\(\int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx\) [216]

Optimal result

Integrand size = 20, antiderivative size = 275 \[ \int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx=-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {6 d^3 \cos (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4}+\frac {6 d^2 (c+d x) \sin (a+b x)}{b^3}-\frac {(c+d x)^3 \sin (a+b x)}{b} \]

[Out]

-2*I*(d*x+c)^3*arctan(exp(I*(b*x+a)))/b+6*d^3*cos(b*x+a)/b^4-3*d*(d*x+c)^2*cos(b*x+a)/b^2+3*I*d*(d*x+c)^2*poly
log(2,-I*exp(I*(b*x+a)))/b^2-3*I*d*(d*x+c)^2*polylog(2,I*exp(I*(b*x+a)))/b^2-6*d^2*(d*x+c)*polylog(3,-I*exp(I*
(b*x+a)))/b^3+6*d^2*(d*x+c)*polylog(3,I*exp(I*(b*x+a)))/b^3-6*I*d^3*polylog(4,-I*exp(I*(b*x+a)))/b^4+6*I*d^3*p
olylog(4,I*exp(I*(b*x+a)))/b^4+6*d^2*(d*x+c)*sin(b*x+a)/b^3-(d*x+c)^3*sin(b*x+a)/b

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4492, 3377, 2718, 4266, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx=-\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \cos (a+b x)}{b^4}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \sin (a+b x)}{b^3}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {3 d (c+d x)^2 \cos (a+b x)}{b^2}-\frac {(c+d x)^3 \sin (a+b x)}{b} \]

[In]

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

[Out]

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4492

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]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6744

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x)^3 \cos (a+b x) \, dx+\int (c+d x)^3 \sec (a+b x) \, dx \\ & = -\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {(3 d) \int (c+d x)^2 \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x)^2 \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}+\frac {(3 d) \int (c+d x)^2 \sin (a+b x) \, dx}{b} \\ & = -\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {\left (6 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \cos (a+b x) \, dx}{b^2} \\ & = -\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}-\frac {3 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \sin (a+b x)}{b^3}-\frac {(c+d x)^3 \sin (a+b x)}{b}+\frac {\left (6 d^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (6 d^3\right ) \int \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (6 d^3\right ) \int \sin (a+b x) \, dx}{b^3} \\ & = -\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {6 d^3 \cos (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \sin (a+b x)}{b^3}-\frac {(c+d x)^3 \sin (a+b x)}{b}-\frac {\left (6 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}+\frac {\left (6 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4} \\ & = -\frac {2 i (c+d x)^3 \arctan \left (e^{i (a+b x)}\right )}{b}+\frac {6 d^3 \cos (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \cos (a+b x)}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^3}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )}{b^4}+\frac {6 d^2 (c+d x) \sin (a+b x)}{b^3}-\frac {(c+d x)^3 \sin (a+b x)}{b} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(557\) vs. \(2(275)=550\).

Time = 1.57 (sec) , antiderivative size = 557, normalized size of antiderivative = 2.03 \[ \int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx=-\frac {2 i b^3 c^3 \arctan \left (e^{i (a+b x)}\right )+3 b^2 c^2 d \cos (a+b x)-6 d^3 \cos (a+b x)+6 b^2 c d^2 x \cos (a+b x)+3 b^2 d^3 x^2 \cos (a+b x)-3 b^3 c^2 d x \log \left (1-i e^{i (a+b x)}\right )-3 b^3 c d^2 x^2 \log \left (1-i e^{i (a+b x)}\right )-b^3 d^3 x^3 \log \left (1-i e^{i (a+b x)}\right )+3 b^3 c^2 d x \log \left (1+i e^{i (a+b x)}\right )+3 b^3 c d^2 x^2 \log \left (1+i e^{i (a+b x)}\right )+b^3 d^3 x^3 \log \left (1+i e^{i (a+b x)}\right )-3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+3 i b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )+6 i d^3 \operatorname {PolyLog}\left (4,-i e^{i (a+b x)}\right )-6 i d^3 \operatorname {PolyLog}\left (4,i e^{i (a+b x)}\right )+b^3 c^3 \sin (a+b x)-6 b c d^2 \sin (a+b x)+3 b^3 c^2 d x \sin (a+b x)-6 b d^3 x \sin (a+b x)+3 b^3 c d^2 x^2 \sin (a+b x)+b^3 d^3 x^3 \sin (a+b x)}{b^4} \]

[In]

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

[Out]

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (250 ) = 500\).

