\(\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx\) [260]
Optimal result
Integrand size = 22, antiderivative size = 228 \[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^3 \sec (a+b x)}{b}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}
\]
[Out]
6*I*d*(d*x+c)^2*arctan(exp(I*(b*x+a)))/b^2-6*d^2*(d*x+c)*cos(b*x+a)/b^3+(d*x+c)^3*cos(b*x+a)/b-6*I*d^2*(d*x+c)
*polylog(2,-I*exp(I*(b*x+a)))/b^3+6*I*d^2*(d*x+c)*polylog(2,I*exp(I*(b*x+a)))/b^3+6*d^3*polylog(3,-I*exp(I*(b*
x+a)))/b^4-6*d^3*polylog(3,I*exp(I*(b*x+a)))/b^4+(d*x+c)^3*sec(b*x+a)/b+6*d^3*sin(b*x+a)/b^4-3*d*(d*x+c)^2*sin
(b*x+a)/b^2
Rubi [A] (verified)
Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of
steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4492, 3377, 2717, 4494, 4266,
2611, 2320, 6724} \[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {(c+d x)^3 \sec (a+b x)}{b}
\]
[In]
Int[(c + d*x)^3*Sin[a + b*x]*Tan[a + b*x]^2,x]
[Out]
((6*I)*d*(c + d*x)^2*ArcTan[E^(I*(a + b*x))])/b^2 - (6*d^2*(c + d*x)*Cos[a + b*x])/b^3 + ((c + d*x)^3*Cos[a +
b*x])/b - ((6*I)*d^2*(c + d*x)*PolyLog[2, (-I)*E^(I*(a + b*x))])/b^3 + ((6*I)*d^2*(c + d*x)*PolyLog[2, I*E^(I*
(a + b*x))])/b^3 + (6*d^3*PolyLog[3, (-I)*E^(I*(a + b*x))])/b^4 - (6*d^3*PolyLog[3, I*E^(I*(a + b*x))])/b^4 +
((c + d*x)^3*Sec[a + b*x])/b + (6*d^3*Sin[a + b*x])/b^4 - (3*d*(c + d*x)^2*Sin[a + b*x])/b^2
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 2717
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 4494
Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 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]
Rubi steps \begin{align*}
\text {integral}& = -\int (c+d x)^3 \sin (a+b x) \, dx+\int (c+d x)^3 \sec (a+b x) \tan (a+b x) \, dx \\ & = \frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {(c+d x)^3 \sec (a+b x)}{b}-\frac {(3 d) \int (c+d x)^2 \cos (a+b x) \, dx}{b}-\frac {(3 d) \int (c+d x)^2 \sec (a+b x) \, dx}{b} \\ & = \frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}+\frac {(c+d x)^3 \cos (a+b x)}{b}+\frac {(c+d x)^3 \sec (a+b x)}{b}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{b^2} \\ & = \frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \sec (a+b x)}{b}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {\left (6 i d^3\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (6 i d^3\right ) \int \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (6 d^3\right ) \int \cos (a+b x) \, dx}{b^3} \\ & = \frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \sec (a+b x)}{b}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2}+\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4}-\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^4} \\ & = \frac {6 i d (c+d x)^2 \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {6 d^2 (c+d x) \cos (a+b x)}{b^3}+\frac {(c+d x)^3 \cos (a+b x)}{b}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {6 d^3 \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )}{b^4}-\frac {6 d^3 \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )}{b^4}+\frac {(c+d x)^3 \sec (a+b x)}{b}+\frac {6 d^3 \sin (a+b x)}{b^4}-\frac {3 d (c+d x)^2 \sin (a+b x)}{b^2} \\
\end{align*}
Mathematica [B] (verified)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of
optimal. \(532\) vs. \(2(228)=456\).
