3.1.29 \(\int (c+d x)^2 \csc ^2(a+b x) \, dx\) [29]

Optimal. Leaf size=83 \[ -\frac {i (c+d x)^2}{b}-\frac {(c+d x)^2 \cot (a+b x)}{b}+\frac {2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {i d^2 \text {Li}_2\left (e^{2 i (a+b x)}\right )}{b^3} \]

[Out]

-I*(d*x+c)^2/b-(d*x+c)^2*cot(b*x+a)/b+2*d*(d*x+c)*ln(1-exp(2*I*(b*x+a)))/b^2-I*d^2*polylog(2,exp(2*I*(b*x+a)))
/b^3

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Rubi [A]
time = 0.10, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4269, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {i d^2 \text {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^3}+\frac {2 d (c+d x) \log \left (1-e^{2 i (a+b x)}\right )}{b^2}-\frac {(c+d x)^2 \cot (a+b x)}{b}-\frac {i (c+d x)^2}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Csc[a + b*x]^2,x]

[Out]

((-I)*(c + d*x)^2)/b - ((c + d*x)^2*Cot[a + b*x])/b + (2*d*(c + d*x)*Log[1 - E^((2*I)*(a + b*x))])/b^2 - (I*d^
2*PolyLog[2, E^((2*I)*(a + b*x))])/b^3

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 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 3798

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 4269

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]

Rubi steps

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(181\) vs. \(2(83)=166\).
time = 2.54, size = 181, normalized size = 2.18 \begin {gather*} \frac {\csc (a) \left (-2 b c d (b x \cos (a)-\log (\sin (a+b x)) \sin (a))+d^2 \left (-b^2 e^{i \tan ^{-1}(\tan (a))} x^2 \cos (a) \sqrt {\sec ^2(a)}-\left (-i b x \left (\pi -2 \tan ^{-1}(\tan (a))\right )-\pi \log \left (1+e^{-2 i b x}\right )-2 \left (b x+\tan ^{-1}(\tan (a))\right ) \log \left (1-e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )+\pi \log (\cos (b x))+2 \tan ^{-1}(\tan (a)) \log \left (\sin \left (b x+\tan ^{-1}(\tan (a))\right )\right )+i \text {Li}_2\left (e^{2 i \left (b x+\tan ^{-1}(\tan (a))\right )}\right )\right ) \sin (a)\right )+b^2 (c+d x)^2 \csc (a+b x) \sin (b x)\right )}{b^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Csc[a]*(-2*b*c*d*(b*x*Cos[a] - Log[Sin[a + b*x]]*Sin[a]) + d^2*(-(b^2*E^(I*ArcTan[Tan[a]])*x^2*Cos[a]*Sqrt[Se
c[a]^2]) - ((-I)*b*x*(Pi - 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E
^((2*I)*(b*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*Poly
Log[2, E^((2*I)*(b*x + ArcTan[Tan[a]]))])*Sin[a]) + b^2*(c + d*x)^2*Csc[a + b*x]*Sin[b*x]))/b^3

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (77 ) = 154\).
time = 0.06, size = 276, normalized size = 3.33

