\(\int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx\) [178]

Optimal result

Integrand size = 18, antiderivative size = 544 \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}-\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}+\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2}+\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))} \]

[Out]

3/2*a*f*ln(a+b*sinh(d*x+c))/(a^2+b^2)^2/d^2+3/2*a^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(
5/2)/d-1/2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d-3/2*a^2*(f*x+e)*ln(1+b*exp(d*x+c)/
(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(5/2)/d+1/2*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d+3/
2*a^2*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(5/2)/d^2-1/2*f*polylog(2,-b*exp(d*x+c)/(a-(a^2
+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-3/2*a^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(5/2)/d^2+1
/2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-1/2*b*(f*x+e)*cosh(d*x+c)/(a^2+b^2)/d/(a
+b*sinh(d*x+c))^2-1/2*f/(a^2+b^2)/d^2/(a+b*sinh(d*x+c))-3/2*a*b*(f*x+e)*cosh(d*x+c)/(a^2+b^2)^2/d/(a+b*sinh(d*
x+c))

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {3406, 3405, 3403, 2296, 2221, 2317, 2438, 2747, 31, 6874, 32} \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 d^2 \left (a^2+b^2\right )^{5/2}}-\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 d^2 \left (a^2+b^2\right )^{5/2}}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 d^2 \left (a^2+b^2\right )^{3/2}}-\frac {f}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 d^2 \left (a^2+b^2\right )^2}+\frac {3 a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{2 d \left (a^2+b^2\right )^{5/2}}-\frac {3 a^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{2 d \left (a^2+b^2\right )^{5/2}}-\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{2 d \left (a^2+b^2\right )^{3/2}}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{2 d \left (a^2+b^2\right )^{3/2}}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right )^2 (a+b \sinh (c+d x))}-\frac {b (e+f x) \cosh (c+d x)}{2 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^2} \]

[In]

Int[(e + f*x)/(a + b*Sinh[c + d*x])^3,x]

[Out]

(3*a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(5/2)*d) - ((e + f*x)*Log[1 +
(b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(3/2)*d) - (3*a^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(2*(a^2 + b^2)^(5/2)*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*(
a^2 + b^2)^(3/2)*d) + (3*a*f*Log[a + b*Sinh[c + d*x]])/(2*(a^2 + b^2)^2*d^2) + (3*a^2*f*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/(2*(a^2 + b^2)^(5/2)*d^2) - (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^
2]))])/(2*(a^2 + b^2)^(3/2)*d^2) - (3*a^2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(2*(a^2 + b^
2)^(5/2)*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(2*(a^2 + b^2)^(3/2)*d^2) - (b*(e + f
*x)*Cosh[c + d*x])/(2*(a^2 + b^2)*d*(a + b*Sinh[c + d*x])^2) - f/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x])) - (
3*a*b*(e + f*x)*Cosh[c + d*x])/(2*(a^2 + b^2)^2*d*(a + b*Sinh[c + d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3405

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[b*(c + d*x)^m*(Cos[
e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f*x]))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[b*d*(m/(f*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/(a + b*Sin[e + f*x])), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3406

