3.1.38 \(\int (c+d x)^2 (a+b \coth (e+f x)) \, dx\) [38]

Optimal. Leaf size=101 \[ \frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )}{f^2}-\frac {b d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{2 f^3} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3803, 3797, 2221, 2611, 2320, 6724} \begin {gather*} \frac {a (c+d x)^3}{3 d}+\frac {b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}+\frac {b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {b (c+d x)^3}{3 d}-\frac {b d^2 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^2*(a + b*Coth[e + f*x]),x]

[Out]

(a*(c + d*x)^3)/(3*d) - (b*(c + d*x)^3)/(3*d) + (b*(c + d*x)^2*Log[1 - E^(2*(e + f*x))])/f + (b*d*(c + d*x)*Po
lyLog[2, E^(2*(e + f*x))])/f^2 - (b*d^2*PolyLog[3, E^(2*(e + f*x))])/(2*f^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 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 3803

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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 (a+b \coth (e+f x)) \, dx &=\int \left (a (c+d x)^2+b (c+d x)^2 \coth (e+f x)\right ) \, dx\\ &=\frac {a (c+d x)^3}{3 d}+b \int (c+d x)^2 \coth (e+f x) \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}-(2 b) \int \frac {e^{2 (e+f x)} (c+d x)^2}{1-e^{2 (e+f x)}} \, dx\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}-\frac {(2 b d) \int (c+d x) \log \left (1-e^{2 (e+f x)}\right ) \, dx}{f}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {\left (b d^2\right ) \int \text {Li}_2\left (e^{2 (e+f x)}\right ) \, dx}{f^2}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (e+f x)}\right )}{2 f^3}\\ &=\frac {a (c+d x)^3}{3 d}-\frac {b (c+d x)^3}{3 d}+\frac {b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )}{f}+\frac {b d (c+d x) \text {Li}_2\left (e^{2 (e+f x)}\right )}{f^2}-\frac {b d^2 \text {Li}_3\left (e^{2 (e+f x)}\right )}{2 f^3}\\ \end {align*}

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Mathematica [A]
time = 1.37, size = 102, normalized size = 1.01 \begin {gather*} \frac {2 f^2 \left ((a-b) f x \left (3 c^2+3 c d x+d^2 x^2\right )+3 b (c+d x)^2 \log \left (1-e^{2 (e+f x)}\right )\right )+6 b d f (c+d x) \text {PolyLog}\left (2,e^{2 (e+f x)}\right )-3 b d^2 \text {PolyLog}\left (3,e^{2 (e+f x)}\right )}{6 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^2*(a + b*Coth[e + f*x]),x]

[Out]

(2*f^2*((a - b)*f*x*(3*c^2 + 3*c*d*x + d^2*x^2) + 3*b*(c + d*x)^2*Log[1 - E^(2*(e + f*x))]) + 6*b*d*f*(c + d*x
)*PolyLog[2, E^(2*(e + f*x))] - 3*b*d^2*PolyLog[3, E^(2*(e + f*x))])/(6*f^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(464\) vs. \(2(95)=190\).
time = 2.15, size = 465, normalized size = 4.60

