3.211 \(\int \frac{(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=216 \[ \frac{3 i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{2 f \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )\right )}{a d^2}-\frac{f \log (\sin (c+d x))}{a d^2}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{3 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d} \]

[Out]

(-3*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)*Cot[c
+ d*x])/(a*d) - (f*Csc[c + d*x])/(2*a*d^2) - ((e + f*x)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (2*f*Log[Sin[c/2
+ Pi/4 + (d*x)/2]])/(a*d^2) - (f*Log[Sin[c + d*x]])/(a*d^2) + (((3*I)/2)*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d
^2) - (((3*I)/2)*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2)

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Rubi [A]  time = 0.283136, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {4535, 4185, 4183, 2279, 2391, 4184, 3475, 3318} \[ \frac{3 i f \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{2 a d^2}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{2 f \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )\right )}{a d^2}-\frac{f \log (\sin (c+d x))}{a d^2}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{d x}{2}+\frac{\pi }{4}\right )}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{3 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

(-3*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) + ((e + f*x)*Cot[c
+ d*x])/(a*d) - (f*Csc[c + d*x])/(2*a*d^2) - ((e + f*x)*Cot[c + d*x]*Csc[c + d*x])/(2*a*d) - (2*f*Log[Sin[c/2
+ Pi/4 + (d*x)/2]])/(a*d^2) - (f*Log[Sin[c + d*x]])/(a*d^2) + (((3*I)/2)*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d
^2) - (((3*I)/2)*f*PolyLog[2, E^(I*(c + d*x))])/(a*d^2)

Rule 4535

Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csc[c + d*x]^(n - 1))/(a +
b*Sin[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4183

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

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

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

Rule 3318

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

Rubi steps

\begin{align*} \int \frac{(e+f x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{\int (e+f x) \csc ^3(c+d x) \, dx}{a}-\int \frac{(e+f x) \csc ^2(c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac{f \csc (c+d x)}{2 a d^2}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\int (e+f x) \csc (c+d x) \, dx}{2 a}-\frac{\int (e+f x) \csc ^2(c+d x) \, dx}{a}+\int \frac{(e+f x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac{(e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}+\frac{\int (e+f x) \csc (c+d x) \, dx}{a}-\frac{f \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{2 a d}+\frac{f \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{2 a d}-\frac{f \int \cot (c+d x) \, dx}{a d}-\int \frac{e+f x}{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac{f \log (\sin (c+d x))}{a d^2}-\frac{\int (e+f x) \csc ^2\left (\frac{1}{2} \left (c+\frac{\pi }{2}\right )+\frac{d x}{2}\right ) \, dx}{2 a}+\frac{(i f) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{2 a d^2}-\frac{(i f) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{2 a d^2}-\frac{f \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac{f \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}\\ &=-\frac{3 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac{f \log (\sin (c+d x))}{a d^2}+\frac{i f \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac{i f \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}+\frac{(i f) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac{(i f) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac{f \int \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right ) \, dx}{a d}\\ &=-\frac{3 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{(e+f x) \cot \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )}{a d}+\frac{(e+f x) \cot (c+d x)}{a d}-\frac{f \csc (c+d x)}{2 a d^2}-\frac{(e+f x) \cot (c+d x) \csc (c+d x)}{2 a d}-\frac{2 f \log \left (\sin \left (\frac{c}{2}+\frac{\pi }{4}+\frac{d x}{2}\right )\right )}{a d^2}-\frac{f \log (\sin (c+d x))}{a d^2}+\frac{3 i f \text{Li}_2\left (-e^{i (c+d x)}\right )}{2 a d^2}-\frac{3 i f \text{Li}_2\left (e^{i (c+d x)}\right )}{2 a d^2}\\ \end{align*}

