3.385 \(\int (c+d x) \sec (a+b x) \sin (3 a+3 b x) \, dx\)

Optimal. Leaf size=107 \[ -\frac{i d \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac{d \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}-\frac{d x}{b}-\frac{i (c+d x)^2}{2 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.180171, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4431, 4404, 2635, 8, 4407, 3719, 2190, 2279, 2391} \[ -\frac{i d \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^2}+\frac{d \sin (a+b x) \cos (a+b x)}{b^2}+\frac{(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}-\frac{d x}{b}-\frac{i (c+d x)^2}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4431

Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && M
emberQ[{Sin, Cos}, F] && MemberQ[{Sec, Csc}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b/d,
1]

Rule 4404

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] - Dist[(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 2635

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] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

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 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*(m + 1)), x
] - Dist[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] &&
IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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 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) \sec (a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x) \cos (a+b x) \sin (a+b x)-(c+d x) \sin ^2(a+b x) \tan (a+b x)\right ) \, dx\\ &=3 \int (c+d x) \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x) \sin ^2(a+b x) \tan (a+b x) \, dx\\ &=\frac{3 (c+d x) \sin ^2(a+b x)}{2 b}-\frac{(3 d) \int \sin ^2(a+b x) \, dx}{2 b}+\int (c+d x) \cos (a+b x) \sin (a+b x) \, dx-\int (c+d x) \tan (a+b x) \, dx\\ &=-\frac{i (c+d x)^2}{2 d}+\frac{3 d \cos (a+b x) \sin (a+b x)}{4 b^2}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}+2 i \int \frac{e^{2 i (a+b x)} (c+d x)}{1+e^{2 i (a+b x)}} \, dx-\frac{d \int \sin ^2(a+b x) \, dx}{2 b}-\frac{(3 d) \int 1 \, dx}{4 b}\\ &=-\frac{3 d x}{4 b}-\frac{i (c+d x)^2}{2 d}+\frac{(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{d \cos (a+b x) \sin (a+b x)}{b^2}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}-\frac{d \int 1 \, dx}{4 b}-\frac{d \int \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=-\frac{d x}{b}-\frac{i (c+d x)^2}{2 d}+\frac{(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{d \cos (a+b x) \sin (a+b x)}{b^2}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}+\frac{(i d) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^2}\\ &=-\frac{d x}{b}-\frac{i (c+d x)^2}{2 d}+\frac{(c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac{i d \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{2 b^2}+\frac{d \cos (a+b x) \sin (a+b x)}{b^2}+\frac{2 (c+d x) \sin ^2(a+b x)}{b}\\ \end{align*}

Mathematica [B]  time = 5.82378, size = 254, normalized size = 2.37 \[ \frac{d \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt{\cot ^2(a)+1}}\right )}{2 b^2 \sqrt{\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}-\frac{d \cos (2 b x) (2 b x \cos (2 a)-\sin (2 a))}{2 b^2}+\frac{d \sin (2 b x) (2 b x \sin (2 a)+\cos (2 a))}{2 b^2}+\frac{c \left (2 \sin ^2(a+b x)+\log (\cos (a+b x))\right )}{b}-\frac{1}{2} d x^2 \tan (a) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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)]) - (d*Cos[2*b*x]*(2*b*x*Cos[2*a] - Sin[2*a]))/(2*b^2) + (d*(Cos[2*
a] + 2*b*x*Sin[2*a])*Sin[2*b*x])/(2*b^2) + (c*(Log[Cos[a + b*x]] + 2*Sin[a + b*x]^2))/b - (d*x^2*Tan[a])/2

________________________________________________________________________________________

Maple [A]  time = 0.324, size = 177, normalized size = 1.7 \begin{align*} -{\frac{i}{2}}d{x}^{2}+icx-{\frac{ \left ( 2\,dxb+id+2\,bc \right ){{\rm e}^{2\,i \left ( bx+a \right ) }}}{4\,{b}^{2}}}-{\frac{ \left ( 2\,dxb-id+2\,bc \right ){{\rm e}^{-2\,i \left ( bx+a \right ) }}}{4\,{b}^{2}}}-2\,{\frac{c\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{b}}+{\frac{c\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) }{b}}-{\frac{2\,idax}{b}}-{\frac{id{a}^{2}}{{b}^{2}}}+{\frac{d\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) x}{b}}-{\frac{{\frac{i}{2}}d{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+2\,{\frac{ad\ln \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(3*b*x+3*a),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{c{\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - \log \left (\cos \left (2 \, b x\right )^{2} + 2 \, \cos \left (2 \, b x\right ) \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, b x\right )^{2} - 2 \, \sin \left (2 \, b x\right ) \sin \left (2 \, a\right ) + \sin \left (2 \, a\right )^{2}\right )\right )}}{2 \, b} - \frac{{\left (i \, b^{2} x^{2} - 2 i \, b x \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 2 \, b x \cos \left (2 \, b x + 2 \, a\right ) - b x \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 ) + i \,{\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - \sin \left (2 \, b x + 2 \, a\right )\right )} d}{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(3*b*x+3*a),x, algorithm="maxima")

[Out]

-1/2*c*(2*cos(2*b*x + 2*a) - log(cos(2*b*x)^2 + 2*cos(2*b*x)*cos(2*a) + cos(2*a)^2 + sin(2*b*x)^2 - 2*sin(2*b*
x)*sin(2*a) + sin(2*a)^2))/b - 1/2*(2*b*x*cos(2*b*x + 2*a) + 4*b^2*integrate(x*sin(2*b*x + 2*a)/(cos(2*b*x + 2
*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1), x) - sin(2*b*x + 2*a))*d/b^2

________________________________________________________________________________________

Fricas [B]  time = 0.621951, size = 937, normalized size = 8.76 \begin{align*} \frac{2 \, b d x - 4 \,{\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} + 2 \, d \cos \left (b x + a\right ) \sin \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 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*d*x - 4*(b*d*x + b*c)*cos(b*x + a)^2 + 2*d*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)) + 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))/b^2

________________________________________________________________________________________

Sympy [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(3*b*x+3*a),x)

[Out]

Timed out

________________________________________________________________________________________

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 (3 \, b x + 3 \, a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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