Integrand size = 16, antiderivative size = 256 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=-\frac {6 d^2 (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b^3}+\frac {(c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {(c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 d^3 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^4}+\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {3 d^3 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^4}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}+\frac {3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}+\frac {3 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}-\frac {3 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4} \] Output:
-6*d^2*(d*x+c)*arctanh(exp(b*x+a))/b^3+(d*x+c)^3*arctanh(exp(b*x+a))/b-3/2 *d*(d*x+c)^2*csch(b*x+a)/b^2-1/2*(d*x+c)^3*coth(b*x+a)*csch(b*x+a)/b-3*d^3 *polylog(2,-exp(b*x+a))/b^4+3/2*d*(d*x+c)^2*polylog(2,-exp(b*x+a))/b^2+3*d ^3*polylog(2,exp(b*x+a))/b^4-3/2*d*(d*x+c)^2*polylog(2,exp(b*x+a))/b^2-3*d ^2*(d*x+c)*polylog(3,-exp(b*x+a))/b^3+3*d^2*(d*x+c)*polylog(3,exp(b*x+a))/ b^3+3*d^3*polylog(4,-exp(b*x+a))/b^4-3*d^3*polylog(4,exp(b*x+a))/b^4
Time = 2.56 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.72 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=-\frac {b^2 (c+d x)^2 (3 d+b (c+d x) \coth (a+b x)) \text {csch}(a+b x)+b^3 c^3 \log \left (1-e^{a+b x}\right )-6 b c d^2 \log \left (1-e^{a+b x}\right )+3 b^3 c^2 d x \log \left (1-e^{a+b x}\right )-6 b d^3 x \log \left (1-e^{a+b x}\right )+3 b^3 c d^2 x^2 \log \left (1-e^{a+b x}\right )+b^3 d^3 x^3 \log \left (1-e^{a+b x}\right )-b^3 c^3 \log \left (1+e^{a+b x}\right )+6 b c d^2 \log \left (1+e^{a+b x}\right )-3 b^3 c^2 d x \log \left (1+e^{a+b x}\right )+6 b d^3 x \log \left (1+e^{a+b x}\right )-3 b^3 c d^2 x^2 \log \left (1+e^{a+b x}\right )-b^3 d^3 x^3 \log \left (1+e^{a+b x}\right )-3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )+3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,e^{a+b x}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,-e^{a+b x}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,e^{a+b x}\right )-6 d^3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )+6 d^3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{2 b^4} \] Input:
Integrate[(c + d*x)^3*Csch[a + b*x]^3,x]
Output:
-1/2*(b^2*(c + d*x)^2*(3*d + b*(c + d*x)*Coth[a + b*x])*Csch[a + b*x] + b^ 3*c^3*Log[1 - E^(a + b*x)] - 6*b*c*d^2*Log[1 - E^(a + b*x)] + 3*b^3*c^2*d* x*Log[1 - E^(a + b*x)] - 6*b*d^3*x*Log[1 - E^(a + b*x)] + 3*b^3*c*d^2*x^2* Log[1 - E^(a + b*x)] + b^3*d^3*x^3*Log[1 - E^(a + b*x)] - b^3*c^3*Log[1 + E^(a + b*x)] + 6*b*c*d^2*Log[1 + E^(a + b*x)] - 3*b^3*c^2*d*x*Log[1 + E^(a + b*x)] + 6*b*d^3*x*Log[1 + E^(a + b*x)] - 3*b^3*c*d^2*x^2*Log[1 + E^(a + b*x)] - b^3*d^3*x^3*Log[1 + E^(a + b*x)] - 3*d*(-2*d^2 + b^2*(c + d*x)^2) *PolyLog[2, -E^(a + b*x)] + 3*d*(-2*d^2 + b^2*(c + d*x)^2)*PolyLog[2, E^(a + b*x)] + 6*b*c*d^2*PolyLog[3, -E^(a + b*x)] + 6*b*d^3*x*PolyLog[3, -E^(a + b*x)] - 6*b*c*d^2*PolyLog[3, E^(a + b*x)] - 6*b*d^3*x*PolyLog[3, E^(a + b*x)] - 6*d^3*PolyLog[4, -E^(a + b*x)] + 6*d^3*PolyLog[4, E^(a + b*x)])/b ^4
Result contains complex when optimal does not.