Mathematica [B]  time = 3.57546, size = 484, normalized size = 2.24 \[ \frac{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (12 f \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \left (i \left (\text{PolyLog}\left (2,-e^{i (c+d x)}\right )-\text{PolyLog}\left (2,e^{i (c+d x)}\right )\right )+(c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )\right )-16 d (e+f x) \sin \left (\frac{1}{2} (c+d x)\right )-d (e+f x) \left (\cot \left (\frac{1}{2} (c+d x)\right )+1\right ) \csc \left (\frac{1}{2} (c+d x)\right )+d (e+f x) \left (\tan \left (\frac{1}{2} (c+d x)\right )+1\right ) \sec \left (\frac{1}{2} (c+d x)\right )-2 \tan \left (\frac{1}{2} (c+d x)\right ) (2 d (e+f x)+f) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 \cot \left (\frac{1}{2} (c+d x)\right ) (2 d (e+f x)-f) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+12 d e \log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+8 f (c+d x) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-16 f \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-8 f \log (\sin (c+d x)) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-12 c f \log \left (\tan \left (\frac{1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{8 a d^2 (\sin (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)*Csc[c + d*x]^3)/(a + a*Sin[c + d*x]),x]

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-(d*(e + f*x)*(1 + Cot[(c + d*x)/2])*Csc[(c + d*x)/2]) - 16*d*(e + f*x
)*Sin[(c + d*x)/2] + 8*f*(c + d*x)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2*(-f + 2*d*(e + f*x))*Cot[(c + d*x
)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 16*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2]) - 8*f*Log[Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*d*e*Log[Tan[(c + d*x)/2]
]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 12*c*f*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) +
12*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - Pol
yLog[2, E^(I*(c + d*x))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 2*(f + 2*d*(e + f*x))*(Cos[(c + d*x)/2] + S
in[(c + d*x)/2])*Tan[(c + d*x)/2] + d*(e + f*x)*Sec[(c + d*x)/2]*(1 + Tan[(c + d*x)/2])))/(8*a*d^2*(1 + Sin[c
+ d*x]))

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Maple [B]  time = 0.227, size = 468, normalized size = 2.2 \begin{align*}{\frac{3\,dfx{{\rm e}^{4\,i \left ( dx+c \right ) }}+3\,de{{\rm e}^{4\,i \left ( dx+c \right ) }}-5\,dfx{{\rm e}^{2\,i \left ( dx+c \right ) }}+3\,idfx{{\rm e}^{3\,i \left ( dx+c \right ) }}-5\,de{{\rm e}^{2\,i \left ( dx+c \right ) }}+f{{\rm e}^{3\,i \left ( dx+c \right ) }}+3\,ide{{\rm e}^{3\,i \left ( dx+c \right ) }}-if{{\rm e}^{4\,i \left ( dx+c \right ) }}+4\,dfx-idfx{{\rm e}^{i \left ( dx+c \right ) }}+4\,de-{{\rm e}^{i \left ( dx+c \right ) }}f-ide{{\rm e}^{i \left ( dx+c \right ) }}+if{{\rm e}^{2\,i \left ( dx+c \right ) }}}{ \left ({{\rm e}^{2\,i \left ( dx+c \right ) }}-1 \right ) ^{2}{d}^{2} \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) a}}-{\frac{3\,fc\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}-1 \right ) }{2\,a{d}^{2}}}-{\frac{{\frac{3\,i}{2}}f{\it polylog} \left ( 2,{{\rm e}^{i \left ( dx+c \right ) }} \right ) }{a{d}^{2}}}+{\frac{{\frac{3\,i}{2}}f{\it polylog} \left ( 2,-{{\rm e}^{i \left ( dx+c \right ) }} \right ) }{a{d}^{2}}}+{\frac{3\,e\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}-1 \right ) }{2\,da}}-{\frac{3\,e\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ) }{2\,da}}+4\,{\frac{f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }} \right ) }{a{d}^{2}}}-{\frac{f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}-1 \right ) }{a{d}^{2}}}-{\frac{f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ) }{a{d}^{2}}}-2\,{\frac{f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+i \right ) }{a{d}^{2}}}+{\frac{3\,\ln \left ( 1-{{\rm e}^{i \left ( dx+c \right ) }} \right ) fx}{2\,da}}+{\frac{3\,\ln \left ( 1-{{\rm e}^{i \left ( dx+c \right ) }} \right ) cf}{2\,a{d}^{2}}}-{\frac{3\,f\ln \left ({{\rm e}^{i \left ( dx+c \right ) }}+1 \right ) x}{2\,da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x)

