Integrand size = 20, antiderivative size = 300 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f} \]
-I*b^2*(d*x+c)^3/f+1/4*a^2*(d*x+c)^4/d+1/2*I*a*b*(d*x+c)^4/d-1/4*b^2*(d*x+ c)^4/d+3*b^2*d*(d*x+c)^2*ln(1+exp(2*I*(f*x+e)))/f^2-2*a*b*(d*x+c)^3*ln(1+e xp(2*I*(f*x+e)))/f-3*I*b^2*d^2*(d*x+c)*polylog(2,-exp(2*I*(f*x+e)))/f^3+3* I*a*b*d*(d*x+c)^2*polylog(2,-exp(2*I*(f*x+e)))/f^2+3/2*b^2*d^3*polylog(3,- exp(2*I*(f*x+e)))/f^4-3*a*b*d^2*(d*x+c)*polylog(3,-exp(2*I*(f*x+e)))/f^3-3 /2*I*a*b*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4+b^2*(d*x+c)^3*tan(f*x+e)/f
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1337\) vs. \(2(300)=600\).
Time = 7.33 (sec) , antiderivative size = 1337, normalized size of antiderivative = 4.46 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx =\text {Too large to display} \]
((I/4)*b^2*d^3*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2* I)*(e + f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x)) ] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e])/(E^ (I*e)*f^4) - ((I/2)*a*b*c*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))* Log[1 + E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((- 2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x) )])*Sec[e])/(E^(I*e)*f^3) - ((I/4)*a*b*d^3*E^(I*e)*((2*f^4*x^4)/E^((2*I)*e ) - (4*I)*(1 + E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-2*I)*(e + f*x))] + 6*(1 + E^((-2*I)*e))*f^2*x^2*PolyLog[2, -E^((-2*I)*(e + f*x))] - (6*I)*(1 + E^( (-2*I)*e))*f*x*PolyLog[3, -E^((-2*I)*(e + f*x))] - 3*(1 + E^((-2*I)*e))*Po lyLog[4, -E^((-2*I)*(e + f*x))])*Sec[e])/f^4 + (3*b^2*c^2*d*Sec[e]*(Cos[e] *Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^2*(Cos[e]^2 + Si n[e]^2)) - (2*a*b*c^3*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x] ] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (3*b^2*c*d^2*Csc[e]*((f^2*x^2 )/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[ 1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - Arc Tan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[ Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[ e]^2])*Sec[e])/(f^3*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a*b*c^2*d*C sc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[...
Time = 0.76 (sec) , antiderivative size = 300, 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, 4205, 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.
\(\displaystyle \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^3 (a+b \tan (e+f x))^2dx\) |
\(\Big \downarrow \) 4205 |
\(\displaystyle \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \tan (e+f x)+b^2 (c+d x)^3 \tan ^2(e+f x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a b (c+d x)^4}{2 d}-\frac {3 i a b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}\) |
((-I)*b^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) + ((I/2)*a*b*(c + d*x)^ 4)/d - (b^2*(c + d*x)^4)/(4*d) + (3*b^2*d*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (2*a*b*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f - ((3*I )*b^2*d^2*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a*b*d*( c + d*x)^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 + (3*b^2*d^3*PolyLog[3, - E^((2*I)*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, -E^((2*I)* (e + f*x))])/f^3 - (((3*I)/2)*a*b*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^ 4 + (b^2*(c + d*x)^3*Tan[e + f*x])/f
3.1.44.3.1 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, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (271 ) = 542\).
Time = 1.60 (sec) , antiderivative size = 952, normalized size of antiderivative = 3.17
-12*I/f^2*b*e^2*d^2*c*a*x+6*I/f^2*b*a*c*d^2*polylog(2,-exp(2*I*(f*x+e)))*x +12*I/f*b*d*c^2*a*e*x+2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/f/(exp(2 *I*(f*x+e))+1)+3/f^2*b^2*c^2*d*ln(exp(2*I*(f*x+e))+1)-6/f^2*b^2*c^2*d*ln(e xp(I*(f*x+e)))+3/f^2*b^2*d^3*ln(exp(2*I*(f*x+e))+1)*x^2-2/f*b*a*c^3*ln(exp (2*I*(f*x+e))+1)+4/f*b*a*c^3*ln(exp(I*(f*x+e)))-6/f^4*b^2*e^2*d^3*ln(exp(I *(f*x+e)))-2*I/f*b^2*d^3*x^3+4*I/f^4*b^2*d^3*e^3+1/2*I*d^3*a*b*x^4-3/2*d*b ^2*c^2*x^2+1/4*d^3*a^2*x^4+1/4/d*a^2*c^4-1/4*d^3*b^2*x^4-b^2*c^3*x-1/4/d*b ^2*c^4-2*I*a*b*c^3*x-1/2*I/d*a*b*c^4+d^2*a^2*c*x^3+3/2*d*a^2*c^2*x^2+a^2*c ^3*x-d^2*b^2*c*x^3-3/2*I*a*b*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4-4/f^4*b* e^3*a*d^3*ln(exp(I*(f*x+e)))-3/f^3*b*d^3*a*polylog(3,-exp(2*I*(f*x+e)))*x+ 6/f^2*b^2*d^2*c*ln(exp(2*I*(f*x+e))+1)*x-2/f*b*d^3*a*ln(exp(2*I*(f*x+e))+1 )*x^3-3/f^3*b*a*c*d^2*polylog(3,-exp(2*I*(f*x+e)))+12/f^3*b^2*e*c*d^2*ln(e xp(I*(f*x+e)))-3*I/f^3*b^2*d^3*polylog(2,-exp(2*I*(f*x+e)))*x+3*I/f^4*b*e^ 4*a*d^3-6*I/f*b^2*d^2*c*x^2-6*I/f^3*b^2*d^2*c*e^2-3*I/f^3*b^2*d^2*c*polylo g(2,-exp(2*I*(f*x+e)))+6*I/f^3*b^2*d^3*e^2*x+2*I*d^2*a*b*c*x^3+3*I*d*a*b*c ^2*x^2+3/2*b^2*d^3*polylog(3,-exp(2*I*(f*x+e)))/f^4-12/f^2*b*e*a*c^2*d*ln( exp(I*(f*x+e)))-6/f*b*d*c^2*a*ln(exp(2*I*(f*x+e))+1)*x-6/f*b*a*c*d^2*ln(ex p(2*I*(f*x+e))+1)*x^2+12/f^3*b*e^2*a*c*d^2*ln(exp(I*(f*x+e)))+6*I/f^2*b*d* c^2*a*e^2+3*I/f^2*b*d*c^2*a*polylog(2,-exp(2*I*(f*x+e)))+3*I/f^2*b*d^3*a*p olylog(2,-exp(2*I*(f*x+e)))*x^2+4*I/f^3*b*e^3*a*d^3*x-8*I/f^3*b*e^3*d^2...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (264) = 528\).
Time = 0.28 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.53 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\frac {{\left (a^{2} - b^{2}\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a^{2} - b^{2}\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a^{2} - b^{2}\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a^{2} - b^{2}\right )} c^{3} f^{4} x + 3 i \, a b d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 i \, a b d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 6 \, {\left (i \, a b d^{3} f^{2} x^{2} + i \, a b c^{2} d f^{2} - i \, b^{2} c d^{2} f + i \, {\left (2 \, a b c d^{2} f^{2} - b^{2} d^{3} f\right )} x\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \, {\left (-i \, a b d^{3} f^{2} x^{2} - i \, a b c^{2} d f^{2} + i \, b^{2} c d^{2} f - i \, {\left (2 \, a b c d^{2} f^{2} - b^{2} d^{3} f\right )} x\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 2 \, {\left (2 \, a b d^{3} f^{3} x^{3} + 2 \, a b c^{3} f^{3} - 3 \, b^{2} c^{2} d f^{2} + 3 \, {\left (2 \, a b c d^{2} f^{3} - b^{2} d^{3} f^{2}\right )} x^{2} + 6 \, {\left (a b c^{2} d f^{3} - b^{2} c d^{2} f^{2}\right )} x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (2 \, a b d^{3} f^{3} x^{3} + 2 \, a b c^{3} f^{3} - 3 \, b^{2} c^{2} d f^{2} + 3 \, {\left (2 \, a b c d^{2} f^{3} - b^{2} d^{3} f^{2}\right )} x^{2} + 6 \, {\left (a b c^{2} d f^{3} - b^{2} c d^{2} f^{2}\right )} x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a b d^{3} f x + 2 \, a b c d^{2} f - b^{2} d^{3}\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 \, {\left (2 \, a b d^{3} f x + 2 \, a b c d^{2} f - b^{2} d^{3}\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, {\left (b^{2} d^{3} f^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} x + b^{2} c^{3} f^{3}\right )} \tan \left (f x + e\right )}{4 \, f^{4}} \]