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*(c + d*x)^m
*Cos[e + f*x]*((a + b*Sin[e + f*x])^(n + 1)/(f*(n + 1)*(a^2 - b^2))), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^
m*(a + b*Sin[e + f*x])^(n + 1), x], x] - Dist[b*((n + 2)/((n + 1)*(a^2 - b^2))), Int[(c + d*x)^m*Sin[e + f*x]*
(a + b*Sin[e + f*x])^(n + 1), x], x] + Dist[b*d*(m/(f*(n + 1)*(a^2 - b^2))), Int[(c + d*x)^(m - 1)*Cos[e + f*x
]*(a + b*Sin[e + f*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && ILtQ[n, -2] &&
IGtQ[m, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}+\frac {a \int \frac {e+f x}{(a+b \sinh (c+d x))^2} \, dx}{a^2+b^2}-\frac {b \int \frac {(e+f x) \sinh (c+d x)}{(a+b \sinh (c+d x))^2} \, dx}{2 \left (a^2+b^2\right )}+\frac {(b f) \int \frac {\cosh (c+d x)}{(a+b \sinh (c+d x))^2} \, dx}{2 \left (a^2+b^2\right ) d} \\ & = -\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {a b (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}+\frac {a^2 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}-\frac {b \int \left (-\frac {a (e+f x)}{b (a+b \sinh (c+d x))^2}+\frac {e+f x}{b (a+b \sinh (c+d x))}\right ) \, dx}{2 \left (a^2+b^2\right )}+\frac {f \text {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {(a b f) \int \frac {\cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {a b (e+f x) \cosh (c+d x)}{\left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}+\frac {\left (2 a^2\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}-\frac {\int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{2 \left (a^2+b^2\right )}+\frac {a \int \frac {e+f x}{(a+b \sinh (c+d x))^2} \, dx}{2 \left (a^2+b^2\right )}+\frac {(a f) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d^2} \\ & = \frac {a f \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}+\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{5/2}}-\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{5/2}}+\frac {a^2 \int \frac {e+f x}{a+b \sinh (c+d x)} \, dx}{2 \left (a^2+b^2\right )^2}-\frac {\int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2+b^2}+\frac {(a b f) \int \frac {\cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{2 \left (a^2+b^2\right )^2 d} \\ & = \frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}-\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}+\frac {a f \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}-\frac {b \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {b \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {(a f) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d^2}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{5/2} d}+\frac {\left (a^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{5/2} d} \\ & = \frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}-\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d}+\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}+\frac {\left (a^2 b\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{5/2}}-\frac {\left (a^2 b\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{5/2}}-\frac {\left (a^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{5/2} d^2}+\frac {\left (a^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{5/2} d^2}+\frac {f \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {f \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{2 \left (a^2+b^2\right )^{3/2} d} \\ & = \frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}-\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}+\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d^2}-\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}+\frac {f \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {f \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (a^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{2 \left (a^2+b^2\right )^{5/2} d}+\frac {\left (a^2 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{2 \left (a^2+b^2\right )^{5/2} d} \\ & = \frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}-\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}+\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d^2}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2} d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))}-\frac {\left (a^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}+\frac {\left (a^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2} \\ & = \frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}-\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}-\frac {3 a^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d}+\frac {(e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d}+\frac {3 a f \log (a+b \sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d^2}+\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}-\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{5/2} d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \left (a^2+b^2\right )^{3/2} d^2}-\frac {b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^2}-\frac {f}{2 \left (a^2+b^2\right ) d^2 (a+b \sinh (c+d x))}-\frac {3 a b (e+f x) \cosh (c+d x)}{2 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.03 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.42 \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=-\frac {-\frac {-3 a \sqrt {-\left (a^2+b^2\right )^2} f (c+d x)+6 a^2 \sqrt {a^2+b^2} f \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )-4 a^2 \sqrt {-a^2-b^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 b^2 \sqrt {-a^2-b^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+6 a^2 \sqrt {-a^2-b^2} f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+4 a^2 \sqrt {-a^2-b^2} c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 b^2 \sqrt {-a^2-b^2} c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 a^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-b^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 a^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+b^2 \sqrt {-a^2-b^2} f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 a \sqrt {-\left (a^2+b^2\right )^2} f \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+\sqrt {-a^2-b^2} \left (2 a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+\sqrt {-a^2-b^2} \left (-2 a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}+\frac {b \left (a^2+b^2\right ) d (e+f x) \cosh (c+d x)}{(a+b \sinh (c+d x))^2}+\frac {\left (a^2+b^2\right ) f+3 a b d (e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}}{2 \left (a^2+b^2\right )^2 d^2} \]

[In]

Integrate[(e + f*x)/(a + b*Sinh[c + d*x])^3,x]

[Out]

-1/2*(-((-3*a*Sqrt[-(a^2 + b^2)^2]*f*(c + d*x) + 6*a^2*Sqrt[a^2 + b^2]*f*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2
- b^2]] - 4*a^2*Sqrt[-a^2 - b^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*b^2*Sqrt[-a^2 - b^2]*d*e
*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 6*a^2*Sqrt[-a^2 - b^2]*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2
+ b^2]] + 4*a^2*Sqrt[-a^2 - b^2]*c*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*b^2*Sqrt[-a^2 - b^2]*c*f
*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*a^2*Sqrt[-a^2 - b^2]*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2])] - b^2*Sqrt[-a^2 - b^2]*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 2*a^2*
Sqrt[-a^2 - b^2]*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + b^2*Sqrt[-a^2 - b^2]*f*(c + d*x)
*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*a*Sqrt[-(a^2 + b^2)^2]*f*Log[2*a*E^(c + d*x) + b*(-1 + E^(
2*(c + d*x)))] + Sqrt[-a^2 - b^2]*(2*a^2 - b^2)*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + Sqrt[-a
^2 - b^2]*(-2*a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/Sqrt[-(a^2 + b^2)^2]) + (b*(a
^2 + b^2)*d*(e + f*x)*Cosh[c + d*x])/(a + b*Sinh[c + d*x])^2 + ((a^2 + b^2)*f + 3*a*b*d*(e + f*x)*Cosh[c + d*x
])/(a + b*Sinh[c + d*x]))/((a^2 + b^2)^2*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1231\) vs. \(2(480)=960\).

Time = 2.05 (sec) , antiderivative size = 1232, normalized size of antiderivative = 2.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(1232\)

[In]

int((f*x+e)/(a+b*sinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

(2*a^2*b*d*f*x*exp(3*d*x+3*c)-b^3*d*f*x*exp(3*d*x+3*c)+6*a^3*d*f*x*exp(2*d*x+2*c)+2*a^2*b*d*e*exp(3*d*x+3*c)-3
*a*b^2*d*f*x*exp(2*d*x+2*c)-b^3*d*e*exp(3*d*x+3*c)+6*a^3*d*e*exp(2*d*x+2*c)-10*a^2*b*d*f*x*exp(d*x+c)-a^2*b*f*
exp(3*d*x+3*c)-3*a*b^2*d*e*exp(2*d*x+2*c)-b^3*d*f*x*exp(d*x+c)-b^3*f*exp(3*d*x+3*c)-2*a^3*f*exp(2*d*x+2*c)-10*
a^2*b*d*e*exp(d*x+c)+3*a*b^2*d*f*x-2*a*b^2*f*exp(2*d*x+2*c)-b^3*d*e*exp(d*x+c)+a^2*b*f*exp(d*x+c)+3*d*e*a*b^2+
b^3*f*exp(d*x+c))/d^2/(a^2+b^2)^2/(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)^2-1/(a^2+b^2)^(5/2)/d^2*b^2*f*c*arctanh(
1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-3/(a^2+b^2)^2/d^2*a*f*ln(exp(d*x+c))+3/2/(a^2+b^2)^2/d^2*a*f*ln(b*ex
p(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/(a^2+b^2)^(5/2)/d*a^2*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/(
a^2+b^2)^(5/2)/d^2*a^2*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(5/2)/d*a^2*f*ln((-b*
exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(5/2)/d*a^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/
2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2)/d^2*a^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1
/2)))*c-1/(a^2+b^2)^(5/2)/d^2*a^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+1/(a^2+b^2)^(5/
2)/d^2*a^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)^(5/2)/d^2*a^2*f*dilog((
b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/(a^2+b^2)^(5/2)/d*b^2*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a
)/(a^2+b^2)^(1/2))-1/2/(a^2+b^2)^(5/2)/d*b^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/
2/(a^2+b^2)^(5/2)/d*b^2*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/2/(a^2+b^2)^(5/2)/d^2*b
^2*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/2/(a^2+b^2)^(5/2)/d^2*b^2*f*ln((b*exp(d*x+
c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/2/(a^2+b^2)^(5/2)/d^2*b^2*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2
)-a)/(-a+(a^2+b^2)^(1/2)))+1/2/(a^2+b^2)^(5/2)/d^2*b^2*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(
1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6396 vs. \(2 (476) = 952\).

Time = 0.35 (sec) , antiderivative size = 6396, normalized size of antiderivative = 11.76 \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)/(a+b*sinh(d*x+c))**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {f x + e}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="maxima")

[Out]

1/2*(4*a^2*d*integrate(x*e^(d*x + c)/(a^4*b*d*e^(2*d*x + 2*c) + 2*a^2*b^3*d*e^(2*d*x + 2*c) + b^5*d*e^(2*d*x +
2*c) + 2*a^5*d*e^(d*x + c) + 4*a^3*b^2*d*e^(d*x + c) + 2*a*b^4*d*e^(d*x + c) - a^4*b*d - 2*a^2*b^3*d - b^5*d)
, x) - 2*b^2*d*integrate(x*e^(d*x + c)/(a^4*b*d*e^(2*d*x + 2*c) + 2*a^2*b^3*d*e^(2*d*x + 2*c) + b^5*d*e^(2*d*x
+ 2*c) + 2*a^5*d*e^(d*x + c) + 4*a^3*b^2*d*e^(d*x + c) + 2*a*b^4*d*e^(d*x + c) - a^4*b*d - 2*a^2*b^3*d - b^5*
d), x) + 3*a*b*(a*log((b*e^(d*x + c) + a - sqrt(a^2 + b^2))/(b*e^(d*x + c) + a + sqrt(a^2 + b^2)))/((a^4*b + 2
*a^2*b^3 + b^5)*sqrt(a^2 + b^2)*d^2) - 2*(d*x + c)/((a^4*b + 2*a^2*b^3 + b^5)*d^2) + log(b*e^(2*d*x + 2*c) + 2
*a*e^(d*x + c) - b)/((a^4*b + 2*a^2*b^3 + b^5)*d^2)) + 2*(3*a*b^2*d*x - (a^2*b*e^(3*c) + b^3*e^(3*c) - (2*a^2*
b*d*e^(3*c) - b^3*d*e^(3*c))*x)*e^(3*d*x) - (2*a^3*e^(2*c) + 2*a*b^2*e^(2*c) - 3*(2*a^3*d*e^(2*c) - a*b^2*d*e^
(2*c))*x)*e^(2*d*x) + (a^2*b*e^c + b^3*e^c - (10*a^2*b*d*e^c + b^3*d*e^c)*x)*e^(d*x))/(a^4*b^2*d^2 + 2*a^2*b^4
*d^2 + b^6*d^2 + (a^4*b^2*d^2*e^(4*c) + 2*a^2*b^4*d^2*e^(4*c) + b^6*d^2*e^(4*c))*e^(4*d*x) + 4*(a^5*b*d^2*e^(3
*c) + 2*a^3*b^3*d^2*e^(3*c) + a*b^5*d^2*e^(3*c))*e^(3*d*x) + 2*(2*a^6*d^2*e^(2*c) + 3*a^4*b^2*d^2*e^(2*c) - b^
6*d^2*e^(2*c))*e^(2*d*x) - 4*(a^5*b*d^2*e^c + 2*a^3*b^3*d^2*e^c + a*b^5*d^2*e^c)*e^(d*x)) - 3*a^2*log((b*e^(d*
x + c) + a - sqrt(a^2 + b^2))/(b*e^(d*x + c) + a + sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)*
d^2))*f + 1/2*e*((2*a^2 - b^2)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2
)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)*d) - 2*(3*a*b^2 + (10*a^2*b + b^3)*e^(-d*x - c) + 3*(2*a^3 - a*b^
2)*e^(-2*d*x - 2*c) - (2*a^2*b - b^3)*e^(-3*d*x - 3*c))/((a^4*b^2 + 2*a^2*b^4 + b^6 + 4*(a^5*b + 2*a^3*b^3 + a
*b^5)*e^(-d*x - c) + 2*(2*a^6 + 3*a^4*b^2 - b^6)*e^(-2*d*x - 2*c) - 4*(a^5*b + 2*a^3*b^3 + a*b^5)*e^(-3*d*x -
3*c) + (a^4*b^2 + 2*a^2*b^4 + b^6)*e^(-4*d*x - 4*c))*d))

Giac [F]

\[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\int { \frac {f x + e}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate((f*x+e)/(a+b*sinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((f*x + e)/(b*sinh(d*x + c) + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x}{(a+b \sinh (c+d x))^3} \, dx=\int \frac {e+f\,x}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^3} \,d x \]

[In]

int((e + f*x)/(a + b*sinh(c + d*x))^3,x)

[Out]

int((e + f*x)/(a + b*sinh(c + d*x))^3, x)