Time = 1.68 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.33
\[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {\sec (a+b x) \left (3 b^3 c^3-6 b c d^2+9 b^3 c^2 d x-6 b d^3 x+9 b^3 c d^2 x^2+3 b^3 d^3 x^3+12 i b^2 c^2 d \arctan \left (e^{i (a+b x)}\right ) \cos (a+b x)+b^3 c^3 \cos (2 (a+b x))-6 b c d^2 \cos (2 (a+b x))+3 b^3 c^2 d x \cos (2 (a+b x))-6 b d^3 x \cos (2 (a+b x))+3 b^3 c d^2 x^2 \cos (2 (a+b x))+b^3 d^3 x^3 \cos (2 (a+b x))-12 b^2 c d^2 x \cos (a+b x) \log \left (1-i e^{i (a+b x)}\right )-6 b^2 d^3 x^2 \cos (a+b x) \log \left (1-i e^{i (a+b x)}\right )+12 b^2 c d^2 x \cos (a+b x) \log \left (1+i e^{i (a+b x)}\right )+6 b^2 d^3 x^2 \cos (a+b x) \log \left (1+i e^{i (a+b x)}\right )-12 i b d^2 (c+d x) \cos (a+b x) \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )+12 i b d^2 (c+d x) \cos (a+b x) \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )+12 d^3 \cos (a+b x) \operatorname {PolyLog}\left (3,-i e^{i (a+b x)}\right )-12 d^3 \cos (a+b x) \operatorname {PolyLog}\left (3,i e^{i (a+b x)}\right )-3 b^2 c^2 d \sin (2 (a+b x))+6 d^3 \sin (2 (a+b x))-6 b^2 c d^2 x \sin (2 (a+b x))-3 b^2 d^3 x^2 \sin (2 (a+b x))\right )}{2 b^4}
\]
[In]
Integrate[(c + d*x)^3*Sin[a + b*x]*Tan[a + b*x]^2,x]
[Out]
(Sec[a + b*x]*(3*b^3*c^3 - 6*b*c*d^2 + 9*b^3*c^2*d*x - 6*b*d^3*x + 9*b^3*c*d^2*x^2 + 3*b^3*d^3*x^3 + (12*I)*b^
2*c^2*d*ArcTan[E^(I*(a + b*x))]*Cos[a + b*x] + b^3*c^3*Cos[2*(a + b*x)] - 6*b*c*d^2*Cos[2*(a + b*x)] + 3*b^3*c
^2*d*x*Cos[2*(a + b*x)] - 6*b*d^3*x*Cos[2*(a + b*x)] + 3*b^3*c*d^2*x^2*Cos[2*(a + b*x)] + b^3*d^3*x^3*Cos[2*(a
+ b*x)] - 12*b^2*c*d^2*x*Cos[a + b*x]*Log[1 - I*E^(I*(a + b*x))] - 6*b^2*d^3*x^2*Cos[a + b*x]*Log[1 - I*E^(I*
(a + b*x))] + 12*b^2*c*d^2*x*Cos[a + b*x]*Log[1 + I*E^(I*(a + b*x))] + 6*b^2*d^3*x^2*Cos[a + b*x]*Log[1 + I*E^
(I*(a + b*x))] - (12*I)*b*d^2*(c + d*x)*Cos[a + b*x]*PolyLog[2, (-I)*E^(I*(a + b*x))] + (12*I)*b*d^2*(c + d*x)
*Cos[a + b*x]*PolyLog[2, I*E^(I*(a + b*x))] + 12*d^3*Cos[a + b*x]*PolyLog[3, (-I)*E^(I*(a + b*x))] - 12*d^3*Co
s[a + b*x]*PolyLog[3, I*E^(I*(a + b*x))] - 3*b^2*c^2*d*Sin[2*(a + b*x)] + 6*d^3*Sin[2*(a + b*x)] - 6*b^2*c*d^2
*x*Sin[2*(a + b*x)] - 3*b^2*d^3*x^2*Sin[2*(a + b*x)]))/(2*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. 676 vs. \(2 (211 ) = 422\).
Time = 3.57 (sec) , antiderivative size = 677, normalized size of antiderivative = 2.97
| | |
method | result | size |
| | |
risch |
\(\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 i b^{2} d^{3} x^{2}+3 b^{3} c^{2} d x +6 i b^{2} c \,d^{2} x +b^{3} c^{3}+3 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b -6 i d^{3}\right ) {\mathrm e}^{i \left (x b +a \right )}}{2 b^{4}}+\frac {\left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}-3 i b^{2} d^{3} x^{2}+3 b^{3} c^{2} d x -6 i b^{2} c \,d^{2} x +b^{3} c^{3}-3 i b^{2} c^{2} d -6 b \,d^{3} x -6 c \,d^{2} b +6 i d^{3}\right ) {\mathrm e}^{-i \left (x b +a \right )}}{2 b^{4}}+\frac {2 \,{\mathrm e}^{i \left (x b +a \right )} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}+\frac {6 d^{2} c \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {6 d^{3} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 i d^{3} a^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 i c \,d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {6 i c \,d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {6 i d^{3} x \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {6 i d^{3} x \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {12 i d^{2} c a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {3 d^{3} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{2}}+\frac {3 d^{3} a^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}+\frac {6 i d \,c^{2} \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {6 d^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {6 d^{2} c \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}-\frac {3 d^{3} a^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}-\frac {3 d^{3} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x^{2}}{b^{2}}-\frac {6 d^{2} c \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}\) |
\(677\) |
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[In]
int((d*x+c)^3*sin(b*x+a)*tan(b*x+a)^2,x,method=_RETURNVERBOSE)
[Out]
1/2*(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*(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))+2*exp(I*(b*x+a))*(d^3*x^3+3*c*d
^2*x^2+3*c^2*d*x+c^3)/b/(exp(2*I*(b*x+a))+1)+6/b^2*d^2*c*ln(1+I*exp(I*(b*x+a)))*x-6*d^3*polylog(3,I*exp(I*(b*x
+a)))/b^4+6*I/b^4*d^3*a^2*arctan(exp(I*(b*x+a)))+6*I*c*d^2*polylog(2,I*exp(I*(b*x+a)))/b^3-6*I*c*d^2*polylog(2
,-I*exp(I*(b*x+a)))/b^3-6*I*d^3*x*polylog(2,-I*exp(I*(b*x+a)))/b^3+6*I*d^3*x*polylog(2,I*exp(I*(b*x+a)))/b^3-1
2*I/b^3*d^2*c*a*arctan(exp(I*(b*x+a)))+3/b^2*d^3*ln(1+I*exp(I*(b*x+a)))*x^2+3/b^4*d^3*a^2*ln(1-I*exp(I*(b*x+a)
))+6*I/b^2*d*c^2*arctan(exp(I*(b*x+a)))+6*d^3*polylog(3,-I*exp(I*(b*x+a)))/b^4-6/b^2*d^2*c*ln(1-I*exp(I*(b*x+a
)))*x+6/b^3*d^2*c*ln(1+I*exp(I*(b*x+a)))*a-3/b^4*d^3*a^2*ln(1+I*exp(I*(b*x+a)))-3/b^2*d^3*ln(1-I*exp(I*(b*x+a)
))*x^2-6/b^3*d^2*c*ln(1-I*exp(I*(b*x+a)))*a
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. 896 vs. \(2 (202) = 404\).
