\(\int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx\) [175]

Optimal result

Integrand size = 18, antiderivative size = 254 \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}-\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {d \log (a+b \sinh (e+f x))}{\left (a^2+b^2\right ) f^2}+\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\frac {a (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {a (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {b (c+d x) \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}+\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac {a d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac {d \log (a+b \sinh (e+f x))}{f^2 \left (a^2+b^2\right )} \]

[In]

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

[Out]

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

Rule 31

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

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]

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

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.76 \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\frac {d \log (a+b \sinh (e+f x))+\frac {a \left (f (c+d x) \left (\log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )-\log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )+d \operatorname {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-d \operatorname {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-\frac {b f (c+d x) \cosh (e+f x)}{a+b \sinh (e+f x)}}{\left (a^2+b^2\right ) f^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(518\) vs. \(2(234)=468\).

Time = 1.53 (sec) , antiderivative size = 519, normalized size of antiderivative = 2.04

method	result	size
risch	\(\frac {2 \left (d x +c \right ) \left (a \,{\mathrm e}^{f x +e}-b \right )}{f \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 f x +2 e}+2 a \,{\mathrm e}^{f x +e}-b \right )}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a^{2}+b^{2}\right )}+\frac {d \ln \left (b \,{\mathrm e}^{2 f x +2 e}+2 a \,{\mathrm e}^{f x +e}-b \right )}{f^{2} \left (a^{2}+b^{2}\right )}-\frac {2 a c \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {a d \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a d \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {a d \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a d \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {a d \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a d \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {2 a d e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(519\)

[In]

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

[Out]

2*(d*x+c)*(a*exp(f*x+e)-b)/f/(a^2+b^2)/(b*exp(2*f*x+2*e)+2*a*exp(f*x+e)-b)-2/f^2/(a^2+b^2)*d*ln(exp(f*x+e))+1/
f^2/(a^2+b^2)*d*ln(b*exp(2*f*x+2*e)+2*a*exp(f*x+e)-b)-2/f/(a^2+b^2)^(3/2)*a*c*arctanh(1/2*(2*b*exp(f*x+e)+2*a)
/(a^2+b^2)^(1/2))+1/f/(a^2+b^2)^(3/2)*a*d*ln((-b*exp(f*x+e)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/f/(a^
2+b^2)^(3/2)*a*d*ln((b*exp(f*x+e)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/f^2/(a^2+b^2)^(3/2)*a*d*ln((-b*e
xp(f*x+e)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*e-1/f^2/(a^2+b^2)^(3/2)*a*d*ln((b*exp(f*x+e)+(a^2+b^2)^(1/2
)+a)/(a+(a^2+b^2)^(1/2)))*e+1/f^2/(a^2+b^2)^(3/2)*a*d*dilog((-b*exp(f*x+e)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1
/2)))-1/f^2/(a^2+b^2)^(3/2)*a*d*dilog((b*exp(f*x+e)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+2/f^2/(a^2+b^2)^(3
/2)*a*d*e*arctanh(1/2*(2*b*exp(f*x+e)+2*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. 1717 vs. \(2 (232) = 464\).

