3.1.24 \(\int (c+d x)^2 \text {csch}(a+b x) \, dx\) [24]

Optimal. Leaf size=99 \[ -\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac {2 d^2 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \]

[Out]

-2*(d*x+c)^2*arctanh(exp(b*x+a))/b-2*d*(d*x+c)*polylog(2,-exp(b*x+a))/b^2+2*d*(d*x+c)*polylog(2,exp(b*x+a))/b^
2+2*d^2*polylog(3,-exp(b*x+a))/b^3-2*d^2*polylog(3,exp(b*x+a))/b^3

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Rubi [A]
time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4267, 2611, 2320, 6724} \begin {gather*} \frac {2 d^2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*Csch[a + b*x],x]

[Out]

(-2*(c + d*x)^2*ArcTanh[E^(a + b*x)])/b - (2*d*(c + d*x)*PolyLog[2, -E^(a + b*x)])/b^2 + (2*d*(c + d*x)*PolyLo
g[2, E^(a + b*x)])/b^2 + (2*d^2*PolyLog[3, -E^(a + b*x)])/b^3 - (2*d^2*PolyLog[3, E^(a + b*x)])/b^3

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int (c+d x)^2 \text {csch}(a+b x) \, dx &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {(2 d) \int (c+d x) \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {(2 d) \int (c+d x) \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\left (2 d^2\right ) \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (2 d^2\right ) \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac {2 (c+d x)^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {2 d (c+d x) \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {2 d (c+d x) \text {Li}_2\left (e^{a+b x}\right )}{b^2}+\frac {2 d^2 \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {2 d^2 \text {Li}_3\left (e^{a+b x}\right )}{b^3}\\ \end {align*}

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Mathematica [A]
time = 1.59, size = 170, normalized size = 1.72 \begin {gather*} \frac {-2 b^2 c^2 \tanh ^{-1}\left (e^{a+b x}\right )+2 b^2 c d x \log \left (1-e^{a+b x}\right )+b^2 d^2 x^2 \log \left (1-e^{a+b x}\right )-2 b^2 c d x \log \left (1+e^{a+b x}\right )-b^2 d^2 x^2 \log \left (1+e^{a+b x}\right )-2 b d (c+d x) \text {PolyLog}\left (2,-e^{a+b x}\right )+2 b d (c+d x) \text {PolyLog}\left (2,e^{a+b x}\right )+2 d^2 \text {PolyLog}\left (3,-e^{a+b x}\right )-2 d^2 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(305\) vs. \(2(94)=188\).
time = 0.77, size = 306, normalized size = 3.09

method	result	size
risch	\(-\frac {2 d^{2} a^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 d c \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 d c \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {d^{2} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {2 d^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b}+\frac {d^{2} \ln \left ({\mathrm e}^{b x +a}+1\right ) a^{2}}{b^{3}}-\frac {2 d^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 d^{2} \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 d^{2} \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 d c \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b}+\frac {2 d c \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{2}}-\frac {2 d c \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b}-\frac {2 d c \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{2}}+\frac {4 d a c \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {2 c^{2} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b}\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (92) = 184\).
time = 0.33, size = 195, normalized size = 1.97 \begin {gather*} -c^{2} {\left (\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} - \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{b}\right )} - \frac {2 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} + \frac {2 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )} c d}{b^{2}} - \frac {{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )} d^{2}}{b^{3}} + \frac {{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )} d^{2}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^2*(log(e^(-b*x - a) + 1)/b - log(e^(-b*x - a) - 1)/b) - 2*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))*
c*d/b^2 + 2*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))*c*d/b^2 - (b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*d
ilog(-e^(b*x + a)) - 2*polylog(3, -e^(b*x + a)))*d^2/b^3 + (b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilog(e^(b*x
+ a)) - 2*polylog(3, e^(b*x + a)))*d^2/b^3

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (92) = 184\).
time = 0.34, size = 242, normalized size = 2.44 \begin {gather*} -\frac {2 \, d^{2} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \, d^{2} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \, {\left (b d^{2} x + b c d\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 2 \, {\left (b d^{2} x + b c d\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*d^2*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 2*d^2*polylog(3, -cosh(b*x + a) - sinh(b*x + a)) - 2*(b*d^
2*x + b*c*d)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 2*(b*d^2*x + b*c*d)*dilog(-cosh(b*x + a) - sinh(b*x + a))
+ (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - (b^2*c^2 - 2*a*b*c*d + a^2*d^
2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*log(-cosh(b*x +
a) - sinh(b*x + a) + 1))/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)**2*csch(a + b*x), 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*csch(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^2*csch(b*x + a), 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}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^2/sinh(a + b*x),x)

[Out]

int((c + d*x)^2/sinh(a + b*x), x)

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