[Out]

(3*d*f*x*exp(4*I*(d*x+c))+3*d*e*exp(4*I*(d*x+c))-5*d*f*x*exp(2*I*(d*x+c))+3*I*d*f*x*exp(3*I*(d*x+c))-5*d*e*exp
(2*I*(d*x+c))+f*exp(3*I*(d*x+c))+3*I*d*e*exp(3*I*(d*x+c))-I*f*exp(4*I*(d*x+c))+4*d*f*x-I*d*f*x*exp(I*(d*x+c))+
4*d*e-exp(I*(d*x+c))*f-I*d*e*exp(I*(d*x+c))+I*f*exp(2*I*(d*x+c)))/(exp(2*I*(d*x+c))-1)^2/d^2/(exp(I*(d*x+c))+I
)/a-3/2/d^2/a*f*c*ln(exp(I*(d*x+c))-1)-3/2*I*f*polylog(2,exp(I*(d*x+c)))/a/d^2+3/2*I*f*polylog(2,-exp(I*(d*x+c
)))/a/d^2+3/2/d/a*e*ln(exp(I*(d*x+c))-1)-3/2/d/a*e*ln(exp(I*(d*x+c))+1)+4/d^2/a*f*ln(exp(I*(d*x+c)))-1/d^2/a*f
*ln(exp(I*(d*x+c))-1)-1/d^2/a*f*ln(exp(I*(d*x+c))+1)-2/d^2/a*f*ln(exp(I*(d*x+c))+I)+3/2/d/a*ln(1-exp(I*(d*x+c)
))*f*x+3/2/d^2/a*ln(1-exp(I*(d*x+c)))*c*f-3/2/d/a*ln(exp(I*(d*x+c))+1)*f*x

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Maxima [B]  time = 4.08092, size = 2817, normalized size = 13.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