Time = 0.31 (sec) , antiderivative size = 896, normalized size of antiderivative = 3.93
\[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\text {Too large to display}
\]
[In]
integrate((d*x+c)^3*sin(b*x+a)*tan(b*x+a)^2,x, algorithm="fricas")
[Out]
1/2*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 6*b^3*c^2*d*x + 2*b^3*c^3 + 6*d^3*cos(b*x + a)*polylog(3, I*cos(b*x + a
) + sin(b*x + a)) - 6*d^3*cos(b*x + a)*polylog(3, I*cos(b*x + a) - sin(b*x + a)) + 6*d^3*cos(b*x + a)*polylog(
3, -I*cos(b*x + a) + sin(b*x + a)) - 6*d^3*cos(b*x + a)*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)*cos(b*x + a)^2 - 6*(-I*b*d^3*x - I*
b*c*d^2)*cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + a)) - 6*(-I*b*d^3*x - I*b*c*d^2)*cos(b*x + a)*dilog(I*c
os(b*x + a) - sin(b*x + a)) - 6*(I*b*d^3*x + I*b*c*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a) + sin(b*x + a)) - 6
*(I*b*d^3*x + I*b*c*d^2)*cos(b*x + a)*dilog(-I*cos(b*x + a) - sin(b*x + a)) - 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2
*d^3)*cos(b*x + a)*log(cos(b*x + a) + I*sin(b*x + a) + I) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)
*log(cos(b*x + a) - I*sin(b*x + a) + I) - 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)
*log(I*cos(b*x + a) + sin(b*x + a) + 1) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)
*log(I*cos(b*x + a) - sin(b*x + a) + 1) - 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a)
*log(-I*cos(b*x + a) + sin(b*x + a) + 1) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cos(b*x + a
)*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log(-cos(b*x +
a) + I*sin(b*x + a) + I) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cos(b*x + a)*log(-cos(b*x + a) - I*sin(b*x +
a) + I) - 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)*sin(b*x + a))/(b^4*cos(b*x + a))
Sympy [F]
\[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\int \left (c + d x\right )^{3} \sin {\left (a + b x \right )} \tan ^{2}{\left (a + b x \right )}\, dx
\]
[In]
integrate((d*x+c)**3*sin(b*x+a)*tan(b*x+a)**2,x)
[Out]
Integral((c + d*x)**3*sin(a + b*x)*tan(a + b*x)**2, 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. 11010 vs. \(2 (202) = 404\).