Time = 2.51 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.28

method	result	size
risch	\(\text {Expression too large to display}\)	\(901\)

[In]

int((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

6/b^3*d^2*c*polylog(3,I*exp(I*(b*x+a)))-1/b^4*a^3*d^3*ln(1+I*exp(I*(b*x+a)))+6/b^3*d^3*polylog(3,I*exp(I*(b*x+
a)))*x-6/b^3*d^2*c*polylog(3,-I*exp(I*(b*x+a)))-1/b*d^3*ln(1+I*exp(I*(b*x+a)))*x^3+1/b*d^3*ln(1-I*exp(I*(b*x+a
)))*x^3+1/b^4*a^3*d^3*ln(1-I*exp(I*(b*x+a)))-6/b^3*d^3*polylog(3,-I*exp(I*(b*x+a)))*x-2*I/b*c^3*arctan(exp(I*(
b*x+a)))-6*I*d^3*polylog(4,-I*exp(I*(b*x+a)))/b^4+6*I*d^3*polylog(4,I*exp(I*(b*x+a)))/b^4-6*I/b^3*c*d^2*a^2*ar
ctan(exp(I*(b*x+a)))+6*I/b^2*c^2*d*a*arctan(exp(I*(b*x+a)))-6*I/b^2*d^2*c*polylog(2,I*exp(I*(b*x+a)))*x+6*I/b^
2*d^2*c*polylog(2,-I*exp(I*(b*x+a)))*x-1/2*I*(d^3*x^3*b^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3-3*I*b^2*d^3*x^
2-6*b*d^3*x-6*I*b^2*c*d^2*x-6*c*d^2*b-3*I*b^2*c^2*d+6*I*d^3)/b^4*exp(-I*(b*x+a))+1/2*I*(d^3*x^3*b^3+3*b^3*c*d^
2*x^2+3*b^3*c^2*d*x+b^3*c^3+3*I*b^2*d^3*x^2-6*b*d^3*x+6*I*b^2*c*d^2*x-6*c*d^2*b+3*I*b^2*c^2*d-6*I*d^3)/b^4*exp
(I*(b*x+a))+3*I/b^2*c^2*d*polylog(2,-I*exp(I*(b*x+a)))-3*I/b^2*c^2*d*polylog(2,I*exp(I*(b*x+a)))-3/b*c^2*d*ln(
1+I*exp(I*(b*x+a)))*x-3/b^2*c^2*d*ln(1+I*exp(I*(b*x+a)))*a+3/b*c^2*d*ln(1-I*exp(I*(b*x+a)))*x+3/b^2*c^2*d*ln(1
-I*exp(I*(b*x+a)))*a+3/b^3*a^2*c*d^2*ln(1+I*exp(I*(b*x+a)))-3/b^3*a^2*c*d^2*ln(1-I*exp(I*(b*x+a)))+3/b*d^2*c*l
n(1-I*exp(I*(b*x+a)))*x^2-3/b*d^2*c*ln(1+I*exp(I*(b*x+a)))*x^2+2*I/b^4*d^3*a^3*arctan(exp(I*(b*x+a)))-3*I/b^2*
d^3*polylog(2,I*exp(I*(b*x+a)))*x^2+3*I/b^2*d^3*polylog(2,-I*exp(I*(b*x+a)))*x^2

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. 1075 vs. \(2 (237) = 474\).

Time = 0.31 (sec) , antiderivative size = 1075, normalized size of antiderivative = 3.91 \[ \int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