(16*d*f*x*cos(5*d*x + 5*c) + 16*I*d*f*x*sin(5*d*x + 5*c) - 16*I*d*e - (8*f*cos(5*d*x + 5*c) + 8*I*f*cos(4*d*x
+ 4*c) - 16*f*cos(3*d*x + 3*c) - 16*I*f*cos(2*d*x + 2*c) + 8*f*cos(d*x + c) + 8*I*f*sin(5*d*x + 5*c) - 8*f*sin
(4*d*x + 4*c) - 16*I*f*sin(3*d*x + 3*c) + 16*f*sin(2*d*x + 2*c) + 8*I*f*sin(d*x + c) + 8*I*f)*arctan2(cos(c) +
sin(d*x), cos(d*x) + sin(c)) - (6*I*d*f*x + 6*I*d*e + 2*(3*d*f*x + 3*d*e + 2*f)*cos(5*d*x + 5*c) + (6*I*d*f*x
+ 6*I*d*e + 4*I*f)*cos(4*d*x + 4*c) - 4*(3*d*f*x + 3*d*e + 2*f)*cos(3*d*x + 3*c) + (-12*I*d*f*x - 12*I*d*e -
8*I*f)*cos(2*d*x + 2*c) + 2*(3*d*f*x + 3*d*e + 2*f)*cos(d*x + c) + (6*I*d*f*x + 6*I*d*e + 4*I*f)*sin(5*d*x + 5
*c) - 2*(3*d*f*x + 3*d*e + 2*f)*sin(4*d*x + 4*c) + (-12*I*d*f*x - 12*I*d*e - 8*I*f)*sin(3*d*x + 3*c) + 4*(3*d*
f*x + 3*d*e + 2*f)*sin(2*d*x + 2*c) + (6*I*d*f*x + 6*I*d*e + 4*I*f)*sin(d*x + c) + 4*I*f)*arctan2(sin(d*x + c)
, cos(d*x + c) + 1) - (-6*I*d*e - 2*(3*d*e - 2*f)*cos(5*d*x + 5*c) + (-6*I*d*e + 4*I*f)*cos(4*d*x + 4*c) + 4*(
3*d*e - 2*f)*cos(3*d*x + 3*c) + (12*I*d*e - 8*I*f)*cos(2*d*x + 2*c) - 2*(3*d*e - 2*f)*cos(d*x + c) + (-6*I*d*e
+ 4*I*f)*sin(5*d*x + 5*c) + 2*(3*d*e - 2*f)*sin(4*d*x + 4*c) + (12*I*d*e - 8*I*f)*sin(3*d*x + 3*c) - 4*(3*d*e
- 2*f)*sin(2*d*x + 2*c) + (-6*I*d*e + 4*I*f)*sin(d*x + c) + 4*I*f)*arctan2(sin(d*x + c), cos(d*x + c) - 1) -
(6*d*f*x*cos(5*d*x + 5*c) + 6*I*d*f*x*cos(4*d*x + 4*c) - 12*d*f*x*cos(3*d*x + 3*c) - 12*I*d*f*x*cos(2*d*x + 2*
c) + 6*d*f*x*cos(d*x + c) + 6*I*d*f*x*sin(5*d*x + 5*c) - 6*d*f*x*sin(4*d*x + 4*c) - 12*I*d*f*x*sin(3*d*x + 3*c
) + 12*d*f*x*sin(2*d*x + 2*c) + 6*I*d*f*x*sin(d*x + c) + 6*I*d*f*x)*arctan2(sin(d*x + c), -cos(d*x + c) + 1) -
(-4*I*d*f*x + 12*I*d*e + 4*f)*cos(4*d*x + 4*c) - (20*d*f*x - 12*d*e + 4*I*f)*cos(3*d*x + 3*c) - (12*I*d*f*x -
20*I*d*e - 4*f)*cos(2*d*x + 2*c) + (12*d*f*x - 4*d*e + 4*I*f)*cos(d*x + c) + (6*f*cos(5*d*x + 5*c) + 6*I*f*co
s(4*d*x + 4*c) - 12*f*cos(3*d*x + 3*c) - 12*I*f*cos(2*d*x + 2*c) + 6*f*cos(d*x + c) + 6*I*f*sin(5*d*x + 5*c) -
6*f*sin(4*d*x + 4*c) - 12*I*f*sin(3*d*x + 3*c) + 12*f*sin(2*d*x + 2*c) + 6*I*f*sin(d*x + c) + 6*I*f)*dilog(-e
^(I*d*x + I*c)) - (6*f*cos(5*d*x + 5*c) + 6*I*f*cos(4*d*x + 4*c) - 12*f*cos(3*d*x + 3*c) - 12*I*f*cos(2*d*x +
2*c) + 6*f*cos(d*x + c) + 6*I*f*sin(5*d*x + 5*c) - 6*f*sin(4*d*x + 4*c) - 12*I*f*sin(3*d*x + 3*c) + 12*f*sin(2
*d*x + 2*c) + 6*I*f*sin(d*x + c) + 6*I*f)*dilog(e^(I*d*x + I*c)) - (3*d*f*x + 3*d*e + (-3*I*d*f*x - 3*I*d*e -
2*I*f)*cos(5*d*x + 5*c) + (3*d*f*x + 3*d*e + 2*f)*cos(4*d*x + 4*c) + (6*I*d*f*x + 6*I*d*e + 4*I*f)*cos(3*d*x +
3*c) - 2*(3*d*f*x + 3*d*e + 2*f)*cos(2*d*x + 2*c) + (-3*I*d*f*x - 3*I*d*e - 2*I*f)*cos(d*x + c) + (3*d*f*x +
3*d*e + 2*f)*sin(5*d*x + 5*c) + (3*I*d*f*x + 3*I*d*e + 2*I*f)*sin(4*d*x + 4*c) - 2*(3*d*f*x + 3*d*e + 2*f)*sin
(3*d*x + 3*c) + (-6*I*d*f*x - 6*I*d*e - 4*I*f)*sin(2*d*x + 2*c) + (3*d*f*x + 3*d*e + 2*f)*sin(d*x + c) + 2*f)*
log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1) + (3*d*f*x + 3*d*e - (3*I*d*f*x + 3*I*d*e - 2*I*f)*c
os(5*d*x + 5*c) + (3*d*f*x + 3*d*e - 2*f)*cos(4*d*x + 4*c) - (-6*I*d*f*x - 6*I*d*e + 4*I*f)*cos(3*d*x + 3*c) -
2*(3*d*f*x + 3*d*e - 2*f)*cos(2*d*x + 2*c) - (3*I*d*f*x + 3*I*d*e - 2*I*f)*cos(d*x + c) + (3*d*f*x + 3*d*e -
2*f)*sin(5*d*x + 5*c) - (-3*I*d*f*x - 3*I*d*e + 2*I*f)*sin(4*d*x + 4*c) - 2*(3*d*f*x + 3*d*e - 2*f)*sin(3*d*x
+ 3*c) - (6*I*d*f*x + 6*I*d*e - 4*I*f)*sin(2*d*x + 2*c) + (3*d*f*x + 3*d*e - 2*f)*sin(d*x + c) - 2*f)*log(cos(
d*x + c)^2 + sin(d*x + c)^2 - 2*cos(d*x + c) + 1) - (-4*I*f*cos(5*d*x + 5*c) + 4*f*cos(4*d*x + 4*c) + 8*I*f*co
s(3*d*x + 3*c) - 8*f*cos(2*d*x + 2*c) - 4*I*f*cos(d*x + c) + 4*f*sin(5*d*x + 5*c) + 4*I*f*sin(4*d*x + 4*c) - 8
*f*sin(3*d*x + 3*c) - 8*I*f*sin(2*d*x + 2*c) + 4*f*sin(d*x + c) + 4*f)*log(cos(d*x)^2 + cos(c)^2 + 2*cos(c)*si
n(d*x) + sin(d*x)^2 + 2*cos(d*x)*sin(c) + sin(c)^2) - (4*d*f*x - 12*d*e + 4*I*f)*sin(4*d*x + 4*c) - (20*I*d*f*
x - 12*I*d*e - 4*f)*sin(3*d*x + 3*c) + (12*d*f*x - 20*d*e + 4*I*f)*sin(2*d*x + 2*c) - (-12*I*d*f*x + 4*I*d*e +
4*f)*sin(d*x + c))/(-4*I*a*d^2*cos(5*d*x + 5*c) + 4*a*d^2*cos(4*d*x + 4*c) + 8*I*a*d^2*cos(3*d*x + 3*c) - 8*a
*d^2*cos(2*d*x + 2*c) - 4*I*a*d^2*cos(d*x + c) + 4*a*d^2*sin(5*d*x + 5*c) + 4*I*a*d^2*sin(4*d*x + 4*c) - 8*a*d
^2*sin(3*d*x + 3*c) - 8*I*a*d^2*sin(2*d*x + 2*c) + 4*a*d^2*sin(d*x + c) + 4*a*d^2)