Time = 1.31 (sec) , antiderivative size = 11010, normalized size of antiderivative = 48.29
\[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\text {Too large to display}
\]
[In]
integrate((d*x+c)^3*sin(b*x+a)*tan(b*x+a)^2,x, algorithm="maxima")
[Out]
1/2*(2*c^3*(1/cos(b*x + a) + cos(b*x + a)) - 6*a*c^2*d*(1/cos(b*x + a) + cos(b*x + a))/b + 6*a^2*c*d^2*(1/cos(
b*x + a) + cos(b*x + a))/b^2 - 2*a^3*d^3*(1/cos(b*x + a) + cos(b*x + a))/b^3 + 3*((b*x + (b*x + a)*cos(2*b*x +
2*a) + a + sin(2*b*x + 2*a))*cos(3*b*x + 3*a)^3 + 6*(b*x + a)*cos(b*x + a)^3 + ((b*x + a)*sin(2*b*x + 2*a) -
cos(2*b*x + 2*a) - 1)*sin(3*b*x + 3*a)^3 + 6*(b*x + a)*cos(b*x + a)*sin(b*x + a)^2 + 2*(4*(b*x + a)*cos(2*b*x
+ 2*a)*cos(b*x + a) + 4*(b*x + a)*cos(b*x + a) + (3*(b*x + a)*sin(b*x + a) + cos(b*x + a))*sin(2*b*x + 2*a))*c
os(3*b*x + 3*a)^2 + ((b*x + a)*cos(b*x + a) - sin(b*x + a))*cos(2*b*x + 2*a)^2 + (8*(b*x + a)*sin(2*b*x + 2*a)
*sin(b*x + a) + (b*x + (b*x + a)*cos(2*b*x + 2*a) + a + sin(2*b*x + 2*a))*cos(3*b*x + 3*a) + 2*(3*(b*x + a)*co
s(b*x + a) - sin(b*x + a))*cos(2*b*x + 2*a) + 6*(b*x + a)*cos(b*x + a) - 2*sin(b*x + a))*sin(3*b*x + 3*a)^2 +
((b*x + a)*cos(b*x + a) - sin(b*x + a))*sin(2*b*x + 2*a)^2 + ((b*x + a)*cos(2*b*x + 2*a)^2 + 13*(b*x + a)*cos(
b*x + a)^2 + (b*x + a)*sin(2*b*x + 2*a)^2 + (b*x + a)*sin(b*x + a)^2 + b*x + (13*(b*x + a)*cos(b*x + a)^2 + (b
*x + a)*sin(b*x + a)^2 + 2*b*x + 2*a)*cos(2*b*x + 2*a) + (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) + cos(b*x + a
)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a) + a)*cos(3*b*x + 3*a) + 2*(3*(b*x + a)*cos(b*x + a)^3 + 3*(b*x + a)*cos
(b*x + a)*sin(b*x + a)^2 + (b*x + a)*cos(b*x + a) - sin(b*x + a))*cos(2*b*x + 2*a) + (b*x + a)*cos(b*x + a) -
((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin
(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x
+ 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*
x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x + a)
^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)
^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*log(cos(b
*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + ((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2
*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin
(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2
*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a)*cos(b*x + a)
+ cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 + 2*
(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x
+ a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) + (((b*x +
a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*cos(3*b*x + 3*a)^2 + 12*(b*x + a)*cos(b*x + a)*sin(b*x + a) + 2*(
((b*x + a)*sin(b*x + a) - cos(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)*cos(b*x + a) + sin(b*x + a))*sin(2*b*x +
2*a) + (b*x + a)*sin(b*x + a) - cos(b*x + a))*cos(3*b*x + 3*a) + (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) - co
s(b*x + a)^2 - sin(b*x + a)^2 - 2)*cos(2*b*x + 2*a) - cos(2*b*x + 2*a)^2 - cos(b*x + a)^2 + ((b*x + a)*cos(b*x
+ a)^2 + 13*(b*x + a)*sin(b*x + a)^2)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2 - sin(b*x + a)^2 - 1)*sin(3*b*x +
3*a) + 6*((b*x + a)*cos(b*x + a)^2*sin(b*x + a) + (b*x + a)*sin(b*x + a)^3)*sin(2*b*x + 2*a) - sin(b*x + a))*
c^2*d/(((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^
2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*si
n(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) +
cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b
*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x
+ 2*a)^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*b)
- 6*((b*x + (b*x + a)*cos(2*b*x + 2*a) + a + sin(2*b*x + 2*a))*cos(3*b*x + 3*a)^3 + 6*(b*x + a)*cos(b*x + a)^
3 + ((b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(3*b*x + 3*a)^3 + 6*(b*x + a)*cos(b*x + a)*sin(b*x