Time = 0.27 (sec) , antiderivative size = 1717, normalized size of antiderivative = 6.76 \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*(a^2*b + b^3)*d*e - 2*(a^2*b + b^3)*c*f - 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e)*cosh(f*x + e)^2 - 2*(
(a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e)*sinh(f*x + e)^2 + (a*b^2*d*cosh(f*x + e)^2 + a*b^2*d*sinh(f*x + e)^2
+ 2*a^2*b*d*cosh(f*x + e) - a*b^2*d + 2*(a*b^2*d*cosh(f*x + e) + a^2*b*d)*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2)
*dilog((a*cosh(f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b +
1) - (a*b^2*d*cosh(f*x + e)^2 + a*b^2*d*sinh(f*x + e)^2 + 2*a^2*b*d*cosh(f*x + e) - a*b^2*d + 2*(a*b^2*d*cosh
(f*x + e) + a^2*b*d)*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(f*x + e) + a*sinh(f*x + e) - (b*cosh(f
*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (a*b^2*d*f*x + a*b^2*d*e - (a*b^2*d*f*x + a*b^2
*d*e)*cosh(f*x + e)^2 - (a*b^2*d*f*x + a*b^2*d*e)*sinh(f*x + e)^2 - 2*(a^2*b*d*f*x + a^2*b*d*e)*cosh(f*x + e)
- 2*(a^2*b*d*f*x + a^2*b*d*e + (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2)*l
og(-(a*cosh(f*x + e) + a*sinh(f*x + e) + (b*cosh(f*x + e) + b*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b) + (
a*b^2*d*f*x + a*b^2*d*e - (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*x + e)^2 - (a*b^2*d*f*x + a*b^2*d*e)*sinh(f*x + e)^
2 - 2*(a^2*b*d*f*x + a^2*b*d*e)*cosh(f*x + e) - 2*(a^2*b*d*f*x + a^2*b*d*e + (a*b^2*d*f*x + a*b^2*d*e)*cosh(f*
x + e))*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(f*x + e) + a*sinh(f*x + e) - (b*cosh(f*x + e) + b*si
nh(f*x + e))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*((a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b^2)*d*e - (a^3 + a*b^2)*c*f)
*cosh(f*x + e) + ((a^2*b + b^3)*d*cosh(f*x + e)^2 + (a^2*b + b^3)*d*sinh(f*x + e)^2 + 2*(a^3 + a*b^2)*d*cosh(f
*x + e) - (a^2*b + b^3)*d + 2*((a^2*b + b^3)*d*cosh(f*x + e) + (a^3 + a*b^2)*d)*sinh(f*x + e) - (a*b^2*d*e - a
*b^2*c*f - (a*b^2*d*e - a*b^2*c*f)*cosh(f*x + e)^2 - (a*b^2*d*e - a*b^2*c*f)*sinh(f*x + e)^2 - 2*(a^2*b*d*e -
a^2*b*c*f)*cosh(f*x + e) - 2*(a^2*b*d*e - a^2*b*c*f + (a*b^2*d*e - a*b^2*c*f)*cosh(f*x + e))*sinh(f*x + e))*sq
rt((a^2 + b^2)/b^2))*log(2*b*cosh(f*x + e) + 2*b*sinh(f*x + e) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + ((a^2*b +
b^3)*d*cosh(f*x + e)^2 + (a^2*b + b^3)*d*sinh(f*x + e)^2 + 2*(a^3 + a*b^2)*d*cosh(f*x + e) - (a^2*b + b^3)*d +
2*((a^2*b + b^3)*d*cosh(f*x + e) + (a^3 + a*b^2)*d)*sinh(f*x + e) + (a*b^2*d*e - a*b^2*c*f - (a*b^2*d*e - a*b
^2*c*f)*cosh(f*x + e)^2 - (a*b^2*d*e - a*b^2*c*f)*sinh(f*x + e)^2 - 2*(a^2*b*d*e - a^2*b*c*f)*cosh(f*x + e) -
2*(a^2*b*d*e - a^2*b*c*f + (a*b^2*d*e - a*b^2*c*f)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 + b^2)/b^2))*log(2*
b*cosh(f*x + e) + 2*b*sinh(f*x + e) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*((a^3 + a*b^2)*d*f*x + 2*(a^3 + a*b
^2)*d*e - (a^3 + a*b^2)*c*f + 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e)*cosh(f*x + e))*sinh(f*x + e))/((a^4*
b + 2*a^2*b^3 + b^5)*f^2*cosh(f*x + e)^2 + (a^4*b + 2*a^2*b^3 + b^5)*f^2*sinh(f*x + e)^2 + 2*(a^5 + 2*a^3*b^2
+ a*b^4)*f^2*cosh(f*x + e) - (a^4*b + 2*a^2*b^3 + b^5)*f^2 + 2*((a^4*b + 2*a^2*b^3 + b^5)*f^2*cosh(f*x + e) +
(a^5 + 2*a^3*b^2 + a*b^4)*f^2)*sinh(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\int { \frac {d x + c}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

(2*a*f*integrate(x*e^(f*x + e)/(a^2*b*f*e^(2*f*x + 2*e) + b^3*f*e^(2*f*x + 2*e) + 2*a^3*f*e^(f*x + e) + 2*a*b^
2*f*e^(f*x + e) - a^2*b*f - b^3*f), x) + b*(a*log((b*e^(f*x + e) + a - sqrt(a^2 + b^2))/(b*e^(f*x + e) + a + s
qrt(a^2 + b^2)))/((a^2*b + b^3)*sqrt(a^2 + b^2)*f^2) - 2*(f*x + e)/((a^2*b + b^3)*f^2) + log(b*e^(2*f*x + 2*e)
+ 2*a*e^(f*x + e) - b)/((a^2*b + b^3)*f^2)) - 2*(a*x*e^(f*x + e) - b*x)/(a^2*b*f + b^3*f - (a^2*b*f*e^(2*e) +
b^3*f*e^(2*e))*e^(2*f*x) - 2*(a^3*f*e^e + a*b^2*f*e^e)*e^(f*x)) - a*log((b*e^(f*x + e) + a - sqrt(a^2 + b^2))
/(b*e^(f*x + e) + a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*f^2))*d + c*(a*log((b*e^(-f*x - e) - a - sqrt(a^2 +
b^2))/(b*e^(-f*x - e) - a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*f) - 2*(a*e^(-f*x - e) + b)/((a^2*b + b^3 +
2*(a^3 + a*b^2)*e^(-f*x - e) - (a^2*b + b^3)*e^(-2*f*x - 2*e))*f))

Giac [F]

\[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\int { \frac {d x + c}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*x+c)/(a+b*sinh(f*x+e))^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx=\int \frac {c+d\,x}{{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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