3.118 \(\int \frac{(c+d x)^2}{a-a \sin (e+f x)} \, dx\)

Optimal. Leaf size=112 \[ -\frac{4 i d^2 \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac{4 d (c+d x) \log \left (1+i e^{i (e+f x)}\right )}{a f^2}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{a f}-\frac{i (c+d x)^2}{a f} \]

[Out]

((-I)*(c + d*x)^2)/(a*f) + (4*d*(c + d*x)*Log[1 + I*E^(I*(e + f*x))])/(a*f^2) - ((4*I)*d^2*PolyLog[2, (-I)*E^(
I*(e + f*x))])/(a*f^3) + ((c + d*x)^2*Tan[e/2 + Pi/4 + (f*x)/2])/(a*f)

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Rubi [A]  time = 0.212183, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3318, 4184, 3717, 2190, 2279, 2391} \[ -\frac{4 i d^2 \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3}+\frac{4 d (c+d x) \log \left (1+i e^{i (e+f x)}\right )}{a f^2}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{f x}{2}+\frac{\pi }{4}\right )}{a f}-\frac{i (c+d x)^2}{a f} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2/(a - a*Sin[e + f*x]),x]

[Out]

((-I)*(c + d*x)^2)/(a*f) + (4*d*(c + d*x)*Log[1 + I*E^(I*(e + f*x))])/(a*f^2) - ((4*I)*d^2*PolyLog[2, (-I)*E^(
I*(e + f*x))])/(a*f^3) + ((c + d*x)^2*Tan[e/2 + Pi/4 + (f*x)/2])/(a*f)

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])

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 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && 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 \frac{(c+d x)^2}{a-a \sin (e+f x)} \, dx &=\frac{\int (c+d x)^2 \csc ^2\left (\frac{1}{2} \left (e-\frac{\pi }{2}\right )+\frac{f x}{2}\right ) \, dx}{2 a}\\ &=\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{(2 d) \int (c+d x) \cot \left (\frac{e}{2}-\frac{\pi }{4}+\frac{f x}{2}\right ) \, dx}{a f}\\ &=-\frac{i (c+d x)^2}{a f}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}-\frac{(4 d) \int \frac{e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )} (c+d x)}{1+i e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}} \, dx}{a f}\\ &=-\frac{i (c+d x)^2}{a f}+\frac{4 d (c+d x) \log \left (1+i e^{i (e+f x)}\right )}{a f^2}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}-\frac{\left (4 d^2\right ) \int \log \left (1+i e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right ) \, dx}{a f^2}\\ &=-\frac{i (c+d x)^2}{a f}+\frac{4 d (c+d x) \log \left (1+i e^{i (e+f x)}\right )}{a f^2}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}+\frac{\left (4 i d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{2 i \left (\frac{e}{2}+\frac{f x}{2}\right )}\right )}{a f^3}\\ &=-\frac{i (c+d x)^2}{a f}+\frac{4 d (c+d x) \log \left (1+i e^{i (e+f x)}\right )}{a f^2}-\frac{4 i d^2 \text{Li}_2\left (-i e^{i (e+f x)}\right )}{a f^3}+\frac{(c+d x)^2 \tan \left (\frac{e}{2}+\frac{\pi }{4}+\frac{f x}{2}\right )}{a f}\\ \end{align*}

Mathematica [A]  time = 0.722677, size = 92, normalized size = 0.82 \[ \frac{f (c+d x) \left (f (c+d x) \tan \left (\frac{1}{4} (2 e+2 f x+\pi )\right )-i f (c+d x)+4 d \log \left (1+i e^{i (e+f x)}\right )\right )-4 i d^2 \text{PolyLog}\left (2,-i e^{i (e+f x)}\right )}{a f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2/(a - a*Sin[e + f*x]),x]

[Out]

((-4*I)*d^2*PolyLog[2, (-I)*E^(I*(e + f*x))] + f*(c + d*x)*((-I)*f*(c + d*x) + 4*d*Log[1 + I*E^(I*(e + f*x))]
+ f*(c + d*x)*Tan[(2*e + Pi + 2*f*x)/4]))/(a*f^3)