+ a)^2 + 2*(4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a) + 4*(b*x + a)*cos(b*x + a) + (3*(b*x + a)*sin(b*x + a) +
cos(b*x + a))*sin(2*b*x + 2*a))*cos(3*b*x + 3*a)^2 + ((b*x + a)*cos(b*x + a) - sin(b*x + a))*cos(2*b*x + 2*a)
^2 + (8*(b*x + a)*sin(2*b*x + 2*a)*sin(b*x + a) + (b*x + (b*x + a)*cos(2*b*x + 2*a) + a + sin(2*b*x + 2*a))*co
s(3*b*x + 3*a) + 2*(3*(b*x + a)*cos(b*x + a) - sin(b*x + a))*cos(2*b*x + 2*a) + 6*(b*x + a)*cos(b*x + a) - 2*s
in(b*x + a))*sin(3*b*x + 3*a)^2 + ((b*x + a)*cos(b*x + a) - sin(b*x + a))*sin(2*b*x + 2*a)^2 + ((b*x + a)*cos(
2*b*x + 2*a)^2 + 13*(b*x + a)*cos(b*x + a)^2 + (b*x + a)*sin(2*b*x + 2*a)^2 + (b*x + a)*sin(b*x + a)^2 + b*x +
(13*(b*x + a)*cos(b*x + a)^2 + (b*x + a)*sin(b*x + a)^2 + 2*b*x + 2*a)*cos(2*b*x + 2*a) + (12*(b*x + a)*cos(b
*x + a)*sin(b*x + a) + cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a) + a)*cos(3*b*x + 3*a) + 2*(3*(b*x + a
)*cos(b*x + a)^3 + 3*(b*x + a)*cos(b*x + a)*sin(b*x + a)^2 + (b*x + a)*cos(b*x + a) - sin(b*x + a))*cos(2*b*x
+ 2*a) + (b*x + a)*cos(b*x + a) - ((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*
x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 +
2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*
b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*
cos(3*b*x + 3*a) + 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)
^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x
+ 3*a) + sin(b*x + a)^2)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + ((cos(2*b*x + 2*a)^2 + si
n(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x +
2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)
^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2
+ 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(
2*b*x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2
*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*log(cos(b*x + a)^2 + sin(b*x + a)^
2 - 2*sin(b*x + a) + 1) + (((b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*cos(3*b*x + 3*a)^2 + 12*(b*x +
a)*cos(b*x + a)*sin(b*x + a) + 2*(((b*x + a)*sin(b*x + a) - cos(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)*cos(b*
x + a) + sin(b*x + a))*sin(2*b*x + 2*a) + (b*x + a)*sin(b*x + a) - cos(b*x + a))*cos(3*b*x + 3*a) + (12*(b*x +
a)*cos(b*x + a)*sin(b*x + a) - cos(b*x + a)^2 - sin(b*x + a)^2 - 2)*cos(2*b*x + 2*a) - cos(2*b*x + 2*a)^2 - c
os(b*x + a)^2 + ((b*x + a)*cos(b*x + a)^2 + 13*(b*x + a)*sin(b*x + a)^2)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2
- sin(b*x + a)^2 - 1)*sin(3*b*x + 3*a) + 6*((b*x + a)*cos(b*x + a)^2*sin(b*x + a) + (b*x + a)*sin(b*x + a)^3)
*sin(2*b*x + 2*a) - sin(b*x + a))*a*c*d^2/(((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)
*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x +
2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 +
2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b
*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b
*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*s
in(3*b*x + 3*a) + sin(b*x + a)^2)*b^2) + 3*((b*x + (b*x + a)*cos(2*b*x + 2*a) + a + sin(2*b*x + 2*a))*cos(3*b*
x + 3*a)^3 + 6*(b*x + a)*cos(b*x + a)^3 + ((b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) - 1)*sin(3*b*x + 3*a)
^3 + 6*(b*x + a)*cos(b*x + a)*sin(b*x + a)^2 + 2*(4*(b*x + a)*cos(2*b*x + 2*a)*cos(b*x + a) + 4*(b*x + a)*cos(
b*x + a) + (3*(b*x + a)*sin(b*x + a) + cos(b*x + a))*sin(2*b*x + 2*a))*cos(3*b*x + 3*a)^2 + ((b*x + a)*cos(b*x
+ a) - sin(b*x + a))*cos(2*b*x + 2*a)^2 + (8*(b*x + a)*sin(2*b*x + 2*a)*sin(b*x + a) + (b*x + (b*x + a)*cos(2
*b*x + 2*a) + a + sin(2*b*x + 2*a))*cos(3*b*x + 3*a) + 2*(3*(b*x + a)*cos(b*x + a) - sin(b*x + a))*cos(2*b*x +
2*a) + 6*(b*x + a)*cos(b*x + a) - 2*sin(b*x + a))*sin(3*b*x + 3*a)^2 + ((b*x + a)*cos(b*x + a) - sin(b*x + a)