method	result	size
risch	\(-\frac {2 i \left (d^{2} x^{2}+2 c d x +c^{2}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )}+\frac {2 d c \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{2}}-\frac {4 d c \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{2}}+\frac {2 d c \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{b^{2}}-\frac {2 i d^{2} x^{2}}{b}-\frac {4 i d^{2} a x}{b^{2}}-\frac {2 i d^{2} a^{2}}{b^{3}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) x}{b^{2}}+\frac {2 d^{2} \ln \left (1-{\mathrm e}^{i \left (b x +a \right )}\right ) a}{b^{3}}-\frac {2 i d^{2} \polylog \left (2, {\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}+\frac {2 d^{2} \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right ) x}{b^{2}}-\frac {2 i d^{2} \polylog \left (2, -{\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}-\frac {2 d^{2} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{b^{3}}+\frac {4 d^{2} a \ln \left ({\mathrm e}^{i \left (b x +a \right )}\right )}{b^{3}}\)	\(276\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (74) = 148\).
time = 0.38, size = 552, normalized size = 6.65 \begin {gather*} -\frac {2 \, b^{2} c^{2} + 2 \, {\left (b d^{2} x + b c d - {\left (b d^{2} x + b c d\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (-i \, b d^{2} x - i \, b c d\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) - 2 \, {\left (b c d \cos \left (2 \, b x + 2 \, a\right ) + i \, b c d \sin \left (2 \, b x + 2 \, a\right ) - b c d\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) - 1\right ) + 2 \, {\left (b d^{2} x \cos \left (2 \, b x + 2 \, a\right ) + i \, b d^{2} x \sin \left (2 \, b x + 2 \, a\right ) - b d^{2} x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x\right )} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, {\left (d^{2} \cos \left (2 \, b x + 2 \, a\right ) + i \, d^{2} \sin \left (2 \, b x + 2 \, a\right ) - d^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 2 \, {\left (d^{2} \cos \left (2 \, b x + 2 \, a\right ) + i \, d^{2} \sin \left (2 \, b x + 2 \, a\right ) - d^{2}\right )} {\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) - {\left (i \, b d^{2} x + i \, b c d + {\left (-i \, b d^{2} x - i \, b c d\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - {\left (i \, b d^{2} x + i \, b c d + {\left (-i \, b d^{2} x - i \, b c d\right )} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b d^{2} x + b c d\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) + 2 \, {\left (i \, b^{2} d^{2} x^{2} + 2 i \, b^{2} c d x\right )} \sin \left (2 \, b x + 2 \, a\right )}{-i \, b^{3} \cos \left (2 \, b x + 2 \, a\right ) + b^{3} \sin \left (2 \, b x + 2 \, a\right ) + i \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^2*c^2 + 2*(b*d^2*x + b*c*d - (b*d^2*x + b*c*d)*cos(2*b*x + 2*a) + (-I*b*d^2*x - I*b*c*d)*sin(2*b*x + 2*a
))*arctan2(sin(b*x + a), cos(b*x + a) + 1) - 2*(b*c*d*cos(2*b*x + 2*a) + I*b*c*d*sin(2*b*x + 2*a) - b*c*d)*arc
tan2(sin(b*x + a), cos(b*x + a) - 1) + 2*(b*d^2*x*cos(2*b*x + 2*a) + I*b*d^2*x*sin(2*b*x + 2*a) - b*d^2*x)*arc
tan2(sin(b*x + a), -cos(b*x + a) + 1) + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x)*cos(2*b*x + 2*a) + 2*(d^2*cos(2*b*x + 2*
a) + I*d^2*sin(2*b*x + 2*a) - d^2)*dilog(-e^(I*b*x + I*a)) + 2*(d^2*cos(2*b*x + 2*a) + I*d^2*sin(2*b*x + 2*a)
- d^2)*dilog(e^(I*b*x + I*a)) - (I*b*d^2*x + I*b*c*d + (-I*b*d^2*x - I*b*c*d)*cos(2*b*x + 2*a) + (b*d^2*x + b*
c*d)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - (I*b*d^2*x + I*b*c*d + (-I*
b*d^2*x - I*b*c*d)*cos(2*b*x + 2*a) + (b*d^2*x + b*c*d)*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2
- 2*cos(b*x + a) + 1) + 2*(I*b^2*d^2*x^2 + 2*I*b^2*c*d*x)*sin(2*b*x + 2*a))/(-I*b^3*cos(2*b*x + 2*a) + b^3*sin
(2*b*x + 2*a) + I*b^3)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (74) = 148\).
time = 0.38, size = 379, normalized size = 4.57 \begin {gather*} \frac {-i \, d^{2} {\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d^{2} {\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + i \, d^{2} {\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) - i \, d^{2} {\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + b c d\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b c d - a d^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) + {\left (b c d - a d^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) - \frac {1}{2} i \, \sin \left (b x + a\right ) + \frac {1}{2}\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) + {\left (b d^{2} x + a d^{2}\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \cos \left (b x + a\right )}{b^{3} \sin \left (b x + a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c + d x\right )^{2} \csc ^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{{\sin \left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/sin(a + b*x)^2,x)

[Out]

int((c + d*x)^2/sin(a + b*x)^2, x)

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