3.218 \(\int (c+d x) \sin (a+b x) \tan (a+b x) \, dx\)

Optimal. Leaf size=103 \[ \frac{i d \text{PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac{i d \text{PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac{d \cos (a+b x)}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0675388, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4407, 3296, 2638, 4181, 2279, 2391} \[ \frac{i d \text{PolyLog}\left (2,-i e^{i (a+b x)}\right )}{b^2}-\frac{i d \text{PolyLog}\left (2,i e^{i (a+b x)}\right )}{b^2}-\frac{d \cos (a+b x)}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4407

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 3296

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 2638

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

Rule 4181

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 2279

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 2391

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]

Rubi steps

\begin{align*} \int (c+d x) \sin (a+b x) \tan (a+b x) \, dx &=-\int (c+d x) \cos (a+b x) \, dx+\int (c+d x) \sec (a+b x) \, dx\\ &=-\frac{2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{(c+d x) \sin (a+b x)}{b}-\frac{d \int \log \left (1-i e^{i (a+b x)}\right ) \, dx}{b}+\frac{d \int \log \left (1+i e^{i (a+b x)}\right ) \, dx}{b}+\frac{d \int \sin (a+b x) \, dx}{b}\\ &=-\frac{2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{d \cos (a+b x)}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}+\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}-\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}\\ &=-\frac{2 i (c+d x) \tan ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{d \cos (a+b x)}{b^2}+\frac{i d \text{Li}_2\left (-i e^{i (a+b x)}\right )}{b^2}-\frac{i d \text{Li}_2\left (i e^{i (a+b x)}\right )}{b^2}-\frac{(c+d x) \sin (a+b x)}{b}\\ \end{align*}

Mathematica [B]  time = 0.423804, size = 213, normalized size = 2.07 \[ \frac{d \left (i \left (\text{PolyLog}\left (2,-e^{i \left (-a-b x+\frac{\pi }{2}\right )}\right )-\text{PolyLog}\left (2,e^{i \left (-a-b x+\frac{\pi }{2}\right )}\right )\right )+\left (-a-b x+\frac{\pi }{2}\right ) \left (\log \left (1-e^{i \left (-a-b x+\frac{\pi }{2}\right )}\right )-\log \left (1+e^{i \left (-a-b x+\frac{\pi }{2}\right )}\right )\right )-\left (\frac{\pi }{2}-a\right ) \log \left (\tan \left (\frac{1}{2} \left (-a-b x+\frac{\pi }{2}\right )\right )\right )\right )}{b^2}-\frac{d \cos (b x) (b x \sin (a)+\cos (a))}{b^2}-\frac{d \sin (b x) (b x \cos (a)-\sin (a))}{b^2}-\frac{c \sin (a+b x)}{b}+\frac{c \tanh ^{-1}(\sin (a+b x))}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*ArcTanh[Sin[a + b*x]])/b + (d*((-a + Pi/2 - b*x)*(Log[1 - E^(I*(-a + Pi/2 - b*x))] - Log[1 + E^(I*(-a + Pi/
2 - b*x))]) - (-a + Pi/2)*Log[Tan[(-a + Pi/2 - b*x)/2]] + I*(PolyLog[2, -E^(I*(-a + Pi/2 - b*x))] - PolyLog[2,
E^(I*(-a + Pi/2 - b*x))])))/b^2 - (d*Cos[b*x]*(Cos[a] + b*x*Sin[a]))/b^2 - (d*(b*x*Cos[a] - Sin[a])*Sin[b*x])
/b^2 - (c*Sin[a + b*x])/b

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Maple [B]  time = 0.227, size = 221, normalized size = 2.2 \begin{align*}{\frac{{\frac{i}{2}} \left ( dxb+bc+id \right ){{\rm e}^{i \left ( bx+a \right ) }}}{{b}^{2}}}-{\frac{{\frac{i}{2}} \left ( dxb+bc-id \right ){{\rm e}^{-i \left ( bx+a \right ) }}}{{b}^{2}}}-{\frac{2\,ic\arctan \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{b}}-{\frac{d\ln \left ( 1+i{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{b}}-{\frac{d\ln \left ( 1+i{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{2}}}+{\frac{d\ln \left ( 1-i{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{b}}+{\frac{d\ln \left ( 1-i{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{2}}}+{\frac{id{\it dilog} \left ( 1+i{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-{\frac{id{\it dilog} \left ( 1-i{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+{\frac{2\,ida\arctan \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*(d*x*b+b*c+I*d)/b^2*exp(I*(b*x+a))-1/2*I*(d*x*b+b*c-I*d)/b^2*exp(-I*(b*x+a))-2*I/b*c*arctan(exp(I*(b*x+a
)))-1/b*d*ln(1+I*exp(I*(b*x+a)))*x-1/b^2*d*ln(1+I*exp(I*(b*x+a)))*a+1/b*d*ln(1-I*exp(I*(b*x+a)))*x+1/b^2*d*ln(
1-I*exp(I*(b*x+a)))*a+I/b^2*d*dilog(1+I*exp(I*(b*x+a)))-I/b^2*d*dilog(1-I*exp(I*(b*x+a)))+2*I/b^2*d*a*arctan(e
xp(I*(b*x+a)))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [B]  time = 0.594791, size = 905, normalized size = 8.79 \begin{align*} -\frac{2 \, d \cos \left (b x + a\right ) + i \, d{\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + i \, d{\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - i \, d{\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - i \, d{\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) -{\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) +{\left (b c - a d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) -{\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) +{\left (b d x + a d\right )} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) -{\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) +{\left (b d x + a d\right )} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) -{\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) +{\left (b c - a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) + 2 \,{\left (b d x + b c\right )} \sin \left (b x + a\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) \sin ^{2}{\left (a + b x \right )} \sec{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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