)*sin(2*b*x + 2*a)^2 + ((b*x + a)*cos(2*b*x + 2*a)^2 + 13*(b*x + a)*cos(b*x + a)^2 + (b*x + a)*sin(2*b*x + 2*a
)^2 + (b*x + a)*sin(b*x + a)^2 + b*x + (13*(b*x + a)*cos(b*x + a)^2 + (b*x + a)*sin(b*x + a)^2 + 2*b*x + 2*a)*
cos(2*b*x + 2*a) + (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) + cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)
+ a)*cos(3*b*x + 3*a) + 2*(3*(b*x + a)*cos(b*x + a)^3 + 3*(b*x + a)*cos(b*x + a)*sin(b*x + a)^2 + (b*x + a)*c
os(b*x + a) - sin(b*x + a))*cos(2*b*x + 2*a) + (b*x + a)*cos(b*x + a) - ((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a
)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos
(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x
+ a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*
x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)
+ cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*s
in(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x
+ a) + 1) + ((cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x
+ a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) +
1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x
+ a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) + 2*
(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin
(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2*b*x + 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)
^2)*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) + (((b*x + a)*sin(2*b*x + 2*a) - cos(2*b*x + 2*a
) - 1)*cos(3*b*x + 3*a)^2 + 12*(b*x + a)*cos(b*x + a)*sin(b*x + a) + 2*(((b*x + a)*sin(b*x + a) - cos(b*x + a)
)*cos(2*b*x + 2*a) + ((b*x + a)*cos(b*x + a) + sin(b*x + a))*sin(2*b*x + 2*a) + (b*x + a)*sin(b*x + a) - cos(b
*x + a))*cos(3*b*x + 3*a) + (12*(b*x + a)*cos(b*x + a)*sin(b*x + a) - cos(b*x + a)^2 - sin(b*x + a)^2 - 2)*cos
(2*b*x + 2*a) - cos(2*b*x + 2*a)^2 - cos(b*x + a)^2 + ((b*x + a)*cos(b*x + a)^2 + 13*(b*x + a)*sin(b*x + a)^2)
*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2 - sin(b*x + a)^2 - 1)*sin(3*b*x + 3*a) + 6*((b*x + a)*cos(b*x + a)^2*si
n(b*x + a) + (b*x + a)*sin(b*x + a)^3)*sin(2*b*x + 2*a) - sin(b*x + a))*a^2*d^3/(((cos(2*b*x + 2*a)^2 + sin(2*
b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*cos(3*b*x + 3*a)^2 + (cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*x + 2*a)
^2 + (cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1)*sin(3*b*x + 3*a)^2 + (cos(b*x + a)^2 +
sin(b*x + a)^2)*sin(2*b*x + 2*a)^2 + 2*(cos(2*b*x + 2*a)^2*cos(b*x + a) + cos(b*x + a)*sin(2*b*x + 2*a)^2 + 2
*cos(2*b*x + 2*a)*cos(b*x + a) + cos(b*x + a))*cos(3*b*x + 3*a) + 2*(cos(b*x + a)^2 + sin(b*x + a)^2)*cos(2*b*
x + 2*a) + cos(b*x + a)^2 + 2*(cos(2*b*x + 2*a)^2*sin(b*x + a) + sin(2*b*x + 2*a)^2*sin(b*x + a) + 2*cos(2*b*x
+ 2*a)*sin(b*x + a) + sin(b*x + a))*sin(3*b*x + 3*a) + sin(b*x + a)^2)*b^3) + ((b*x + a)^3*d^3 - 6*b*c*d^2 +
6*(a + I)*d^3 + 3*(b*c*d^2 - (a + I)*d^3)*(b*x + a)^2 + ((b*x + a)^3*d^3 - 6*b*c*d^2 + 6*(a - I)*d^3 + 3*(b*c*
d^2 - (a - I)*d^3)*(b*x + a)^2 - 6*(-I*b*c*d^2 + (I*a + 1)*d^3)*(b*x + a))*cos(3*b*x + 3*a)^2 + 6*((b*x + a)^3
*d^3 - 2*b*c*d^2 - 2*(b*x + a)*d^3 + 2*a*d^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2)*cos(b*x + a)^2 - ((b*x + a)^3*
d^3 - 6*b*c*d^2 + 6*(a - I)*d^3 + 3*(b*c*d^2 - (a - I)*d^3)*(b*x + a)^2 + 6*(I*b*c*d^2 + (-I*a - 1)*d^3)*(b*x
+ a))*sin(3*b*x + 3*a)^2 - 12*(-I*(b*x + a)^3*d^3 + 2*I*b*c*d^2 + 2*I*(b*x + a)*d^3 - 2*I*a*d^3 + 3*(-I*b*c*d^
2 + I*a*d^3)*(b*x + a)^2)*cos(b*x + a)*sin(b*x + a) - 6*((b*x + a)^3*d^3 - 2*b*c*d^2 - 2*(b*x + a)*d^3 + 2*a*d
^3 + 3*(b*c*d^2 - a*d^3)*(b*x + a)^2)*sin(b*x + a)^2 - 6*(I*b*c*d^2 + (-I*a + 1)*d^3)*(b*x + a) - 6*((-I*(b*x
+ a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a) + (-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*co
s(2*b*x + 2*a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) + ((-I*(
b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(b*x + a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*
x + a))*sin(b*x + a))*cos(2*b*x + 2*a) + (-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(b*x + a
) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2
*b*x + 2*a) + (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*sin(3*b*x + 3*a) + (((
b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(b*x + a) + (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b
*x + a))*sin(b*x + a))*sin(2*b*x + 2*a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(b*x + a))*arct
an2(cos(b*x + a), sin(b*x + a) + 1) - 6*((-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a) + (-I*(b*x +
a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x
+ a))*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) + ((-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(b*x
+ a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(b*x + a))*cos(2*b*x + 2*a) + (-I*(b*x + a)^2*d^3
+ 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(b*x + a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + ((b*x
+ a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(
b*x + a))*sin(2*b*x + 2*a))*sin(3*b*x + 3*a) + (((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(b*x + a)
+ (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(b*x + a))*sin(2*b*x + 2*a) + ((b*x + a)^2*d^3 +
2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(b*x + a))*arctan2(cos(b*x + a), -sin(b*x + a) + 1) + ((7*(b*x + a)^3*d^3 -
18*b*c*d^2 + 6*(3*a - I)*d^3 + 3*(7*b*c*d^2 - (7*a - I)*d^3)*(b*x + a)^2 - 6*(-I*b*c*d^2 + (I*a + 3)*d^3)*(b*
x + a))*cos(b*x + a) + (7*I*(b*x + a)^3*d^3 - 18*I*b*c*d^2 - 6*(-3*I*a - 1)*d^3 - 3*(-7*I*b*c*d^2 + (7*I*a + 1
)*d^3)*(b*x + a)^2 - 6*(b*c*d^2 - (a - 3*I)*d^3)*(b*x + a))*sin(b*x + a))*cos(3*b*x + 3*a) + ((b*x + a)^3*d^3
- 6*b*c*d^2 + 6*(a + I)*d^3 + 3*(b*c*d^2 - (a + I)*d^3)*(b*x + a)^2 - 6*(I*b*c*d^2 + (-I*a + 1)*d^3)*(b*x + a)
)*cos(2*b*x + 2*a) - 12*((-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3 + (-I*b*c*d^2 - I*(b*x + a)*d^3 + I*a*d^3)*co
s(2*b*x + 2*a) + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) + ((-I*b*c*d^2 - I*(b*x
+ a)*d^3 + I*a*d^3)*cos(b*x + a) + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(b*x + a))*cos(2*b*x + 2*a) + (-I*b*c*
d^2 - I*(b*x + a)*d^3 + I*a*d^3)*cos(b*x + a) + (b*c*d^2 + (b*x + a)*d^3 - a*d^3 + (b*c*d^2 + (b*x + a)*d^3 -
a*d^3)*cos(2*b*x + 2*a) + (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*sin(2*b*x + 2*a))*sin(3*b*x + 3*a) + ((b*c*d
^2 + (b*x + a)*d^3 - a*d^3)*cos(b*x + a) + (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*sin(b*x + a))*sin(2*b*x + 2
*a) + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(b*x + a))*dilog(I*e^(I*b*x + I*a)) - 12*((I*b*c*d^2 + I*(b*x + a)*
d^3 - I*a*d^3 + (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*cos(2*b*x + 2*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*s
in(2*b*x + 2*a))*cos(3*b*x + 3*a) + ((I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*cos(b*x + a) - (b*c*d^2 + (b*x +
a)*d^3 - a*d^3)*sin(b*x + a))*cos(2*b*x + 2*a) + (I*b*c*d^2 + I*(b*x + a)*d^3 - I*a*d^3)*cos(b*x + a) - (b*c*d
^2 + (b*x + a)*d^3 - a*d^3 + (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(2*b*x + 2*a) - (-I*b*c*d^2 - I*(b*x + a)*d^
3 + I*a*d^3)*sin(2*b*x + 2*a))*sin(3*b*x + 3*a) - ((b*c*d^2 + (b*x + a)*d^3 - a*d^3)*cos(b*x + a) - (-I*b*c*d^
2 - I*(b*x + a)*d^3 + I*a*d^3)*sin(b*x + a))*sin(2*b*x + 2*a) - (b*c*d^2 + (b*x + a)*d^3 - a*d^3)*sin(b*x + a)
)*dilog(-I*e^(I*b*x + I*a)) - 3*(((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a) + ((b*x + a)^2*d^3 + 2*(b*c*
d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) + (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(2*b*x +
2*a))*cos(3*b*x + 3*a) + (((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(b*x + a) + (I*(b*x + a)^2*d^3
+ 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*sin(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*
(b*x + a))*cos(b*x + a) + (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a) + (I*(b*x + a)^2*d^3 + 2*(I*b
*c*d^2 - I*a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x +
2*a))*sin(3*b*x + 3*a) + ((I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*d^3)*(b*x + a))*cos(b*x + a) - ((b*x + a)^2*
d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(b*x + a))*sin(2*b*x + 2*a) + (I*(b*x + a)^2*d^3 + 2*(I*b*c*d^2 - I*a*
d^3)*(b*x + a))*sin(b*x + a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + 3*(((b*x + a)^2*d^3
+ 2*(b*c*d^2 - a*d^3)*(b*x + a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*
x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*cos(3*b*x + 3*a) + (((b*x + a)^2*d^3 + 2*
(b*c*d^2 - a*d^3)*(b*x + a))*cos(b*x + a) - (-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(b*x
+ a))*cos(2*b*x + 2*a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*cos(b*x + a) - (-I*(b*x + a)^2*d^3
+ 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a) + (-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(2*b*x + 2
*a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin(2*b*x + 2*a))*sin(3*b*x + 3*a) - ((-I*(b*x + a)^2*
d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*cos(b*x + a) + ((b*x + a)^2*d^3 + 2*(b*c*d^2 - a*d^3)*(b*x + a))*sin
(b*x + a))*sin(2*b*x + 2*a) - (-I*(b*x + a)^2*d^3 + 2*(-I*b*c*d^2 + I*a*d^3)*(b*x + a))*sin(b*x + a))*log(cos(
b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1) - 12*(d^3*cos(b*x + a) + I*d^3*sin(b*x + a) + (d^3*cos(2*b*x
+ 2*a) + I*d^3*sin(2*b*x + 2*a) + d^3)*cos(3*b*x + 3*a) + (d^3*cos(b*x + a) + I*d^3*sin(b*x + a))*cos(2*b*x +
2*a) + (I*d^3*cos(2*b*x + 2*a) - d^3*sin(2*b*x + 2*a) + I*d^3)*sin(3*b*x + 3*a) + (I*d^3*cos(b*x + a) - d^3*s
in(b*x + a))*sin(2*b*x + 2*a))*polylog(3, I*e^(I*b*x + I*a)) + 12*(d^3*cos(b*x + a) + I*d^3*sin(b*x + a) + (d^
3*cos(2*b*x + 2*a) + I*d^3*sin(2*b*x + 2*a) + d^3)*cos(3*b*x + 3*a) + (d^3*cos(b*x + a) + I*d^3*sin(b*x + a))*
cos(2*b*x + 2*a) - (-I*d^3*cos(2*b*x + 2*a) + d^3*sin(2*b*x + 2*a) - I*d^3)*sin(3*b*x + 3*a) - (-I*d^3*cos(b*x
+ a) + d^3*sin(b*x + a))*sin(2*b*x + 2*a))*polylog(3, -I*e^(I*b*x + I*a)) - (2*(-I*(b*x + a)^3*d^3 + 6*I*b*c*
d^2 + 6*(-I*a - 1)*d^3 + 3*(-I*b*c*d^2 + (I*a + 1)*d^3)*(b*x + a)^2 + 6*(b*c*d^2 - (a - I)*d^3)*(b*x + a))*cos
(3*b*x + 3*a) - (7*I*(b*x + a)^3*d^3 - 18*I*b*c*d^2 - 6*(-3*I*a - 1)*d^3 - 3*(-7*I*b*c*d^2 + (7*I*a + 1)*d^3)*
(b*x + a)^2 - 6*(b*c*d^2 - (a - 3*I)*d^3)*(b*x + a))*cos(b*x + a) + (7*(b*x + a)^3*d^3 - 18*b*c*d^2 + 6*(3*a -
I)*d^3 + 3*(7*b*c*d^2 - (7*a - I)*d^3)*(b*x + a)^2 + 6*(I*b*c*d^2 + (-I*a - 3)*d^3)*(b*x + a))*sin(b*x + a))*
sin(3*b*x + 3*a) + (I*(b*x + a)^3*d^3 - 6*I*b*c*d^2 - 6*(-I*a + 1)*d^3 - 3*(-I*b*c*d^2 + (I*a - 1)*d^3)*(b*x +
a)^2 + 6*(b*c*d^2 - (a + I)*d^3)*(b*x + a))*sin(2*b*x + 2*a))/(b^3*cos(b*x + a) + I*b^3*sin(b*x + a) + (b^3*c
os(2*b*x + 2*a) + I*b^3*sin(2*b*x + 2*a) + b^3)*cos(3*b*x + 3*a) + (b^3*cos(b*x + a) + I*b^3*sin(b*x + a))*cos
(2*b*x + 2*a) + (I*b^3*cos(2*b*x + 2*a) - b^3*sin(2*b*x + 2*a) + I*b^3)*sin(3*b*x + 3*a) + (I*b^3*cos(b*x + a)
- b^3*sin(b*x + a))*sin(2*b*x + 2*a)))/b
Giac [F]
\[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \sin \left (b x + a\right ) \tan \left (b x + a\right )^{2} \,d x }
\]
[In]
integrate((d*x+c)^3*sin(b*x+a)*tan(b*x+a)^2,x, algorithm="giac")
[Out]
integrate((d*x + c)^3*sin(b*x + a)*tan(b*x + a)^2, x)
Mupad [F(-1)]
Timed out. \[
\int (c+d x)^3 \sin (a+b x) \tan ^2(a+b x) \, dx=\int \sin \left (a+b\,x\right )\,{\mathrm {tan}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^3 \,d x
\]
[In]
int(sin(a + b*x)*tan(a + b*x)^2*(c + d*x)^3,x)
[Out]
int(sin(a + b*x)*tan(a + b*x)^2*(c + d*x)^3, x)