Time = 1.21 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {3042, 26, 4674, 26, 3042, 26, 4670, 2715, 2838, 3011, 7163, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (c+d x)^3 \text {csch}^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i (c+d x)^3 \csc (i a+i b x)^3dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int (c+d x)^3 \csc (i a+i b x)^3dx\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle -i \left (-\frac {3 d^2 \int -i (c+d x) \text {csch}(a+b x)dx}{b^2}+\frac {1}{2} \int -i (c+d x)^3 \text {csch}(a+b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {3 i d^2 \int (c+d x) \text {csch}(a+b x)dx}{b^2}-\frac {1}{2} i \int (c+d x)^3 \text {csch}(a+b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {3 i d^2 \int i (c+d x) \csc (i a+i b x)dx}{b^2}-\frac {1}{2} i \int i (c+d x)^3 \csc (i a+i b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (-\frac {3 d^2 \int (c+d x) \csc (i a+i b x)dx}{b^2}+\frac {1}{2} \int (c+d x)^3 \csc (i a+i b x)dx-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle -i \left (-\frac {3 d^2 \left (\frac {i d \int \log \left (1-e^{a+b x}\right )dx}{b}-\frac {i d \int \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b^2}+\frac {1}{2} \left (\frac {3 i d \int (c+d x)^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i d \int (c+d x)^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -i \left (-\frac {3 d^2 \left (\frac {i d \int e^{-a-b x} \log \left (1-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {i d \int e^{-a-b x} \log \left (1+e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b^2}+\frac {1}{2} \left (\frac {3 i d \int (c+d x)^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i d \int (c+d x)^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -i \left (\frac {1}{2} \left (\frac {3 i d \int (c+d x)^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i d \int (c+d x)^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -i \left (\frac {1}{2} \left (-\frac {3 i d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle -i \left (\frac {1}{2} \left (-\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {d \int \operatorname {PolyLog}\left (3,-e^{a+b x}\right )dx}{b}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {d \int \operatorname {PolyLog}\left (3,e^{a+b x}\right )dx}{b}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -i \left (\frac {1}{2} \left (-\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {d \int e^{-a-b x} \operatorname {PolyLog}\left (3,-e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {d \int e^{-a-b x} \operatorname {PolyLog}\left (3,e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -i \left (-\frac {3 d^2 \left (\frac {2 i (c+d x) \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i d \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i d \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}+\frac {1}{2} \left (\frac {2 i (c+d x)^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {d \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^2}\right )}{b}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}\right )-\frac {3 i d (c+d x)^2 \text {csch}(a+b x)}{2 b^2}-\frac {i (c+d x)^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
Input:
Int[(c + d*x)^3*Csch[a + b*x]^3,x]
Output:
(-I)*((((-3*I)/2)*d*(c + d*x)^2*Csch[a + b*x])/b^2 - ((I/2)*(c + d*x)^3*Co th[a + b*x]*Csch[a + b*x])/b - (3*d^2*(((2*I)*(c + d*x)*ArcTanh[E^(a + b*x )])/b + (I*d*PolyLog[2, -E^(a + b*x)])/b^2 - (I*d*PolyLog[2, E^(a + b*x)]) /b^2))/b^2 + (((2*I)*(c + d*x)^3*ArcTanh[E^(a + b*x)])/b - ((3*I)*d*(-(((c + d*x)^2*PolyLog[2, -E^(a + b*x)])/b) + (2*d*(((c + d*x)*PolyLog[3, -E^(a + b*x)])/b - (d*PolyLog[4, -E^(a + b*x)])/b^2))/b))/b + ((3*I)*d*(-(((c + d*x)^2*PolyLog[2, E^(a + b*x)])/b) + (2*d*(((c + d*x)*PolyLog[3, E^(a + b *x)])/b - (d*PolyLog[4, E^(a + b*x)])/b^2))/b))/b)/2)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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]
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] + Simp[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]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[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]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(238)=476\).