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Maple [B]  time = 0.087, size = 254, normalized size = 2.3 \begin{align*} 2\,{\frac{{d}^{2}{x}^{2}+2\,cdx+{c}^{2}}{af \left ({{\rm e}^{i \left ( fx+e \right ) }}-i \right ) }}+4\,{\frac{\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}-i \right ) cd}{a{f}^{2}}}-4\,{\frac{\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) cd}{a{f}^{2}}}-{\frac{2\,i{d}^{2}{x}^{2}}{af}}-{\frac{4\,i{d}^{2}ex}{a{f}^{2}}}-{\frac{2\,i{d}^{2}{e}^{2}}{{f}^{3}a}}+4\,{\frac{{d}^{2}\ln \left ( 1+i{{\rm e}^{i \left ( fx+e \right ) }} \right ) x}{a{f}^{2}}}+4\,{\frac{{d}^{2}\ln \left ( 1+i{{\rm e}^{i \left ( fx+e \right ) }} \right ) e}{{f}^{3}a}}-{\frac{4\,i{d}^{2}{\it polylog} \left ( 2,-i{{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}a}}-4\,{\frac{{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }}-i \right ) }{{f}^{3}a}}+4\,{\frac{{d}^{2}e\ln \left ({{\rm e}^{i \left ( fx+e \right ) }} \right ) }{{f}^{3}a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^2/(a-a*sin(f*x+e)),x)

[Out]

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

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Maxima [B]  time = 1.35062, size = 427, normalized size = 3.81 \begin{align*} \frac{-2 i \, c^{2} f^{2} +{\left (4 \, c d f \cos \left (f x + e\right ) + 4 i \, c d f \sin \left (f x + e\right ) - 4 i \, c d f\right )} \arctan \left (\sin \left (f x + e\right ) - 1, \cos \left (f x + e\right )\right ) +{\left (4 \, d^{2} f x \cos \left (f x + e\right ) + 4 i \, d^{2} f x \sin \left (f x + e\right ) - 4 i \, d^{2} f x\right )} \arctan \left (\cos \left (f x + e\right ), -\sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x\right )} \cos \left (f x + e\right ) -{\left (4 \, d^{2} \cos \left (f x + e\right ) + 4 i \, d^{2} \sin \left (f x + e\right ) - 4 i \, d^{2}\right )}{\rm Li}_2\left (-i \, e^{\left (i \, f x + i \, e\right )}\right ) -{\left (2 \, d^{2} f x + 2 \, c d f -{\left (-2 i \, d^{2} f x - 2 i \, c d f\right )} \cos \left (f x + e\right ) - 2 \,{\left (d^{2} f x + c d f\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right ) +{\left (-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x\right )} \sin \left (f x + e\right )}{-i \, a f^{3} \cos \left (f x + e\right ) + a f^{3} \sin \left (f x + e\right ) - a f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a-a*sin(f*x+e)),x, algorithm="maxima")

[Out]

(-2*I*c^2*f^2 + (4*c*d*f*cos(f*x + e) + 4*I*c*d*f*sin(f*x + e) - 4*I*c*d*f)*arctan2(sin(f*x + e) - 1, cos(f*x
+ e)) + (4*d^2*f*x*cos(f*x + e) + 4*I*d^2*f*x*sin(f*x + e) - 4*I*d^2*f*x)*arctan2(cos(f*x + e), -sin(f*x + e)
+ 1) - 2*(d^2*f^2*x^2 + 2*c*d*f^2*x)*cos(f*x + e) - (4*d^2*cos(f*x + e) + 4*I*d^2*sin(f*x + e) - 4*I*d^2)*dilo
g(-I*e^(I*f*x + I*e)) - (2*d^2*f*x + 2*c*d*f - (-2*I*d^2*f*x - 2*I*c*d*f)*cos(f*x + e) - 2*(d^2*f*x + c*d*f)*s
in(f*x + e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 - 2*sin(f*x + e) + 1) + (-2*I*d^2*f^2*x^2 - 4*I*c*d*f^2*x)*si
n(f*x + e))/(-I*a*f^3*cos(f*x + e) + a*f^3*sin(f*x + e) - a*f^3)

