3.1.28 \(\int (c+d x)^3 \text {csch}^2(a+b x) \, dx\) [28]

Optimal. Leaf size=103 \[ -\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}-\frac {3 d^3 \text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^4} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4269, 3797, 2221, 2611, 2320, 6724} \begin {gather*} -\frac {3 d^3 \text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {(c+d x)^3}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Csch[a + b*x]^2,x]

[Out]

-((c + d*x)^3/b) - ((c + d*x)^3*Coth[a + b*x])/b + (3*d*(c + d*x)^2*Log[1 - E^(2*(a + b*x))])/b^2 + (3*d^2*(c
+ d*x)*PolyLog[2, E^(2*(a + b*x))])/b^3 - (3*d^3*PolyLog[3, E^(2*(a + b*x))])/(2*b^4)

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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(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*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, 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]

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)^3 \text {csch}^2(a+b x) \, dx &=-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {(3 d) \int (c+d x)^2 \coth (a+b x) \, dx}{b}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}-\frac {(6 d) \int \frac {e^{2 (a+b x)} (c+d x)^2}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac {\left (6 d^2\right ) \int (c+d x) \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac {\left (3 d^3\right ) \int \text {Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=-\frac {(c+d x)^3}{b}-\frac {(c+d x)^3 \coth (a+b x)}{b}+\frac {3 d (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac {3 d^2 (c+d x) \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac {3 d^3 \text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^4}\\ \end {align*}

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Mathematica [A]
time = 1.36, size = 133, normalized size = 1.29 \begin {gather*} \frac {d \left (-\frac {4 b^3 e^{2 a} x \left (3 c^2+3 c d x+d^2 x^2\right )}{-1+e^{2 a}}+6 b^2 (c+d x)^2 \log \left (1-e^{2 (a+b x)}\right )+6 b d (c+d x) \text {PolyLog}\left (2,e^{2 (a+b x)}\right )-3 d^2 \text {PolyLog}\left (3,e^{2 (a+b x)}\right )\right )}{2 b^4}+\frac {(c+d x)^3 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Csch[a + b*x]^2,x]

[Out]

(d*((-4*b^3*E^(2*a)*x*(3*c^2 + 3*c*d*x + d^2*x^2))/(-1 + E^(2*a)) + 6*b^2*(c + d*x)^2*Log[1 - E^(2*(a + b*x))]
+ 6*b*d*(c + d*x)*PolyLog[2, E^(2*(a + b*x))] - 3*d^2*PolyLog[3, E^(2*(a + b*x))]))/(2*b^4) + ((c + d*x)^3*Cs
ch[a]*Csch[a + b*x]*Sinh[b*x])/b

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(472\) vs. \(2(101)=202\).
time = 0.94, size = 473, normalized size = 4.59

method	result	size
risch	\(-\frac {12 d^{2} a c x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {6 d^{2} c \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{2}}-\frac {6 d^{2} a c \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}+\frac {12 d^{2} a c \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 d^{3} \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 d^{3} \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {2 d^{3} x^{3}}{b}+\frac {4 d^{3} a^{3}}{b^{4}}-\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{\left ({\mathrm e}^{2 b x +2 a}-1\right ) b}+\frac {3 d^{3} \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}-\frac {6 d^{2} c \,x^{2}}{b}-\frac {6 d^{2} c \,a^{2}}{b^{3}}+\frac {6 d^{3} a^{2} x}{b^{3}}+\frac {6 d^{3} \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {6 d^{2} c \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{2} c \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {6 d^{3} \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {3 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}+\frac {3 d^{3} a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {6 d^{3} a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}-\frac {3 d^{3} \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}-\frac {6 d \,c^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}\)	\(473\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^3*csch(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

-12/b^2*d^2*a*c*x+6/b^2*d^2*c*ln(1-exp(b*x+a))*x+6/b^3*d^2*c*ln(1-exp(b*x+a))*a+6/b^2*d^2*c*ln(exp(b*x+a)+1)*x
-6/b^3*d^2*a*c*ln(exp(b*x+a)-1)+12/b^3*d^2*a*c*ln(exp(b*x+a))-2/b*d^3*x^3+4/b^4*d^3*a^3-6/b^4*d^3*polylog(3,ex
p(b*x+a))-6/b^4*d^3*polylog(3,-exp(b*x+a))-2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/(exp(2*b*x+2*a)-1)/b+3/b^2*d^
3*ln(exp(b*x+a)+1)*x^2+6/b^3*d^3*polylog(2,-exp(b*x+a))*x+3/b^2*d*c^2*ln(exp(b*x+a)+1)-6/b*d^2*c*x^2-6/b^3*d^2
*c*a^2+6/b^3*d^3*a^2*x+3/b^2*d*c^2*ln(exp(b*x+a)-1)+3/b^4*d^3*a^2*ln(exp(b*x+a)-1)-6/b^4*d^3*a^2*ln(exp(b*x+a)
)+6/b^3*d^2*c*polylog(2,exp(b*x+a))+6/b^3*d^2*c*polylog(2,-exp(b*x+a))+3/b^2*d^3*ln(1-exp(b*x+a))*x^2-3/b^4*d^
3*ln(1-exp(b*x+a))*a^2+6/b^3*d^3*polylog(2,exp(b*x+a))*x-6/b^2*d*c^2*ln(exp(b*x+a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (100) = 200\).
time = 0.41, size = 320, normalized size = 3.11 \begin {gather*} -3 \, c^{2} d {\left (\frac {2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac {6 \, {\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^{2}}{b^{3}} + \frac {6 \, {\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^{2}}{b^{3}} + \frac {2 \, c^{3}}{b {\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} - \frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2}\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} + \frac {3 \, {\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^{3}}{b^{4}} + \frac {3 \, {\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^{3}}{b^{4}} - \frac {2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2}\right )}}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*csch(b*x+a)^2,x, algorithm="maxima")