1/2*(6*I*d^3*polylog(4, I*cos(b*x + a) + sin(b*x + a)) + 6*I*d^3*polylog(4, I*cos(b*x + a) - sin(b*x + a)) - 6
*I*d^3*polylog(4, -I*cos(b*x + a) + sin(b*x + a)) - 6*I*d^3*polylog(4, -I*cos(b*x + a) - sin(b*x + a)) - 6*(b^
2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d - 2*d^3)*cos(b*x + a) - 3*(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)) - 3*(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)) - 3*(-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)) -
3*(-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)) + (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) - (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) + (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) - (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) + (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) - (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*c
os(b*x + a) - sin(b*x + a) + 1) + (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*si
n(b*x + a) + I) - (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)
- 6*(b*d^3*x + b*c*d^2)*polylog(3, I*cos(b*x + a) + sin(b*x + a)) + 6*(b*d^3*x + b*c*d^2)*polylog(3, I*cos(b*x
+ a) - sin(b*x + a)) - 6*(b*d^3*x + b*c*d^2)*polylog(3, -I*cos(b*x + a) + sin(b*x + a)) + 6*(b*d^3*x + b*c*d^
2)*polylog(3, -I*cos(b*x + a) - sin(b*x + a)) - 2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + b^3*c^3 - 6*b*c*d^2 + 3*(b^
3*c^2*d - 2*b*d^3)*x)*sin(b*x + a))/b^4

Sympy [F]

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

[In]

integrate((d*x+c)**3*sec(b*x+a)*sin(b*x+a)**2,x)

[Out]

Integral((c + d*x)**3*sin(a + b*x)**2*sec(a + b*x), x)

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. 934 vs. \(2 (237) = 474\).

Time = 0.52 (sec) , antiderivative size = 934, normalized size of antiderivative = 3.40 \[ \int (c+d x)^3 \sin (a+b x) \tan (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

1/2*(c^3*(log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1) - 2*sin(b*x + a)) - 3*a*c^2*d*(log(sin(b*x + a) + 1) -
log(sin(b*x + a) - 1) - 2*sin(b*x + a))/b + 3*a^2*c*d^2*(log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1) - 2*si
n(b*x + a))/b^2 - a^3*d^3*(log(sin(b*x + a) + 1) - log(sin(b*x + a) - 1) - 2*sin(b*x + a))/b^3 + (12*I*d^3*pol
ylog(4, I*e^(I*b*x + I*a)) - 12*I*d^3*polylog(4, -I*e^(I*b*x + I*a)) - 2*(I*(b*x + a)^3*d^3 + 3*(I*b*c*d^2 - I
*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2*d^3)*(b*x + a))*arctan2(cos(b*x + a), sin(b*x + a
) + 1) - 2*(I*(b*x + a)^3*d^3 + 3*(I*b*c*d^2 - I*a*d^3)*(b*x + a)^2 + 3*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*a^2*d
^3)*(b*x + a))*arctan2(cos(b*x + a), -sin(b*x + a) + 1) - 6*(b^2*c^2*d - 2*a*b*c*d^2 + (b*x + a)^2*d^3 + (a^2
- 2)*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(b*x + a) - 6*(I*b^2*c^2*d - 2*I*a*b*c*d^2 + I*(b*x + a)^2*d^3 +
I*a^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*dilog(I*e^(I*b*x + I*a)) - 6*(-I*b^2*c^2*d + 2*I*a*b*c*d^2 - I*
(b*x + a)^2*d^3 - I*a^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*dilog(-I*e^(I*b*x + I*a)) + ((b*x + a)^3*d^3
+ 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*(b*x + a))*log(cos(b*x + a)^2 + sin
(b*x + a)^2 + 2*sin(b*x + a) + 1) - ((b*x + a)^3*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*
c*d^2 + a^2*d^3)*(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) + 12*(b*c*d^2 + (b*x + a
)*d^3 - a*d^3)*polylog(3, I*e^(I*b*x + I*a)) - 12*(b*c*d^2 + (b*x + a)*d^3 - a*d^3)*polylog(3, -I*e^(I*b*x + I
*a)) - 2*((b*x + a)^3*d^3 - 6*b*c*d^2 + 6*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2 + 3*(b^2*c^2*d - 2*a*b*c*d^2
+ (a^2 - 2)*d^3)*(b*x + a))*sin(b*x + a))/b^3)/b

Giac [F]

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

[In]

integrate((d*x+c)^3*sec(b*x+a)*sin(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*sec(b*x + a)*sin(b*x + a)^2, x)

Mupad [F(-1)]

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

[In]

int((sin(a + b*x)^2*(c + d*x)^3)/cos(a + b*x),x)

[Out]

int((sin(a + b*x)^2*(c + d*x)^3)/cos(a + b*x), x)