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Fricas [B]  time = 2.37466, size = 3522, normalized size = 16.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(8*(d*f*x + d*e)*cos(d*x + c)^3 - 4*d*f*x + 2*(3*d*f*x + 3*d*e - f)*cos(d*x + c)^2 - 4*d*e - 6*(d*f*x + d*
e)*cos(d*x + c) + (-3*I*f*cos(d*x + c)^3 - 3*I*f*cos(d*x + c)^2 + 3*I*f*cos(d*x + c) + (-3*I*f*cos(d*x + c)^2
+ 3*I*f)*sin(d*x + c) + 3*I*f)*dilog(cos(d*x + c) + I*sin(d*x + c)) + (3*I*f*cos(d*x + c)^3 + 3*I*f*cos(d*x +
c)^2 - 3*I*f*cos(d*x + c) + (3*I*f*cos(d*x + c)^2 - 3*I*f)*sin(d*x + c) - 3*I*f)*dilog(cos(d*x + c) - I*sin(d*
x + c)) + (-3*I*f*cos(d*x + c)^3 - 3*I*f*cos(d*x + c)^2 + 3*I*f*cos(d*x + c) + (-3*I*f*cos(d*x + c)^2 + 3*I*f)
*sin(d*x + c) + 3*I*f)*dilog(-cos(d*x + c) + I*sin(d*x + c)) + (3*I*f*cos(d*x + c)^3 + 3*I*f*cos(d*x + c)^2 -
3*I*f*cos(d*x + c) + (3*I*f*cos(d*x + c)^2 - 3*I*f)*sin(d*x + c) - 3*I*f)*dilog(-cos(d*x + c) - I*sin(d*x + c)
) - ((3*d*f*x + 3*d*e + 2*f)*cos(d*x + c)^3 - 3*d*f*x + (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c)^2 - 3*d*e - (3*d*
f*x + 3*d*e + 2*f)*cos(d*x + c) - (3*d*f*x - (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c)^2 + 3*d*e + 2*f)*sin(d*x + c
) - 2*f)*log(cos(d*x + c) + I*sin(d*x + c) + 1) - ((3*d*f*x + 3*d*e + 2*f)*cos(d*x + c)^3 - 3*d*f*x + (3*d*f*x
+ 3*d*e + 2*f)*cos(d*x + c)^2 - 3*d*e - (3*d*f*x + 3*d*e + 2*f)*cos(d*x + c) - (3*d*f*x - (3*d*f*x + 3*d*e +
2*f)*cos(d*x + c)^2 + 3*d*e + 2*f)*sin(d*x + c) - 2*f)*log(cos(d*x + c) - I*sin(d*x + c) + 1) + ((3*d*e - (3*c
+ 2)*f)*cos(d*x + c)^3 + (3*d*e - (3*c + 2)*f)*cos(d*x + c)^2 - 3*d*e + (3*c + 2)*f - (3*d*e - (3*c + 2)*f)*c
os(d*x + c) + ((3*d*e - (3*c + 2)*f)*cos(d*x + c)^2 - 3*d*e + (3*c + 2)*f)*sin(d*x + c))*log(-1/2*cos(d*x + c)
+ 1/2*I*sin(d*x + c) + 1/2) + ((3*d*e - (3*c + 2)*f)*cos(d*x + c)^3 + (3*d*e - (3*c + 2)*f)*cos(d*x + c)^2 -
3*d*e + (3*c + 2)*f - (3*d*e - (3*c + 2)*f)*cos(d*x + c) + ((3*d*e - (3*c + 2)*f)*cos(d*x + c)^2 - 3*d*e + (3*
c + 2)*f)*sin(d*x + c))*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2) + 3*((d*f*x + c*f)*cos(d*x + c)^3 -
d*f*x + (d*f*x + c*f)*cos(d*x + c)^2 - c*f - (d*f*x + c*f)*cos(d*x + c) - (d*f*x - (d*f*x + c*f)*cos(d*x + c)^
2 + c*f)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + 1) + 3*((d*f*x + c*f)*cos(d*x + c)^3 - d*f*x + (d*
f*x + c*f)*cos(d*x + c)^2 - c*f - (d*f*x + c*f)*cos(d*x + c) - (d*f*x - (d*f*x + c*f)*cos(d*x + c)^2 + c*f)*si
n(d*x + c))*log(-cos(d*x + c) - I*sin(d*x + c) + 1) - 4*(f*cos(d*x + c)^3 + f*cos(d*x + c)^2 - f*cos(d*x + c)
+ (f*cos(d*x + c)^2 - f)*sin(d*x + c) - f)*log(sin(d*x + c) + 1) + 2*(2*d*f*x - 4*(d*f*x + d*e)*cos(d*x + c)^2
+ 2*d*e - (d*f*x + d*e - f)*cos(d*x + c) + f)*sin(d*x + c) + 2*f)/(a*d^2*cos(d*x + c)^3 + a*d^2*cos(d*x + c)^
2 - a*d^2*cos(d*x + c) - a*d^2 + (a*d^2*cos(d*x + c)^2 - a*d^2)*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{e \csc ^{3}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx + \int \frac{f x \csc ^{3}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csc(d*x+c)**3/(a+a*sin(d*x+c)),x)

[Out]

(Integral(e*csc(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f*x*csc(c + d*x)**3/(sin(c + d*x) + 1), x))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )} \csc \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csc(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*csc(d*x + c)^3/(a*sin(d*x + c) + a), x)