 2*a^2*b^2 + b^4) - b/(a^3 + a*b^2 + (a^2*b + b^3)*tan(f*x + e)))*c^3 - (3*(a^3 - I*a^2*b + a*b^2 - I*b^3)*(f*x + e)^4*d^3 - 24*(I*a* b^2 - b^3)*d^3*e^3 - 72*(-I*a*b^2 + b^3)*c*d^2*e^2*f - 12*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^3*e - (a^3 - I*a^2*b + a*b^2 - I*b^3)*c*d^2*f)*(f*x + e )^3 + 18*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^3*e^2 - 2*(a^3 - I*a^2*b + a*b ^2 - I*b^3)*c*d^2*e*f + (a^3 - I*a^2*b + a*b^2 - I*b^3)*c^2*d*f^2)*(f*x + e)^2 - 12*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d^3*e^3 - 3*(a^3 - I*a^2*b + a* b^2 - I*b^3)*c*d^2*e^2*f)*(f*x + e) - 12*(2*(I*a^2*b - a*b^2)*d^3*e^3 + 3* (-I*a*b^2 + b^3)*d^3*e^2 + 3*(-I*a*b^2 + b^3)*c^2*d*f^2 + 6*((-I*a^2*b + a *b^2)*c*d^2*e^2 + (I*a*b^2 - b^3)*c*d^2*e)*f + (2*(I*a^2*b + a*b^2)*d^3*e^ 3 + 3*(-I*a*b^2 - b^3)*d^3*e^2 + 3*(-I*a*b^2 - b^3)*c^2*d*f^2 + 6*((-I*a^2 *b - a*b^2)*c*d^2*e^2 + (I*a*b^2 + b^3)*c*d^2*e)*f)*cos(2*f*x + 2*e) - (2* (a^2*b - I*a*b^2)*d^3*e^3 - 3*(a*b^2 - I*b^3)*d^3*e^2 - 3*(a*b^2 - I*b^3)* c^2*d*f^2 - 6*((a^2*b - I*a*b^2)*c*d^2*e^2 - (a*b^2 - I*b^3)*c*d^2*e)*f)*s in(2*f*x + 2*e))*arctan2(-b*cos(2*f*x + 2*e) + a*sin(2*f*x + 2*e) + b, ...
\[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^3}{(a+b \tan (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \]