3.3.22 \(\int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx\) [222]

3.3.22.1 Optimal result

Integrand size = 22, antiderivative size = 251 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=-\frac {3 d^3 x}{8 b^3}+\frac {(c+d x)^3}{4 b}+\frac {i (c+d x)^4}{4 d}-\frac {(c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac {3 i d^3 \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^4}+\frac {3 d^3 \cos (a+b x) \sin (a+b x)}{8 b^4}-\frac {3 d (c+d x)^2 \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac {3 d^2 (c+d x) \sin ^2(a+b x)}{4 b^3}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b} \]

output

-3/8*d^3*x/b^3+1/4*(d*x+c)^3/b+1/4*I*(d*x+c)^4/d-(d*x+c)^3*ln(1+exp(2*I*(b 
*x+a)))/b+3/2*I*d*(d*x+c)^2*polylog(2,-exp(2*I*(b*x+a)))/b^2-3/2*d^2*(d*x+ 
c)*polylog(3,-exp(2*I*(b*x+a)))/b^3-3/4*I*d^3*polylog(4,-exp(2*I*(b*x+a))) 
/b^4+3/8*d^3*cos(b*x+a)*sin(b*x+a)/b^4-3/4*d*(d*x+c)^2*cos(b*x+a)*sin(b*x+ 
a)/b^2+3/4*d^2*(d*x+c)*sin(b*x+a)^2/b^3-1/2*(d*x+c)^3*sin(b*x+a)^2/b
 

3.3.22.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1731\) vs. \(2(251)=502\).

Time = 6.53 (sec) , antiderivative size = 1731, normalized size of antiderivative = 6.90 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx =\text {Too large to display} \]

input

Integrate[(c + d*x)^3*Sin[a + b*x]^2*Tan[a + b*x],x]
 

output

((-1/4*I)*c*d^2*(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2 
*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I)*(a + b*x) 
)] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b 
^3*E^(I*a)) - ((I/8)*d^3*E^(I*a)*((2*b^4*x^4)/E^((2*I)*a) - (4*I)*b^3*(1 + 
 E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))] + 6*b^2*(1 + E^((-2*I)*a) 
)*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (6*I)*b*(1 + E^((-2*I)*a))*x*Pol 
yLog[3, -E^((-2*I)*(a + b*x))] - 3*(1 + E^((-2*I)*a))*PolyLog[4, -E^((-2*I 
)*(a + b*x))])*Sec[a])/b^4 - (c^3*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b*x] - Sin 
[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (3*c^2*d*Csc[a]*( 
(b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - 
 Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b 
*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - 
 ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[ 
1 + Cot[a]^2])*Sec[a])/(2*b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) + Sec[ 
a]*(Cos[2*a + 2*b*x]/(64*b^4) - ((I/64)*Sin[2*a + 2*b*x])/b^4)*(8*b^3*c^3* 
Cos[a] - (12*I)*b^2*c^2*d*Cos[a] - 12*b*c*d^2*Cos[a] + (6*I)*d^3*Cos[a] + 
24*b^3*c^2*d*x*Cos[a] - (24*I)*b^2*c*d^2*x*Cos[a] - 12*b*d^3*x*Cos[a] + 24 
*b^3*c*d^2*x^2*Cos[a] - (12*I)*b^2*d^3*x^2*Cos[a] + 8*b^3*d^3*x^3*Cos[a] + 
 (32*I)*b^4*c^3*x*Cos[a + 2*b*x] + (48*I)*b^4*c^2*d*x^2*Cos[a + 2*b*x] + ( 
32*I)*b^4*c*d^2*x^3*Cos[a + 2*b*x] + (8*I)*b^4*d^3*x^4*Cos[a + 2*b*x] -...
 

3.3.22.3 Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {4907, 3042, 4202, 2620, 3011, 4904, 3042, 3792, 17, 3042, 3115, 24, 7163, 2720, 7143}

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.

\(\displaystyle \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx\)

\(\Big \downarrow \) 4907

\(\displaystyle \int (c+d x)^3 \tan (a+b x)dx-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^3 \tan (a+b x)dx-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx\)

\(\Big \downarrow \) 4202

\(\displaystyle -2 i \int \frac {e^{2 i (a+b x)} (c+d x)^3}{1+e^{2 i (a+b x)}}dx-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2620

\(\displaystyle -2 i \left (\frac {3 i d \int (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3011

\(\displaystyle -2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\int (c+d x)^3 \cos (a+b x) \sin (a+b x)dx+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 4904

\(\displaystyle -2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )+\frac {3 d \int (c+d x)^2 \sin ^2(a+b x)dx}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )+\frac {3 d \int (c+d x)^2 \sin (a+b x)^2dx}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^2dx+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \sin ^2(a+b x)dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {3 d \left (-\frac {d^2 \int \sin (a+b x)^2dx}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {3 d \left (-\frac {d^2 \left (\frac {\int 1dx}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 24

\(\displaystyle -2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )dx}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )+\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 7163

\(\displaystyle -2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {i d \int \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )dx}{2 b}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )+\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 2720

\(\displaystyle -2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \int e^{-2 i (a+b x)} \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )de^{2 i (a+b x)}}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )+\frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {3 d \left (\frac {d (c+d x) \sin ^2(a+b x)}{2 b^2}-\frac {d^2 \left (\frac {x}{2}-\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}-\frac {(c+d x)^2 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^3}{6 d}\right )}{2 b}-2 i \left (\frac {3 i d \left (\frac {i (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b}-\frac {i d \left (\frac {d \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{4 b^2}-\frac {i (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b}\right )}{b}\right )}{2 b}-\frac {i (c+d x)^3 \log \left (1+e^{2 i (a+b x)}\right )}{2 b}\right )-\frac {(c+d x)^3 \sin ^2(a+b x)}{2 b}+\frac {i (c+d x)^4}{4 d}\)

input

Int[(c + d*x)^3*Sin[a + b*x]^2*Tan[a + b*x],x]
 

output

((I/4)*(c + d*x)^4)/d - (2*I)*(((-1/2*I)*(c + d*x)^3*Log[1 + E^((2*I)*(a + 
 b*x))])/b + (((3*I)/2)*d*(((I/2)*(c + d*x)^2*PolyLog[2, -E^((2*I)*(a + b* 
x))])/b - (I*d*(((-1/2*I)*(c + d*x)*PolyLog[3, -E^((2*I)*(a + b*x))])/b + 
(d*PolyLog[4, -E^((2*I)*(a + b*x))])/(4*b^2)))/b))/b) - ((c + d*x)^3*Sin[a 
 + b*x]^2)/(2*b) + (3*d*((c + d*x)^3/(6*d) - ((c + d*x)^2*Cos[a + b*x]*Sin 
[a + b*x])/(2*b) + (d*(c + d*x)*Sin[a + b*x]^2)/(2*b^2) - (d^2*(x/2 - (Cos 
[a + b*x]*Sin[a + b*x])/(2*b)))/(2*b^2)))/(2*b)
 

3.3.22.3.1 Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 4904

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x 
_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) 
, x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 4907

Int[((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b 
_.)*(x_)]^(p_.), x_Symbol] :> -Int[(c + d*x)^m*Sin[a + b*x]^n*Tan[a + b*x]^ 
(p - 2), x] + Int[(c + d*x)^m*Sin[a + b*x]^(n - 2)*Tan[a + b*x]^p, x] /; Fr 
eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.3.22.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (220 ) = 440\).

Time = 3.11 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.59

method	result	size
risch	\(\frac {6 i d \,c^{2} x a}{b}+\frac {3 i c \,d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {6 i c \,d^{2} a^{2} x}{b^{2}}-i c^{3} x -\frac {i c^{4}}{4 d}-\frac {c^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b}+\frac {2 c^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b}+\frac {6 c \,d^{2} a^{2} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {6 c^{2} d a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}-\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b}-\frac {3 c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{2}}{b}+\frac {3 i d \,c^{2} a^{2}}{b^{2}}+\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x^{2}}{2 b^{2}}-\frac {4 i c \,d^{2} a^{3}}{b^{3}}+\frac {3 i d \,c^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{2}}+\frac {2 i d^{3} a^{3} x}{b^{3}}+i d^{2} c \,x^{3}+\frac {3 i d \,c^{2} x^{2}}{2}-\frac {3 c \,d^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{3}}-\frac {d^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x^{3}}{b}-\frac {3 d^{3} \operatorname {polylog}\left (3, -{\mathrm e}^{2 i \left (x b +a \right )}\right ) x}{2 b^{3}}-\frac {3 i d^{3} \operatorname {polylog}\left (4, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{4 b^{4}}-\frac {2 d^{3} a^{3} \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {3 i d^{3} a^{4}}{2 b^{4}}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}-6 i b^{2} d^{3} x^{2}+12 b^{3} c^{2} d x -12 i b^{2} c \,d^{2} x +4 b^{3} c^{3}-6 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b +3 i d^{3}\right ) {\mathrm e}^{-2 i \left (x b +a \right )}}{32 b^{4}}+\frac {\left (4 d^{3} x^{3} b^{3}+12 b^{3} c \,d^{2} x^{2}+6 i b^{2} d^{3} x^{2}+12 b^{3} c^{2} d x +12 i b^{2} c \,d^{2} x +4 b^{3} c^{3}+6 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b -3 i d^{3}\right ) {\mathrm e}^{2 i \left (x b +a \right )}}{32 b^{4}}+\frac {i d^{3} x^{4}}{4}\)	\(650\)

input

int((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
 

output

-3/4*I*d^3*polylog(4,-exp(2*I*(b*x+a)))/b^4+3/2*I*d*c^2*x^2-1/b*c^3*ln(exp 
(2*I*(b*x+a))+1)-3/b*d*c^2*ln(exp(2*I*(b*x+a))+1)*x-3/b*c*d^2*ln(exp(2*I*( 
b*x+a))+1)*x^2+2*I/b^3*d^3*a^3*x+3/2*I/b^2*d^3*polylog(2,-exp(2*I*(b*x+a)) 
)*x^2+3/2*I/b^2*d*c^2*polylog(2,-exp(2*I*(b*x+a)))+3*I/b^2*d*c^2*a^2-4*I/b 
^3*c*d^2*a^3+1/4*I*d^3*x^4+6/b^3*c*d^2*a^2*ln(exp(I*(b*x+a)))-6/b^2*c^2*d* 
a*ln(exp(I*(b*x+a)))-1/b*d^3*ln(exp(2*I*(b*x+a))+1)*x^3-3/2/b^3*d^3*polylo 
g(3,-exp(2*I*(b*x+a)))*x-3/2/b^3*c*d^2*polylog(3,-exp(2*I*(b*x+a)))+3/2*I/ 
b^4*d^3*a^4-I*c^3*x-1/4*I/d*c^4+2/b*c^3*ln(exp(I*(b*x+a)))-2/b^4*d^3*a^3*l 
n(exp(I*(b*x+a)))-6*I/b^2*c*d^2*a^2*x+6*I/b*d*c^2*x*a+3*I/b^2*c*d^2*polylo 
g(2,-exp(2*I*(b*x+a)))*x+1/32*(4*d^3*x^3*b^3-6*I*b^2*d^3*x^2+12*b^3*c*d^2* 
x^2-12*I*b^2*c*d^2*x+12*b^3*c^2*d*x-6*I*b^2*c^2*d+4*b^3*c^3-6*b*d^3*x+3*I* 
d^3-6*c*d^2*b)/b^4*exp(-2*I*(b*x+a))+1/32*(4*d^3*x^3*b^3+6*I*b^2*d^3*x^2+1 
2*b^3*c*d^2*x^2+12*I*b^2*c*d^2*x+12*b^3*c^2*d*x+6*I*b^2*c^2*d+4*b^3*c^3-6* 
b*d^3*x-3*I*d^3-6*c*d^2*b)/b^4*exp(2*I*(b*x+a))+I*d^2*c*x^3
 

3.3.22.5 Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (216) = 432\).

Time = 0.32 (sec) , antiderivative size = 1134, normalized size of antiderivative = 4.52 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

input

integrate((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^3,x, algorithm="fricas")
 

output

-1/8*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 - 24*I*d^3*polylog(4, I*cos(b*x + a) 
 + sin(b*x + a)) + 24*I*d^3*polylog(4, I*cos(b*x + a) - sin(b*x + a)) + 24 
*I*d^3*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) - 24*I*d^3*polylog(4, -I 
*cos(b*x + a) - sin(b*x + a)) - 2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3 
*c^3 - 3*b*c*d^2 + 3*(2*b^3*c^2*d - b*d^3)*x)*cos(b*x + a)^2 + 3*(2*b^2*d^ 
3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d - d^3)*cos(b*x + a)*sin(b*x + a) + 3*( 
2*b^3*c^2*d - b*d^3)*x + 12*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d 
)*dilog(I*cos(b*x + a) + sin(b*x + a)) + 12*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^ 
2*x - I*b^2*c^2*d)*dilog(I*cos(b*x + a) - sin(b*x + a)) + 12*(-I*b^2*d^3*x 
^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(-I*cos(b*x + a) + sin(b*x + a)) 
+ 12*(I*b^2*d^3*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(-I*cos(b*x + a) 
 - sin(b*x + a)) + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*l 
og(cos(b*x + a) + I*sin(b*x + a) + I) + 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2 
*b*c*d^2 - a^3*d^3)*log(cos(b*x + a) - I*sin(b*x + a) + I) + 4*(b^3*d^3*x^ 
3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3* 
d^3)*log(I*cos(b*x + a) + sin(b*x + a) + 1) + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2 
*x^2 + 3*b^3*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*log(I*cos( 
b*x + a) - sin(b*x + a) + 1) + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^ 
2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*log(-I*cos(b*x + a) + sin 
(b*x + a) + 1) + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*a...
 

3.3.22.6 Sympy [F]

\[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{3} \sin ^{3}{\left (a + b x \right )} \sec {\left (a + b x \right )}\, dx \]

input

integrate((d*x+c)**3*sec(b*x+a)*sin(b*x+a)**3,x)
 

output

Integral((c + d*x)**3*sin(a + b*x)**3*sec(a + b*x), x)
 

3.3.22.7 Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (216) = 432\).

Time = 0.44 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.76 \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=-\frac {24 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} c^{3} - \frac {72 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a c^{2} d}{b} + \frac {72 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a^{2} c d^{2}}{b^{2}} - \frac {24 \, {\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a^{3} d^{3}}{b^{3}} + \frac {-12 i \, {\left (b x + a\right )}^{4} d^{3} - 48 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}^{3} + 48 i \, d^{3} {\rm Li}_{4}(-e^{\left (2 i \, b x + 2 i \, a\right )}) - 72 \, {\left (i \, b^{2} c^{2} d - 2 i \, a b c d^{2} + i \, a^{2} d^{3}\right )} {\left (b x + a\right )}^{2} - 16 \, {\left (-4 i \, {\left (b x + a\right )}^{3} d^{3} + 9 \, {\left (-i \, b c d^{2} + i \, a d^{3}\right )} {\left (b x + a\right )}^{2} + 9 \, {\left (-i \, b^{2} c^{2} d + 2 i \, a b c d^{2} - i \, a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (2 \, {\left (b x + a\right )}^{3} d^{3} - 3 \, b c d^{2} + 3 \, a d^{3} + 6 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 3 \, {\left (2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + {\left (2 \, a^{2} - 1\right )} d^{3}\right )} {\left (b x + a\right )}\right )} \cos \left (2 \, b x + 2 \, a\right ) - 24 \, {\left (3 i \, b^{2} c^{2} d - 6 i \, a b c d^{2} + 4 i \, {\left (b x + a\right )}^{2} d^{3} + 3 i \, a^{2} d^{3} + 6 \, {\left (i \, b c d^{2} - i \, a d^{3}\right )} {\left (b x + a\right )}\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 8 \, {\left (4 \, {\left (b x + a\right )}^{3} d^{3} + 9 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}^{2} + 9 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} {\left (b x + a\right )}\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 24 \, {\left (3 \, b c d^{2} + 4 \, {\left (b x + a\right )} d^{3} - 3 \, a d^{3}\right )} {\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) + 9 \, {\left (2 \, b^{2} c^{2} d - 4 \, a b c d^{2} + 2 \, {\left (b x + a\right )}^{2} d^{3} + {\left (2 \, a^{2} - 1\right )} d^{3} + 4 \, {\left (b c d^{2} - a d^{3}\right )} {\left (b x + a\right )}\right )} \sin \left (2 \, b x + 2 \, a\right )}{b^{3}}}{48 \, b} \]

input

integrate((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^3,x, algorithm="maxima")
 

output

-1/48*(24*(sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1))*c^3 - 72*(sin(b*x + a 
)^2 + log(sin(b*x + a)^2 - 1))*a*c^2*d/b + 72*(sin(b*x + a)^2 + log(sin(b* 
x + a)^2 - 1))*a^2*c*d^2/b^2 - 24*(sin(b*x + a)^2 + log(sin(b*x + a)^2 - 1 
))*a^3*d^3/b^3 + (-12*I*(b*x + a)^4*d^3 - 48*(I*b*c*d^2 - I*a*d^3)*(b*x + 
a)^3 + 48*I*d^3*polylog(4, -e^(2*I*b*x + 2*I*a)) - 72*(I*b^2*c^2*d - 2*I*a 
*b*c*d^2 + I*a^2*d^3)*(b*x + a)^2 - 16*(-4*I*(b*x + a)^3*d^3 + 9*(-I*b*c*d 
^2 + I*a*d^3)*(b*x + a)^2 + 9*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*a^2*d^3)*( 
b*x + a))*arctan2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 6*(2*(b*x + a) 
^3*d^3 - 3*b*c*d^2 + 3*a*d^3 + 6*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(2*b^2* 
c^2*d - 4*a*b*c*d^2 + (2*a^2 - 1)*d^3)*(b*x + a))*cos(2*b*x + 2*a) - 24*(3 
*I*b^2*c^2*d - 6*I*a*b*c*d^2 + 4*I*(b*x + a)^2*d^3 + 3*I*a^2*d^3 + 6*(I*b* 
c*d^2 - I*a*d^3)*(b*x + a))*dilog(-e^(2*I*b*x + 2*I*a)) + 8*(4*(b*x + a)^3 
*d^3 + 9*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 9*(b^2*c^2*d - 2*a*b*c*d^2 + a^2* 
d^3)*(b*x + a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x 
+ 2*a) + 1) + 24*(3*b*c*d^2 + 4*(b*x + a)*d^3 - 3*a*d^3)*polylog(3, -e^(2* 
I*b*x + 2*I*a)) + 9*(2*b^2*c^2*d - 4*a*b*c*d^2 + 2*(b*x + a)^2*d^3 + (2*a^ 
2 - 1)*d^3 + 4*(b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x + 2*a))/b^3)/b
 

3.3.22.8 Giac [F]

\[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{3} \,d x } \]

input

integrate((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^3,x, algorithm="giac")
 

output

integrate((d*x + c)^3*sec(b*x + a)*sin(b*x + a)^3, x)
 

3.3.22.9 Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 \sin ^2(a+b x) \tan (a+b x) \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^3}{\cos \left (a+b\,x\right )} \,d x \]

input

int((sin(a + b*x)^3*(c + d*x)^3)/cos(a + b*x),x)
 

output

int((sin(a + b*x)^3*(c + d*x)^3)/cos(a + b*x), x)