method	result	size
risch	\(\frac {2 b d c \polylog \left (2, {\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b c d \polylog \left (2, -{\mathrm e}^{f x +e}\right )}{f^{2}}+\frac {2 b \,d^{2} \polylog \left (2, {\mathrm e}^{f x +e}\right ) x}{f^{2}}-\frac {b \,d^{2} e^{2} \ln \left (1-{\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {2 b d e c \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{2}}-b c d \,x^{2}+b \,c^{2} x -\frac {b \,d^{2} x^{3}}{3}-\frac {2 b \,c^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f}+\frac {b \,c^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f}+\frac {2 b c d \ln \left ({\mathrm e}^{f x +e}+1\right ) x}{f}+\frac {2 b d c \ln \left (1-{\mathrm e}^{f x +e}\right ) x}{f}+\frac {2 b d c \ln \left (1-{\mathrm e}^{f x +e}\right ) e}{f^{2}}+\frac {4 b d e c \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2}}-\frac {4 b c d e x}{f}-\frac {2 b c d \,e^{2}}{f^{2}}+d a c \,x^{2}+a \,c^{2} x +\frac {2 b \,e^{2} d^{2} x}{f^{2}}+\frac {b \,d^{2} \ln \left (1-{\mathrm e}^{f x +e}\right ) x^{2}}{f}+\frac {b \,d^{2} \ln \left ({\mathrm e}^{f x +e}+1\right ) x^{2}}{f}+\frac {2 b \,d^{2} \polylog \left (2, -{\mathrm e}^{f x +e}\right ) x}{f^{2}}+\frac {b \,c^{3}}{3 d}+\frac {d^{2} a \,x^{3}}{3}+\frac {c^{3} a}{3 d}-\frac {2 b \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {b \,d^{2} e^{2} \ln \left ({\mathrm e}^{f x +e}-1\right )}{f^{3}}+\frac {4 b \,e^{3} d^{2}}{3 f^{3}}-\frac {2 b \,d^{2} \polylog \left (3, {\mathrm e}^{f x +e}\right )}{f^{3}}-\frac {2 b \,d^{2} \polylog \left (3, -{\mathrm e}^{f x +e}\right )}{f^{3}}+\frac {b \,c^{2} \ln \left ({\mathrm e}^{f x +e}+1\right )}{f}\)	\(465\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (97) = 194\).
time = 0.33, size = 251, normalized size = 2.49 \begin {gather*} \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{3} \, b d^{2} x^{3} + a c d x^{2} + b c d x^{2} + a c^{2} x + \frac {b c^{2} \log \left (\sinh \left (f x + e\right )\right )}{f} + \frac {2 \, {\left (f x \log \left (e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (f x + e\right )}\right )\right )} b c d}{f^{2}} + \frac {2 \, {\left (f x \log \left (-e^{\left (f x + e\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (f x + e\right )}\right )\right )} b c d}{f^{2}} + \frac {{\left (f^{2} x^{2} \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (-e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (f x + e\right )})\right )} b d^{2}}{f^{3}} + \frac {{\left (f^{2} x^{2} \log \left (-e^{\left (f x + e\right )} + 1\right ) + 2 \, f x {\rm Li}_2\left (e^{\left (f x + e\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (f x + e\right )})\right )} b d^{2}}{f^{3}} - \frac {2 \, {\left (b d^{2} f^{3} x^{3} + 3 \, b c d f^{3} x^{2}\right )}}{3 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (97) = 194\).
time = 0.38, size = 404, normalized size = 4.00 \begin {gather*} \frac {{\left (a - b\right )} d^{2} f^{3} x^{3} + 3 \, {\left (a - b\right )} c d f^{3} x^{2} + 3 \, {\left (a - b\right )} c^{2} f^{3} x - 6 \, b d^{2} {\rm polylog}\left (3, \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) - 6 \, b d^{2} {\rm polylog}\left (3, -\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} {\rm Li}_2\left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 6 \, {\left (b d^{2} f x + b c d f\right )} {\rm Li}_2\left (-\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right ) + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + b c^{2} f^{2}\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right ) + 3 \, {\left (b c^{2} f^{2} - 2 \, b c d f \cosh \left (1\right ) + b d^{2} \cosh \left (1\right )^{2} + b d^{2} \sinh \left (1\right )^{2} - 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 1\right ) + 3 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f \cosh \left (1\right ) - b d^{2} \cosh \left (1\right )^{2} - b d^{2} \sinh \left (1\right )^{2} + 2 \, {\left (b c d f - b d^{2} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \log \left (-\cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) + 1\right )}{3 \, f^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((a - b)*d^2*f^3*x^3 + 3*(a - b)*c*d*f^3*x^2 + 3*(a - b)*c^2*f^3*x - 6*b*d^2*polylog(3, cosh(f*x + cosh(1)
+ sinh(1)) + sinh(f*x + cosh(1) + sinh(1))) - 6*b*d^2*polylog(3, -cosh(f*x + cosh(1) + sinh(1)) - sinh(f*x +
cosh(1) + sinh(1))) + 6*(b*d^2*f*x + b*c*d*f)*dilog(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(
1))) + 6*(b*d^2*f*x + b*c*d*f)*dilog(-cosh(f*x + cosh(1) + sinh(1)) - sinh(f*x + cosh(1) + sinh(1))) + 3*(b*d^
2*f^2*x^2 + 2*b*c*d*f^2*x + b*c^2*f^2)*log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)) + 1)
+ 3*(b*c^2*f^2 - 2*b*c*d*f*cosh(1) + b*d^2*cosh(1)^2 + b*d^2*sinh(1)^2 - 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*
log(cosh(f*x + cosh(1) + sinh(1)) + sinh(f*x + cosh(1) + sinh(1)) - 1) + 3*(b*d^2*f^2*x^2 + 2*b*c*d*f^2*x + 2*
b*c*d*f*cosh(1) - b*d^2*cosh(1)^2 - b*d^2*sinh(1)^2 + 2*(b*c*d*f - b*d^2*cosh(1))*sinh(1))*log(-cosh(f*x + cos
h(1) + sinh(1)) - sinh(f*x + cosh(1) + sinh(1)) + 1))/f^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**2*(a+b*coth(f*x+e)),x)

[Out]

Integral((a + b*coth(e + f*x))*(c + d*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*(a+b*coth(f*x+e)),x, algorithm="giac")

[Out]

integrate((d*x + c)^2*(b*coth(f*x + e) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(e + f*x))*(c + d*x)^2,x)

[Out]

int((a + b*coth(e + f*x))*(c + d*x)^2, x)

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