Time = 0.25 (sec) , antiderivative size = 876, normalized size of antiderivative = 3.42
Input:
int((d*x+c)^3*csch(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
3*d^3*polylog(4,-exp(b*x+a))/b^4-3*d^3*polylog(4,exp(b*x+a))/b^4-3*d^3*pol ylog(2,-exp(b*x+a))/b^4+3*d^3*polylog(2,exp(b*x+a))/b^4+6/b^4*d^3*a*arctan h(exp(b*x+a))-6/b^3*d^2*c*arctanh(exp(b*x+a))+3/b^3*d^3*ln(1-exp(b*x+a))*x +3/b^4*d^3*ln(1-exp(b*x+a))*a-3/b^3*d^3*ln(exp(b*x+a)+1)*x-3/b^4*d^3*ln(ex p(b*x+a)+1)*a-3/2/b*c*d^2*ln(1-exp(b*x+a))*x^2+3/2/b^3*c*d^2*ln(1-exp(b*x+ a))*a^2-3/b^2*c*d^2*polylog(2,exp(b*x+a))*x+3/2/b*c*d^2*ln(exp(b*x+a)+1)*x ^2-3/2/b^3*c*d^2*ln(exp(b*x+a)+1)*a^2+3/b^2*c*d^2*polylog(2,-exp(b*x+a))*x +3/2/b*c^2*d*ln(exp(b*x+a)+1)*x+3/2/b^2*c^2*d*ln(exp(b*x+a)+1)*a-3/2/b*c^2 *d*ln(1-exp(b*x+a))*x-3/2/b^2*c^2*d*ln(1-exp(b*x+a))*a-3/b^2*d*a*c^2*arcta nh(exp(b*x+a))+3/b^3*d^2*a^2*c*arctanh(exp(b*x+a))-exp(b*x+a)*(exp(2*b*x+2 *a)*b*d^3*x^3+3*exp(2*b*x+2*a)*b*c*d^2*x^2+3*exp(2*b*x+2*a)*b*c^2*d*x+b*d^ 3*x^3+3*exp(2*b*x+2*a)*d^3*x^2+exp(2*b*x+2*a)*b*c^3+3*b*c*d^2*x^2+6*exp(2* b*x+2*a)*c*d^2*x+3*b*c^2*d*x+3*exp(2*b*x+2*a)*c^2*d-3*d^3*x^2+b*c^3-6*c*d^ 2*x-3*c^2*d)/b^2/(exp(2*b*x+2*a)-1)^2+1/b*c^3*arctanh(exp(b*x+a))-1/2/b*d^ 3*ln(1-exp(b*x+a))*x^3-1/2/b^4*d^3*ln(1-exp(b*x+a))*a^3-3/2/b^2*d^3*polylo g(2,exp(b*x+a))*x^2+3/b^3*d^3*polylog(3,exp(b*x+a))*x+1/2/b*d^3*ln(exp(b*x +a)+1)*x^3+1/2/b^4*d^3*ln(exp(b*x+a)+1)*a^3+3/2/b^2*d^3*polylog(2,-exp(b*x +a))*x^2-3/b^3*d^3*polylog(3,-exp(b*x+a))*x-3/2/b^2*c^2*d*polylog(2,exp(b* x+a))+3/2/b^2*c^2*d*polylog(2,-exp(b*x+a))+3/b^3*c*d^2*polylog(3,exp(b*x+a ))-3/b^3*c*d^2*polylog(3,-exp(b*x+a))-1/b^4*d^3*a^3*arctanh(exp(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 4008 vs. \(2 (234) = 468\).
Time = 0.13 (sec) , antiderivative size = 4008, normalized size of antiderivative = 15.66 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^3*csch(b*x+a)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\int \left (c + d x\right )^{3} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \] Input:
integrate((d*x+c)**3*csch(b*x+a)**3,x)
Output:
Integral((c + d*x)**3*csch(a + b*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 605 vs. \(2 (234) = 468\).
Time = 0.26 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.36 \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^3*csch(b*x+a)^3,x, algorithm="maxima")
Output:
1/2*c^3*(log(e^(-b*x - a) + 1)/b - log(e^(-b*x - a) - 1)/b + 2*(e^(-b*x - a) + e^(-3*b*x - 3*a))/(b*(2*e^(-2*b*x - 2*a) - e^(-4*b*x - 4*a) - 1))) + 3/2*(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)))*c*d^2/b^3 - 3/2*(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)))*c*d^2/b^3 - 3*c*d^2*log(e^( b*x + a) + 1)/b^3 + 3*c*d^2*log(e^(b*x + a) - 1)/b^3 - ((b*d^3*x^3*e^(3*a) + 3*c^2*d*e^(3*a) + 3*(b*c*d^2 + d^3)*x^2*e^(3*a) + 3*(b*c^2*d + 2*c*d^2) *x*e^(3*a))*e^(3*b*x) + (b*d^3*x^3*e^a - 3*c^2*d*e^a + 3*(b*c*d^2 - d^3)*x ^2*e^a + 3*(b*c^2*d - 2*c*d^2)*x*e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) - 2*b^ 2*e^(2*b*x + 2*a) + b^2) + 1/2*(b^3*x^3*log(e^(b*x + a) + 1) + 3*b^2*x^2*d ilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))*d^3/b^4 - 1/2*(b^3*x^3*log(-e^(b*x + a) + 1) + 3*b^2*x^2*dilog(e^( b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^(b*x + a)))*d^3 /b^4 + 3/2*(b^2*c^2*d - 2*d^3)*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^4 - 3/2*(b^2*c^2*d - 2*d^3)*(b*x*log(-e^(b*x + a) + 1) + dilog(e^( b*x + a)))/b^4
\[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \operatorname {csch}\left (b x + a\right )^{3} \,d x } \] Input:
integrate((d*x+c)^3*csch(b*x+a)^3,x, algorithm="giac")
Output:
integrate((d*x + c)^3*csch(b*x + a)^3, x)
Timed out. \[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \] Input:
int((c + d*x)^3/sinh(a + b*x)^3,x)
Output:
int((c + d*x)^3/sinh(a + b*x)^3, x)
\[ \int (c+d x)^3 \text {csch}^3(a+b x) \, dx=\text {too large to display} \] Input:
int((d*x+c)^3*csch(b*x+a)^3,x)
Output:
( - 48*e**(5*a + 4*b*x)*int((e**(b*x)*x**3)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**4*d**3 - 144*e**(5*a + 4*b*x)*int ((e**(b*x)*x**2)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b* x) - 1),x)*b**4*c*d**2 - 192*e**(5*a + 4*b*x)*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**3*d**3 - 14 4*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**4*c**2*d - 384*e**(5*a + 4*b*x)*int((e**( b*x)*x)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x )*b**3*c*d**2 - 128*e**(5*a + 4*b*x)*int((e**(b*x)*x)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**2*d**3 - 9*e**(4*a + 4* b*x)*log(e**(a + b*x) - 1)*b**3*c**3 - 36*e**(4*a + 4*b*x)*log(e**(a + b*x ) - 1)*b**2*c**2*d - 24*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1)*b*c*d**2 - 8*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1)*d**3 + 9*e**(4*a + 4*b*x)*log(e** (a + b*x) + 1)*b**3*c**3 + 36*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1)*b**2* c**2*d + 24*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1)*b*c*d**2 + 8*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1)*d**3 - 18*e**(3*a + 3*b*x)*b**3*c**3 - 72*e** (3*a + 3*b*x)*b**2*c**2*d - 48*e**(3*a + 3*b*x)*b*c*d**2 - 16*e**(3*a + 3* b*x)*d**3 + 96*e**(3*a + 2*b*x)*int((e**(b*x)*x**3)/(e**(6*a + 6*b*x) - 3* e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**4*d**3 + 288*e**(3*a + 2* b*x)*int((e**(b*x)*x**2)/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**...