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Fricas [B]  time = 1.87766, size = 1195, normalized size = 10.67 \begin{align*} \frac{d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} +{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} \cos \left (f x + e\right ) -{\left (-2 i \, d^{2} \cos \left (f x + e\right ) + 2 i \, d^{2} \sin \left (f x + e\right ) - 2 i \, d^{2}\right )}{\rm Li}_2\left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) -{\left (2 i \, d^{2} \cos \left (f x + e\right ) - 2 i \, d^{2} \sin \left (f x + e\right ) + 2 i \, d^{2}\right )}{\rm Li}_2\left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - 2 \,{\left (d^{2} e - c d f +{\left (d^{2} e - c d f\right )} \cos \left (f x + e\right ) -{\left (d^{2} e - c d f\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) + 2 \,{\left (d^{2} f x + d^{2} e +{\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right ) -{\left (d^{2} f x + d^{2} e\right )} \sin \left (f x + e\right )\right )} \log \left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (d^{2} f x + d^{2} e +{\left (d^{2} f x + d^{2} e\right )} \cos \left (f x + e\right ) -{\left (d^{2} f x + d^{2} e\right )} \sin \left (f x + e\right )\right )} \log \left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) - 2 \,{\left (d^{2} e - c d f +{\left (d^{2} e - c d f\right )} \cos \left (f x + e\right ) -{\left (d^{2} e - c d f\right )} \sin \left (f x + e\right )\right )} \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) +{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} \sin \left (f x + e\right )}{a f^{3} \cos \left (f x + e\right ) - a f^{3} \sin \left (f x + e\right ) + a f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a-a*sin(f*x+e)),x, algorithm="fricas")

[Out]

(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2 + (d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*cos(f*x + e) - (-2*I*d^2*cos(f*x
+ e) + 2*I*d^2*sin(f*x + e) - 2*I*d^2)*dilog(I*cos(f*x + e) + sin(f*x + e)) - (2*I*d^2*cos(f*x + e) - 2*I*d^2*
sin(f*x + e) + 2*I*d^2)*dilog(-I*cos(f*x + e) + sin(f*x + e)) - 2*(d^2*e - c*d*f + (d^2*e - c*d*f)*cos(f*x + e
) - (d^2*e - c*d*f)*sin(f*x + e))*log(cos(f*x + e) - I*sin(f*x + e) + I) + 2*(d^2*f*x + d^2*e + (d^2*f*x + d^2
*e)*cos(f*x + e) - (d^2*f*x + d^2*e)*sin(f*x + e))*log(I*cos(f*x + e) - sin(f*x + e) + 1) + 2*(d^2*f*x + d^2*e
+ (d^2*f*x + d^2*e)*cos(f*x + e) - (d^2*f*x + d^2*e)*sin(f*x + e))*log(-I*cos(f*x + e) - sin(f*x + e) + 1) -
2*(d^2*e - c*d*f + (d^2*e - c*d*f)*cos(f*x + e) - (d^2*e - c*d*f)*sin(f*x + e))*log(-cos(f*x + e) - I*sin(f*x
+ e) + I) + (d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*sin(f*x + e))/(a*f^3*cos(f*x + e) - a*f^3*sin(f*x + e) + a*f
^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{c^{2}}{\sin{\left (e + f x \right )} - 1}\, dx + \int \frac{d^{2} x^{2}}{\sin{\left (e + f x \right )} - 1}\, dx + \int \frac{2 c d x}{\sin{\left (e + f x \right )} - 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2/(a-a*sin(f*x+e)),x)

[Out]

-(Integral(c**2/(sin(e + f*x) - 1), x) + Integral(d**2*x**2/(sin(e + f*x) - 1), x) + Integral(2*c*d*x/(sin(e +
f*x) - 1), x))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^2/(a-a*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate(-(d*x + c)^2/(a*sin(f*x + e) - a), x)