1/4*((a^2 - b^2)*d^3*f^4*x^4 + 4*(a^2 - b^2)*c*d^2*f^4*x^3 + 6*(a^2 - b^2) *c^2*d*f^4*x^2 + 4*(a^2 - b^2)*c^3*f^4*x + 3*I*a*b*d^3*polylog(4, (tan(f*x + e)^2 + 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 3*I*a*b*d^3*polylo g(4, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 6*(I* a*b*d^3*f^2*x^2 + I*a*b*c^2*d*f^2 - I*b^2*c*d^2*f + I*(2*a*b*c*d^2*f^2 - b ^2*d^3*f)*x)*dilog(2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1) + 1) - 6*(- I*a*b*d^3*f^2*x^2 - I*a*b*c^2*d*f^2 + I*b^2*c*d^2*f - I*(2*a*b*c*d^2*f^2 - b^2*d^3*f)*x)*dilog(2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1) + 1) - 2 *(2*a*b*d^3*f^3*x^3 + 2*a*b*c^3*f^3 - 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f^3 - b^2*d^3*f^2)*x^2 + 6*(a*b*c^2*d*f^3 - b^2*c*d^2*f^2)*x)*log(-2*(I*tan(f *x + e) - 1)/(tan(f*x + e)^2 + 1)) - 2*(2*a*b*d^3*f^3*x^3 + 2*a*b*c^3*f^3 - 3*b^2*c^2*d*f^2 + 3*(2*a*b*c*d^2*f^3 - b^2*d^3*f^2)*x^2 + 6*(a*b*c^2*d*f ^3 - b^2*c*d^2*f^2)*x)*log(-2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 3*(2*a*b*d^3*f*x + 2*a*b*c*d^2*f - b^2*d^3)*polylog(3, (tan(f*x + e)^2 + 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 3*(2*a*b*d^3*f*x + 2*a*b*c* d^2*f - b^2*d^3)*polylog(3, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f *x + e)^2 + 1)) + 4*(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*f^3*x^2 + 3*b^2*c^2*d*f ^3*x + b^2*c^3*f^3)*tan(f*x + e))/f^4
\[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{3}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2525 vs. \(2 (264) = 528\).
Time = 1.73 (sec) , antiderivative size = 2525, normalized size of antiderivative = 8.42 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\text {Too large to display} \]
1/4*(4*(f*x + e)*a^2*c^3 + (f*x + e)^4*a^2*d^3/f^3 - 4*(f*x + e)^3*a^2*d^3 *e/f^3 + 6*(f*x + e)^2*a^2*d^3*e^2/f^3 - 4*(f*x + e)*a^2*d^3*e^3/f^3 + 4*( f*x + e)^3*a^2*c*d^2/f^2 - 12*(f*x + e)^2*a^2*c*d^2*e/f^2 + 12*(f*x + e)*a ^2*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^2*c^2*d/f - 12*(f*x + e)*a^2*c^2*d*e/f + 8*a*b*c^3*log(sec(f*x + e)) - 8*a*b*d^3*e^3*log(sec(f*x + e))/f^3 + 24*a *b*c*d^2*e^2*log(sec(f*x + e))/f^2 - 24*a*b*c^2*d*e*log(sec(f*x + e))/f + 4*(3*(2*a*b + I*b^2)*(f*x + e)^4*d^3 - 24*b^2*d^3*e^3 + 72*b^2*c*d^2*e^2*f - 72*b^2*c^2*d*e*f^2 + 24*b^2*c^3*f^3 - 12*((2*a*b + I*b^2)*d^3*e - (2*a* b + I*b^2)*c*d^2*f)*(f*x + e)^3 + 18*((2*a*b + I*b^2)*d^3*e^2 - 2*(2*a*b + I*b^2)*c*d^2*e*f + (2*a*b + I*b^2)*c^2*d*f^2)*(f*x + e)^2 + 12*(-I*b^2*d^ 3*e^3 + 3*I*b^2*c*d^2*e^2*f - 3*I*b^2*c^2*d*e*f^2 + I*b^2*c^3*f^3)*(f*x + e) - 4*(8*(f*x + e)^3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*c*d^2*e*f - 9*b^2*c ^2*d*f^2 - 9*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 18*(a*b *d^3*e^2 + a*b*c^2*d*f^2 + b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f*x + e) + (8*(f*x + e)^3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*c*d^2*e*f - 9*b^2* c^2*d*f^2 - 9*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 18*(a* b*d^3*e^2 + a*b*c^2*d*f^2 + b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f* x + e))*cos(2*f*x + 2*e) - (-8*I*(f*x + e)^3*a*b*d^3 + 9*I*b^2*d^3*e^2 - 1 8*I*b^2*c*d^2*e*f + 9*I*b^2*c^2*d*f^2 + 9*(2*I*a*b*d^3*e - 2*I*a*b*c*d^2*f + I*b^2*d^3)*(f*x + e)^2 + 18*(-I*a*b*d^3*e^2 - I*a*b*c^2*d*f^2 - I*b^...
\[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \]