\(\int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx\) [261]

Optimal result

Integrand size = 22, antiderivative size = 145 \[ \int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 d^2 \cos (a+b x)}{b^3}+\frac {(c+d x)^2 \cos (a+b x)}{b}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^2 \sec (a+b x)}{b}-\frac {2 d (c+d x) \sin (a+b x)}{b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4492, 3377, 2718, 4494, 4266, 2317, 2438} \[ \int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {4 i d (c+d x) \arctan \left (e^{i (a+b x)}\right )}{b^2}-\frac {2 i d^2 \operatorname {PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^3}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^3}-\frac {2 d^2 \cos (a+b x)}{b^3}-\frac {2 d (c+d x) \sin (a+b x)}{b^2}+\frac {(c+d x)^2 \cos (a+b x)}{b}+\frac {(c+d x)^2 \sec (a+b x)}{b} \]

[In]

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

[Out]

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(362\) vs. \(2(145)=290\).

Time = 3.48 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.50 \[ \int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {-4 b c d \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )-4 d^2 \arctan (\cot (a)) \text {arctanh}\left (\sin (a)+\cos (a) \tan \left (\frac {b x}{2}\right )\right )+\frac {2 d^2 \csc (a) \left ((b x-\arctan (\cot (a))) \left (\log \left (1-e^{i (b x-\arctan (\cot (a)))}\right )-\log \left (1+e^{i (b x-\arctan (\cot (a)))}\right )\right )+i \operatorname {PolyLog}\left (2,-e^{i (b x-\arctan (\cot (a)))}\right )-i \operatorname {PolyLog}\left (2,e^{i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {\csc ^2(a)}}+b^2 (c+d x)^2 \sec (a)+\cos (b x) \left (\left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a)-2 b d (c+d x) \sin (a)\right )-\left (2 b d (c+d x) \cos (a)+\left (-2 d^2+b^2 (c+d x)^2\right ) \sin (a)\right ) \sin (b x)+\frac {b^2 (c+d x)^2 \sin \left (\frac {b x}{2}\right )}{\left (\cos \left (\frac {a}{2}\right )-\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (a+b x)\right )-\sin \left (\frac {1}{2} (a+b x)\right )\right )}-\frac {b^2 (c+d x)^2 \sin \left (\frac {b x}{2}\right )}{\left (\cos \left (\frac {a}{2}\right )+\sin \left (\frac {a}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (a+b x)\right )+\sin \left (\frac {1}{2} (a+b x)\right )\right )}}{b^3} \]

[In]

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

[Out]

(-4*b*c*d*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] - 4*d^2*ArcTan[Cot[a]]*ArcTanh[Sin[a] + Cos[a]*Tan[(b*x)/2]] +
(2*d^2*Csc[a]*((b*x - ArcTan[Cot[a]])*(Log[1 - E^(I*(b*x - ArcTan[Cot[a]]))] - Log[1 + E^(I*(b*x - ArcTan[Cot
[a]]))]) + I*PolyLog[2, -E^(I*(b*x - ArcTan[Cot[a]]))] - I*PolyLog[2, E^(I*(b*x - ArcTan[Cot[a]]))]))/Sqrt[Csc
[a]^2] + b^2*(c + d*x)^2*Sec[a] + Cos[b*x]*((-2*d^2 + b^2*(c + d*x)^2)*Cos[a] - 2*b*d*(c + d*x)*Sin[a]) - (2*b
*d*(c + d*x)*Cos[a] + (-2*d^2 + b^2*(c + d*x)^2)*Sin[a])*Sin[b*x] + (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2]
- Sin[a/2])*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - (b^2*(c + d*x)^2*Sin[(b*x)/2])/((Cos[a/2] + Sin[a/2])*(Co
s[(a + b*x)/2] + Sin[(a + b*x)/2])))/b^3

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (134 ) = 268\).

Time = 2.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.38

method	result	size
risch	\(\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x +2 i b \,d^{2} x +b^{2} c^{2}+2 i b c d -2 d^{2}\right ) {\mathrm e}^{i \left (x b +a \right )}}{2 b^{3}}+\frac {\left (x^{2} d^{2} b^{2}+2 b^{2} c d x -2 i b \,d^{2} x +b^{2} c^{2}-2 i b c d -2 d^{2}\right ) {\mathrm e}^{-i \left (x b +a \right )}}{2 b^{3}}+\frac {2 \,{\mathrm e}^{i \left (x b +a \right )} \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}+\frac {4 i d c \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{2}}+\frac {2 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1+i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}-\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) x}{b^{2}}-\frac {2 d^{2} \ln \left (1-i {\mathrm e}^{i \left (x b +a \right )}\right ) a}{b^{3}}-\frac {2 i d^{2} \operatorname {dilog}\left (1+i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {2 i d^{2} \operatorname {dilog}\left (1-i {\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}-\frac {4 i d^{2} a \arctan \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}\)	\(345\)

[In]

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

[Out]

1/2*(x^2*d^2*b^2+2*b^2*c*d*x+b^2*c^2+2*I*b*d^2*x-2*d^2+2*I*b*c*d)/b^3*exp(I*(b*x+a))+1/2*(x^2*d^2*b^2+2*b^2*c*
d*x+b^2*c^2-2*I*b*d^2*x-2*d^2-2*I*b*c*d)/b^3*exp(-I*(b*x+a))+2*exp(I*(b*x+a))*(d^2*x^2+2*c*d*x+c^2)/b/(exp(2*I
*(b*x+a))+1)+4*I/b^2*d*c*arctan(exp(I*(b*x+a)))+2/b^2*d^2*ln(1+I*exp(I*(b*x+a)))*x+2/b^3*d^2*ln(1+I*exp(I*(b*x
+a)))*a-2/b^2*d^2*ln(1-I*exp(I*(b*x+a)))*x-2/b^3*d^2*ln(1-I*exp(I*(b*x+a)))*a-2*I/b^3*d^2*dilog(1+I*exp(I*(b*x
+a)))+2*I/b^3*d^2*dilog(1-I*exp(I*(b*x+a)))-4*I/b^3*d^2*a*arctan(exp(I*(b*x+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. 511 vs. \(2 (127) = 254\).

Time = 0.28 (sec) , antiderivative size = 511, normalized size of antiderivative = 3.52 \[ \int (c+d x)^2 \sin (a+b x) \tan ^2(a+b x) \, dx=\frac {b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d^{2} \cos \left (b x + a\right ) {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} - 2 \, d^{2}\right )} \cos \left (b x + a\right )^{2} - {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) + {\left (b d^{2} x + a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) + {\left (b c d - a d^{2}\right )} \cos \left (b x + a\right ) \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 2 \, {\left (b d^{2} x + b c d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{b^{3} \cos \left (b x + a\right )} \]

[In]

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

[Out]

(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + I*d^2*cos(b*x + a)*dilog(I*cos(b*x + a) + sin(b*x + a)) + I*d^2*cos(b*x
+ a)*dilog(I*cos(b*x + a) - sin(b*x + a)) - I*d^2*cos(b*x + a)*dilog(-I*cos(b*x + a) + sin(b*x + a)) - I*d^2*
cos(b*x + a)*dilog(-I*cos(b*x + a) - sin(b*x + a)) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a
)^2 - (b*c*d - a*d^2)*cos(b*x + a)*log(cos(b*x + a) + I*sin(b*x + a) + I) + (b*c*d - a*d^2)*cos(b*x + a)*log(c
os(b*x + a) - I*sin(b*x + a) + I) - (b*d^2*x + a*d^2)*cos(b*x + a)*log(I*cos(b*x + a) + sin(b*x + a) + 1) + (b
*d^2*x + a*d^2)*cos(b*x + a)*log(I*cos(b*x + a) - sin(b*x + a) + 1) - (b*d^2*x + a*d^2)*cos(b*x + a)*log(-I*co
s(b*x + a) + sin(b*x + a) + 1) + (b*d^2*x + a*d^2)*cos(b*x + a)*log(-I*cos(b*x + a) - sin(b*x + a) + 1) - (b*c
*d - a*d^2)*cos(b*x + a)*log(-cos(b*x + a) + I*sin(b*x + a) + I) + (b*c*d - a*d^2)*cos(b*x + a)*log(-cos(b*x +
a) - I*sin(b*x + a) + I) - 2*(b*d^2*x + b*c*d)*cos(b*x + a)*sin(b*x + a))/(b^3*cos(b*x + a))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

1/2*(2*((3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*cos(2*b*x + 3*a)*cos(b*x + 2*a) + (3*b^2*d^2*x^2 + 6
*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*sin(2*b*x + 3*a)*sin(b*x + 2*a) + (3*b^2*d^2*x^2*cos(a) + 6*b^2*c*d*x*cos(a) +
3*b^2*c^2*cos(a) - 2*d^2*cos(a))*cos(b*x + 2*a) + (3*b^2*d^2*x^2*sin(a) + 6*b^2*c*d*x*sin(a) + 3*b^2*c^2*sin(
a) - 2*d^2*sin(a))*sin(b*x + 2*a))*cos(3*b*x + 3*a)^2 + ((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x
+ a) - 2*(b*d^2*x + b*c*d)*sin(b*x + a))*cos(2*b*x + 3*a)^2 + 2*((3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2
*d^2)*cos(2*b*x + 3*a)*cos(b*x + 2*a) + (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*sin(2*b*x + 3*a)*sin
(b*x + 2*a) + (3*b^2*d^2*x^2*cos(a) + 6*b^2*c*d*x*cos(a) + 3*b^2*c^2*cos(a) - 2*d^2*cos(a))*cos(b*x + 2*a) + (
3*b^2*d^2*x^2*sin(a) + 6*b^2*c*d*x*sin(a) + 3*b^2*c^2*sin(a) - 2*d^2*sin(a))*sin(b*x + 2*a))*sin(3*b*x + 3*a)^
2 + ((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a) - 2*(b*d^2*x + b*c*d)*sin(b*x + a))*sin(2*b*x
+ 3*a)^2 + ((b^2*d^2*x^2*cos(a) + b^2*c^2*cos(a) + 2*b*c*d*sin(a) - 2*d^2*cos(a) + 2*(b^2*c*d*cos(a) + b*d^2*s
in(a))*x + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(2*b*x + 3*a) + 2*(b*d^2*x + b*c*d)*sin(2*b*x + 3*
a))*cos(3*b*x + 3*a)^2 + (b^2*d^2*x^2*cos(a) + b^2*c^2*cos(a) + 2*b*c*d*sin(a) - 2*d^2*cos(a) + 2*(b^2*c*d*cos
(a) + b*d^2*sin(a))*x)*cos(b*x + a)^2 + (b^2*d^2*x^2*cos(a) + b^2*c^2*cos(a) + 2*b*c*d*sin(a) - 2*d^2*cos(a) +
2*(b^2*c*d*cos(a) + b*d^2*sin(a))*x + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(2*b*x + 3*a) + 2*(b*d
^2*x + b*c*d)*sin(2*b*x + 3*a))*sin(3*b*x + 3*a)^2 + (b^2*d^2*x^2*cos(a) + b^2*c^2*cos(a) + 2*b*c*d*sin(a) - 2
*d^2*cos(a) + 2*(b^2*c*d*cos(a) + b*d^2*sin(a))*x)*sin(b*x + a)^2 + 2*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 -
2*d^2)*cos(2*b*x + 3*a)*cos(b*x + a) + 2*(b*d^2*x + b*c*d)*cos(b*x + a)*sin(2*b*x + 3*a) + (b^2*d^2*x^2*cos(a)
+ b^2*c^2*cos(a) + 2*b*c*d*sin(a) - 2*d^2*cos(a) + 2*(b^2*c*d*cos(a) + b*d^2*sin(a))*x)*cos(b*x + a))*cos(3*b
*x + 3*a) + ((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a)^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c
^2 - 2*d^2)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(2*b*x + 3*
a)*sin(b*x + a) + 2*(b*d^2*x + b*c*d)*sin(2*b*x + 3*a)*sin(b*x + a) + (b^2*d^2*x^2*cos(a) + b^2*c^2*cos(a) + 2
*b*c*d*sin(a) - 2*d^2*cos(a) + 2*(b^2*c*d*cos(a) + b*d^2*sin(a))*x)*sin(b*x + a))*sin(3*b*x + 3*a) + 2*((b*d^2
*x + b*c*d)*cos(b*x + a)^2 + (b*d^2*x + b*c*d)*sin(b*x + a)^2)*sin(2*b*x + 3*a))*cos(3*b*x + 4*a) + ((cos(a)^2
+ sin(a)^2)*b^2*d^2*x^2 + 2*(cos(a)^2 + sin(a)^2)*b^2*c*d*x + (cos(a)^2 + sin(a)^2)*b^2*c^2 - 2*(cos(a)^2 + s
in(a)^2)*d^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(2*b*x + 3*a)^2 + 4*(3*b^2*d^2*x^2*cos(a) + 6*
b^2*c*d*x*cos(a) + 3*b^2*c^2*cos(a) - 2*d^2*cos(a))*cos(b*x + 2*a)*cos(b*x + a) + (b^2*d^2*x^2 + 2*b^2*c*d*x +
b^2*c^2 - 2*d^2)*sin(2*b*x + 3*a)^2 + 4*(3*b^2*d^2*x^2*sin(a) + 6*b^2*c*d*x*sin(a) + 3*b^2*c^2*sin(a) - 2*d^2
*sin(a))*cos(b*x + a)*sin(b*x + 2*a) + 2*(b^2*d^2*x^2*cos(a) + 2*b^2*c*d*x*cos(a) + b^2*c^2*cos(a) + 2*(3*b^2*
d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*cos(b*x + 2*a)*cos(b*x + a) - 2*d^2*cos(a))*cos(2*b*x + 3*a) + 2*(b
^2*d^2*x^2*sin(a) + 2*b^2*c*d*x*sin(a) + b^2*c^2*sin(a) + 2*(3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*
cos(b*x + a)*sin(b*x + 2*a) - 2*d^2*sin(a))*sin(2*b*x + 3*a))*cos(3*b*x + 3*a) + 2*(((3*b^2*d^2*x^2 + 6*b^2*c*
d*x + 3*b^2*c^2 - 2*d^2)*cos(b*x + a)^2 + (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*sin(b*x + a)^2)*co
s(b*x + 2*a) + (b^2*d^2*x^2*cos(a) + 2*b^2*c*d*x*cos(a) + b^2*c^2*cos(a) - 2*d^2*cos(a))*cos(b*x + a) - 2*(b*d
^2*x*cos(a) + b*c*d*cos(a))*sin(b*x + a))*cos(2*b*x + 3*a) + 2*((3*b^2*d^2*x^2*cos(a) + 6*b^2*c*d*x*cos(a) + 3
*b^2*c^2*cos(a) - 2*d^2*cos(a))*cos(b*x + a)^2 + (3*b^2*d^2*x^2*cos(a) + 6*b^2*c*d*x*cos(a) + 3*b^2*c^2*cos(a)
- 2*d^2*cos(a))*sin(b*x + a)^2)*cos(b*x + 2*a) + ((cos(a)^2 + sin(a)^2)*b^2*d^2*x^2 + 2*(cos(a)^2 + sin(a)^2)
*b^2*c*d*x + (cos(a)^2 + sin(a)^2)*b^2*c^2 - 2*(cos(a)^2 + sin(a)^2)*d^2)*cos(b*x + a) - 8*((cos(a)^2 + sin(a)
^2)*b^3*d^2*cos(b*x + a)^2 + (cos(a)^2 + sin(a)^2)*b^3*d^2*sin(b*x + a)^2 + (b^3*d^2*cos(2*b*x + 3*a)^2 + 2*b^
3*d^2*cos(2*b*x + 3*a)*cos(a) + b^3*d^2*sin(2*b*x + 3*a)^2 + 2*b^3*d^2*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + s
in(a)^2)*b^3*d^2)*cos(3*b*x + 3*a)^2 + (b^3*d^2*cos(b*x + a)^2 + b^3*d^2*sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 +
(b^3*d^2*cos(2*b*x + 3*a)^2 + 2*b^3*d^2*cos(2*b*x + 3*a)*cos(a) + b^3*d^2*sin(2*b*x + 3*a)^2 + 2*b^3*d^2*sin(2
*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3*d^2)*sin(3*b*x + 3*a)^2 + (b^3*d^2*cos(b*x + a)^2 + b^3*d^2*sin
(b*x + a)^2)*sin(2*b*x + 3*a)^2 + 2*(b^3*d^2*cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*b^3*d^2*cos(2*b*x + 3*a)*cos(
b*x + a)*cos(a) + b^3*d^2*cos(b*x + a)*sin(2*b*x + 3*a)^2 + 2*b^3*d^2*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (
cos(a)^2 + sin(a)^2)*b^3*d^2*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(b^3*d^2*cos(b*x + a)^2*cos(a) + b^3*d^2*cos(a
)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(b^3*d^2*cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*d^2*cos(2*b*x + 3*a)*c
os(a)*sin(b*x + a) + b^3*d^2*sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*d^2*sin(2*b*x + 3*a)*sin(b*x + a)*sin(a)
+ (cos(a)^2 + sin(a)^2)*b^3*d^2*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(b^3*d^2*cos(b*x + a)^2*sin(a) + b^3*d^2*si
n(b*x + a)^2*sin(a))*sin(2*b*x + 3*a))*integrate((x*cos(2*b*x + 2*a)*cos(b*x + a) + x*sin(2*b*x + 2*a)*sin(b*x
+ a) + x*cos(b*x + a))/(b*cos(2*b*x + 2*a)^2 + b*sin(2*b*x + 2*a)^2 + 2*b*cos(2*b*x + 2*a) + b), x) - 2*((cos
(a)^2 + sin(a)^2)*b*c*d*cos(b*x + a)^2 + (cos(a)^2 + sin(a)^2)*b*c*d*sin(b*x + a)^2 + (b*c*d*cos(2*b*x + 3*a)^
2 + 2*b*c*d*cos(2*b*x + 3*a)*cos(a) + b*c*d*sin(2*b*x + 3*a)^2 + 2*b*c*d*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 +
sin(a)^2)*b*c*d)*cos(3*b*x + 3*a)^2 + (b*c*d*cos(b*x + a)^2 + b*c*d*sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (b*c
*d*cos(2*b*x + 3*a)^2 + 2*b*c*d*cos(2*b*x + 3*a)*cos(a) + b*c*d*sin(2*b*x + 3*a)^2 + 2*b*c*d*sin(2*b*x + 3*a)*
sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d)*sin(3*b*x + 3*a)^2 + (b*c*d*cos(b*x + a)^2 + b*c*d*sin(b*x + a)^2)*sin(2
*b*x + 3*a)^2 + 2*(b*c*d*cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*b*c*d*cos(2*b*x + 3*a)*cos(b*x + a)*cos(a) + b*c*
d*cos(b*x + a)*sin(2*b*x + 3*a)^2 + 2*b*c*d*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d
*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(b*c*d*cos(b*x + a)^2*cos(a) + b*c*d*cos(a)*sin(b*x + a)^2)*cos(2*b*x + 3*
a) + 2*(b*c*d*cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*b*c*d*cos(2*b*x + 3*a)*cos(a)*sin(b*x + a) + b*c*d*sin(2*b*x
+ 3*a)^2*sin(b*x + a) + 2*b*c*d*sin(2*b*x + 3*a)*sin(b*x + a)*sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d*sin(b*x +
a))*sin(3*b*x + 3*a) + 2*(b*c*d*cos(b*x + a)^2*sin(a) + b*c*d*sin(b*x + a)^2*sin(a))*sin(2*b*x + 3*a))*log(cos
(b*x + a)^2 + sin(b*x + a)^2 + 2*sin(b*x + a) + 1) + 2*((cos(a)^2 + sin(a)^2)*b*c*d*cos(b*x + a)^2 + (cos(a)^2
+ sin(a)^2)*b*c*d*sin(b*x + a)^2 + (b*c*d*cos(2*b*x + 3*a)^2 + 2*b*c*d*cos(2*b*x + 3*a)*cos(a) + b*c*d*sin(2*
b*x + 3*a)^2 + 2*b*c*d*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d)*cos(3*b*x + 3*a)^2 + (b*c*d*cos(
b*x + a)^2 + b*c*d*sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (b*c*d*cos(2*b*x + 3*a)^2 + 2*b*c*d*cos(2*b*x + 3*a)*c
os(a) + b*c*d*sin(2*b*x + 3*a)^2 + 2*b*c*d*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d)*sin(3*b*x +
3*a)^2 + (b*c*d*cos(b*x + a)^2 + b*c*d*sin(b*x + a)^2)*sin(2*b*x + 3*a)^2 + 2*(b*c*d*cos(2*b*x + 3*a)^2*cos(b*
x + a) + 2*b*c*d*cos(2*b*x + 3*a)*cos(b*x + a)*cos(a) + b*c*d*cos(b*x + a)*sin(2*b*x + 3*a)^2 + 2*b*c*d*cos(b*
x + a)*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d*cos(b*x + a))*cos(3*b*x + 3*a) + 2*(b*c*d*cos(b*x
+ a)^2*cos(a) + b*c*d*cos(a)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(b*c*d*cos(2*b*x + 3*a)^2*sin(b*x + a) + 2*
b*c*d*cos(2*b*x + 3*a)*cos(a)*sin(b*x + a) + b*c*d*sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*b*c*d*sin(2*b*x + 3*a)*
sin(b*x + a)*sin(a) + (cos(a)^2 + sin(a)^2)*b*c*d*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(b*c*d*cos(b*x + a)^2*sin
(a) + b*c*d*sin(b*x + a)^2*sin(a))*sin(2*b*x + 3*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*sin(b*x + a) + 1)
+ ((b^2*d^2*x^2*sin(a) + b^2*c^2*sin(a) - 2*b*c*d*cos(a) - 2*d^2*sin(a) + 2*(b^2*c*d*sin(a) - b*d^2*cos(a))*x
- 2*(b*d^2*x + b*c*d)*cos(2*b*x + 3*a) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin(2*b*x + 3*a))*cos(
3*b*x + 3*a)^2 + (b^2*d^2*x^2*sin(a) + b^2*c^2*sin(a) - 2*b*c*d*cos(a) - 2*d^2*sin(a) + 2*(b^2*c*d*sin(a) - b*
d^2*cos(a))*x)*cos(b*x + a)^2 + (b^2*d^2*x^2*sin(a) + b^2*c^2*sin(a) - 2*b*c*d*cos(a) - 2*d^2*sin(a) + 2*(b^2*
c*d*sin(a) - b*d^2*cos(a))*x - 2*(b*d^2*x + b*c*d)*cos(2*b*x + 3*a) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2
*d^2)*sin(2*b*x + 3*a))*sin(3*b*x + 3*a)^2 + (b^2*d^2*x^2*sin(a) + b^2*c^2*sin(a) - 2*b*c*d*cos(a) - 2*d^2*sin
(a) + 2*(b^2*c*d*sin(a) - b*d^2*cos(a))*x)*sin(b*x + a)^2 - 2*(2*(b*d^2*x + b*c*d)*cos(2*b*x + 3*a)*cos(b*x +
a) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a)*sin(2*b*x + 3*a) - (b^2*d^2*x^2*sin(a) + b^2*c
^2*sin(a) - 2*b*c*d*cos(a) - 2*d^2*sin(a) + 2*(b^2*c*d*sin(a) - b*d^2*cos(a))*x)*cos(b*x + a))*cos(3*b*x + 3*a
) - 2*((b*d^2*x + b*c*d)*cos(b*x + a)^2 + (b*d^2*x + b*c*d)*sin(b*x + a)^2)*cos(2*b*x + 3*a) - 2*(2*(b*d^2*x +
b*c*d)*cos(2*b*x + 3*a)*sin(b*x + a) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin(2*b*x + 3*a)*sin(b*x
+ a) - (b^2*d^2*x^2*sin(a) + b^2*c^2*sin(a) - 2*b*c*d*cos(a) - 2*d^2*sin(a) + 2*(b^2*c*d*sin(a) - b*d^2*cos(a
))*x)*sin(b*x + a))*sin(3*b*x + 3*a) + ((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*cos(b*x + a)^2 + (b^2*d^
2*x^2 + 2*b^2*c*d*x + b^2*c^2 - 2*d^2)*sin(b*x + a)^2)*sin(2*b*x + 3*a))*sin(3*b*x + 4*a) - 2*((cos(a)^2 + sin
(a)^2)*b*d^2*x + (cos(a)^2 + sin(a)^2)*b*c*d + (b*d^2*x + b*c*d)*cos(2*b*x + 3*a)^2 + (b*d^2*x + b*c*d)*sin(2*