[Out]

-3*c^2*d*(2*x*e^(2*b*x + 2*a)/(b*e^(2*b*x + 2*a) - b) - log((e^(b*x + a) + 1)*e^(-a))/b^2 - log((e^(b*x + a) -
1)*e^(-a))/b^2) + 6*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))*c*d^2/b^3 + 6*(b*x*log(-e^(b*x + a) + 1)
+ dilog(e^(b*x + a)))*c*d^2/b^3 + 2*c^3/(b*(e^(-2*b*x - 2*a) - 1)) - 2*(d^3*x^3 + 3*c*d^2*x^2)/(b*e^(2*b*x +
2*a) - 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^3/b^4
+ 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^3/b^4 - 2*(b^3*d^
3*x^3 + 3*b^3*c*d^2*x^2)/b^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1159 vs. \(2 (100) = 200\).
time = 0.35, size = 1159, normalized size = 11.25 \begin {gather*} -\frac {2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} + 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \cosh \left (b x + a\right )^{2} + 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (b d^{3} x + b c d^{2} - {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b d^{3} x + b c d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (b d^{3} x + b c d^{2} - {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b d^{3} x + b c d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3} - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3} - {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + 2 \, a b c d^{2} - a^{2} d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left (d^{3} \cosh \left (b x + a\right )^{2} + 2 \, d^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{3} \sinh \left (b x + a\right )^{2} - d^{3}\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (d^{3} \cosh \left (b x + a\right )^{2} + 2 \, d^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{3} \sinh \left (b x + a\right )^{2} - d^{3}\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{4} \cosh \left (b x + a\right )^{2} + 2 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )^{2} - b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*csch(b*x+a)^2,x, algorithm="fricas")

[Out]

-(2*b^3*c^3 - 6*a*b^2*c^2*d + 6*a^2*b*c*d^2 - 2*a^3*d^3 + 2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*cosh(b*x + a)^2 + 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*cosh(b*x + a)*sinh(b*x + a) + 2*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3
*c^2*d*x + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3)*sinh(b*x + a)^2 + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^
2)*cosh(b*x + a)^2 - 2*(b*d^3*x + b*c*d^2)*cosh(b*x + a)*sinh(b*x + a) - (b*d^3*x + b*c*d^2)*sinh(b*x + a)^2)*
dilog(cosh(b*x + a) + sinh(b*x + a)) + 6*(b*d^3*x + b*c*d^2 - (b*d^3*x + b*c*d^2)*cosh(b*x + a)^2 - 2*(b*d^3*x
+ b*c*d^2)*cosh(b*x + a)*sinh(b*x + a) - (b*d^3*x + b*c*d^2)*sinh(b*x + a)^2)*dilog(-cosh(b*x + a) - sinh(b*x
+ a)) + 3*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cosh(b*x + a)^
2 - 2*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2*c^2*d)*cosh(b*x + a)*sinh(b*x + a) - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + b
^2*c^2*d)*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 3*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3 - (b^
2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cosh(b*x + a)^2 - 2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*cosh(b*x + a)*sinh(b*
x + a) - (b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 3*(b^2*
d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3 - (b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cosh(b
*x + a)^2 - 2*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*cosh(b*x + a)*sinh(b*x + a) - (b^2*d^3*x^2
+ 2*b^2*c*d^2*x + 2*a*b*c*d^2 - a^2*d^3)*sinh(b*x + a)^2)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 6*(d^3*co
sh(b*x + a)^2 + 2*d^3*cosh(b*x + a)*sinh(b*x + a) + d^3*sinh(b*x + a)^2 - d^3)*polylog(3, cosh(b*x + a) + sinh
(b*x + a)) + 6*(d^3*cosh(b*x + a)^2 + 2*d^3*cosh(b*x + a)*sinh(b*x + a) + d^3*sinh(b*x + a)^2 - d^3)*polylog(3
, -cosh(b*x + a) - sinh(b*x + a)))/(b^4*cosh(b*x + a)^2 + 2*b^4*cosh(b*x + a)*sinh(b*x + a) + b^4*sinh(b*x + a
)^2 - b^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c + d x\right )^{3} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**3*csch(b*x+a)**2,x)

[Out]

Integral((c + d*x)**3*csch(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)^3*csch(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3*csch(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 )}^3}{{\mathrm {sinh}\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)^3/sinh(a + b*x)^2,x)

[Out]

int((c + d*x)^3/sinh(a + b*x)^2, x)

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