b*x + 3*a)^2 - 2*(3*b^2*d^2*x^2*cos(a) + 6*b^2*c*d*x*cos(a) + 3*b^2*c^2*cos(a) - 2*d^2*cos(a))*cos(b*x + 2*a)*
sin(b*x + a) - 2*(3*b^2*d^2*x^2*sin(a) + 6*b^2*c*d*x*sin(a) + 3*b^2*c^2*sin(a) - 2*d^2*sin(a))*sin(b*x + 2*a)*
sin(b*x + a) + 2*(b*d^2*x*cos(a) + b*c*d*cos(a) - (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*cos(b*x +
2*a)*sin(b*x + a))*cos(2*b*x + 3*a) + 2*(b*d^2*x*sin(a) + b*c*d*sin(a) - (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*
c^2 - 2*d^2)*sin(b*x + 2*a)*sin(b*x + a))*sin(2*b*x + 3*a))*sin(3*b*x + 3*a) + 2*((b^2*d^2*x^2*sin(a) + 2*b^2*
c*d*x*sin(a) + b^2*c^2*sin(a) - 2*d^2*sin(a))*cos(b*x + a) + ((3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2
)*cos(b*x + a)^2 + (3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3*b^2*c^2 - 2*d^2)*sin(b*x + a)^2)*sin(b*x + 2*a) - 2*(b*d^2
*x*sin(a) + b*c*d*sin(a))*sin(b*x + a))*sin(2*b*x + 3*a) + 2*((3*b^2*d^2*x^2*sin(a) + 6*b^2*c*d*x*sin(a) + 3*b
^2*c^2*sin(a) - 2*d^2*sin(a))*cos(b*x + a)^2 + (3*b^2*d^2*x^2*sin(a) + 6*b^2*c*d*x*sin(a) + 3*b^2*c^2*sin(a) -
2*d^2*sin(a))*sin(b*x + a)^2)*sin(b*x + 2*a) - 2*((cos(a)^2 + sin(a)^2)*b*d^2*x + (cos(a)^2 + sin(a)^2)*b*c*d
)*sin(b*x + a))/((cos(a)^2 + sin(a)^2)*b^3*cos(b*x + a)^2 + (cos(a)^2 + sin(a)^2)*b^3*sin(b*x + a)^2 + (b^3*co
s(2*b*x + 3*a)^2 + 2*b^3*cos(2*b*x + 3*a)*cos(a) + b^3*sin(2*b*x + 3*a)^2 + 2*b^3*sin(2*b*x + 3*a)*sin(a) + (c
os(a)^2 + sin(a)^2)*b^3)*cos(3*b*x + 3*a)^2 + (b^3*cos(b*x + a)^2 + b^3*sin(b*x + a)^2)*cos(2*b*x + 3*a)^2 + (
b^3*cos(2*b*x + 3*a)^2 + 2*b^3*cos(2*b*x + 3*a)*cos(a) + b^3*sin(2*b*x + 3*a)^2 + 2*b^3*sin(2*b*x + 3*a)*sin(a
) + (cos(a)^2 + sin(a)^2)*b^3)*sin(3*b*x + 3*a)^2 + (b^3*cos(b*x + a)^2 + b^3*sin(b*x + a)^2)*sin(2*b*x + 3*a)
^2 + 2*(b^3*cos(2*b*x + 3*a)^2*cos(b*x + a) + 2*b^3*cos(2*b*x + 3*a)*cos(b*x + a)*cos(a) + b^3*cos(b*x + a)*si
n(2*b*x + 3*a)^2 + 2*b^3*cos(b*x + a)*sin(2*b*x + 3*a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3*cos(b*x + a))*cos(3*
b*x + 3*a) + 2*(b^3*cos(b*x + a)^2*cos(a) + b^3*cos(a)*sin(b*x + a)^2)*cos(2*b*x + 3*a) + 2*(b^3*cos(2*b*x + 3
*a)^2*sin(b*x + a) + 2*b^3*cos(2*b*x + 3*a)*cos(a)*sin(b*x + a) + b^3*sin(2*b*x + 3*a)^2*sin(b*x + a) + 2*b^3*
sin(2*b*x + 3*a)*sin(b*x + a)*sin(a) + (cos(a)^2 + sin(a)^2)*b^3*sin(b*x + a))*sin(3*b*x + 3*a) + 2*(b^3*cos(b
*x + a)^2*sin(a) + b^3*sin(b*x + a)^2*sin(a))*sin(2*b*x + 3*a))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sin(a + b*x)*tan(a + b*x)^2*(c + d*x)^2,x)

[Out]

int(sin(a + b*x)*tan(a + b*x)^2*(c + d*x)^2, x)