\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx\) [131]

Optimal result

Integrand size = 15, antiderivative size = 128 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=-\frac {\left (2 a^2 A-A b^2+3 a b B\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))} \]

[Out]

-(2*A*a^2-A*b^2+3*B*a*b)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)-1/2*(A*b-B*a)*cosh(x)/(a^2
+b^2)/(a+b*sinh(x))^2-1/2*(3*A*a*b-B*a^2+2*B*b^2)*cosh(x)/(a^2+b^2)^2/(a+b*sinh(x))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2833, 12, 2739, 632, 212} \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=-\frac {\left (2 a^2 A+3 a b B-A b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {\cosh (x) \left (a^2 (-B)+3 a A b+2 b^2 B\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\cosh (x) (A b-a B)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2} \]

[In]

Int[(A + B*Sinh[x])/(a + b*Sinh[x])^3,x]

[Out]

-(((2*a^2*A - A*b^2 + 3*a*b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - ((A*b - a*B)*C
osh[x])/(2*(a^2 + b^2)*(a + b*Sinh[x])^2) - ((3*a*A*b - a^2*B + 2*b^2*B)*Cosh[x])/(2*(a^2 + b^2)^2*(a + b*Sinh
[x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2833

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\int \frac {-2 (a A+b B)+(A b-a B) \sinh (x)}{(a+b \sinh (x))^2} \, dx}{2 \left (a^2+b^2\right )} \\ & = -\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\int \frac {2 a^2 A-A b^2+3 a b B}{a+b \sinh (x)} \, dx}{2 \left (a^2+b^2\right )^2} \\ & = -\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\left (2 a^2 A-A b^2+3 a b B\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{2 \left (a^2+b^2\right )^2} \\ & = -\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\left (2 a^2 A-A b^2+3 a b B\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2} \\ & = -\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {\left (2 \left (2 a^2 A-A b^2+3 a b B\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2} \\ & = -\frac {\left (2 a^2 A-A b^2+3 a b B\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\frac {\frac {2 \left (2 a^2 A-A b^2+3 a b B\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {\left (a^2+b^2\right ) (-A b+a B) \cosh (x)}{(a+b \sinh (x))^2}+\frac {\left (-3 a A b+a^2 B-2 b^2 B\right ) \cosh (x)}{a+b \sinh (x)}}{2 \left (a^2+b^2\right )^2} \]

[In]

Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^3,x]

[Out]

((2*(2*a^2*A - A*b^2 + 3*a*b*B)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + ((a^2 + b^2)*(-
(A*b) + a*B)*Cosh[x])/(a + b*Sinh[x])^2 + ((-3*a*A*b + a^2*B - 2*b^2*B)*Cosh[x])/(a + b*Sinh[x]))/(2*(a^2 + b^
2)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(118)=236\).

Time = 0.66 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.45

method	result	size
default	\(-\frac {2 \left (-\frac {b \left (5 A \,a^{2} b +2 A \,b^{3}-3 a^{3} B \right ) \tanh \left (\frac {x}{2}\right )^{3}}{2 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (4 A \,a^{4} b -7 a^{2} A \,b^{3}-2 A \,b^{5}-2 B \,a^{5}+5 a^{3} B \,b^{2}-2 B a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b \left (11 A \,a^{2} b +2 A \,b^{3}-5 a^{3} B +4 B a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}+\frac {4 A \,a^{2} b +A \,b^{3}-2 a^{3} B +B a \,b^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,b^{2}+3 b B a \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}\)	\(314\)
risch	\(\frac {2 A \,a^{2} b^{2} {\mathrm e}^{3 x}-A \,b^{4} {\mathrm e}^{3 x}+3 B a \,b^{3} {\mathrm e}^{3 x}+6 A \,a^{3} b \,{\mathrm e}^{2 x}-3 A a \,b^{3} {\mathrm e}^{2 x}-2 B \,a^{4} {\mathrm e}^{2 x}+5 B \,a^{2} b^{2} {\mathrm e}^{2 x}-2 B \,b^{4} {\mathrm e}^{2 x}-10 A \,a^{2} b^{2} {\mathrm e}^{x}-A \,b^{4} {\mathrm e}^{x}+4 B \,a^{3} b \,{\mathrm e}^{x}-5 B a \,b^{3} {\mathrm e}^{x}+3 A a \,b^{3}-B \,a^{2} b^{2}+2 B \,b^{4}}{b \left (a^{2}+b^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) A \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) b B a}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) A \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) b B a}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\)	\(586\)

[In]

int((A+B*sinh(x))/(a+b*sinh(x))^3,x,method=_RETURNVERBOSE)

[Out]

-2*(-1/2*b*(5*A*a^2*b+2*A*b^3-3*B*a^3)/a/(a^4+2*a^2*b^2+b^4)*tanh(1/2*x)^3-1/2*(4*A*a^4*b-7*A*a^2*b^3-2*A*b^5-
2*B*a^5+5*B*a^3*b^2-2*B*a*b^4)/(a^4+2*a^2*b^2+b^4)/a^2*tanh(1/2*x)^2+1/2*b*(11*A*a^2*b+2*A*b^3-5*B*a^3+4*B*a*b
^2)/(a^4+2*a^2*b^2+b^4)/a*tanh(1/2*x)+1/2*(4*A*a^2*b+A*b^3-2*B*a^3+B*a*b^2)/(a^4+2*a^2*b^2+b^4))/(tanh(1/2*x)^
2*a-2*b*tanh(1/2*x)-a)^2+(2*A*a^2-A*b^2+3*B*a*b)/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2
*x)-2*b)/(a^2+b^2)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1614 vs. \(2 (119) = 238\).

Time = 0.36 (sec) , antiderivative size = 1614, normalized size of antiderivative = 12.61 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="fricas")

[Out]

-1/2*(2*B*a^4*b^2 - 6*A*a^3*b^3 - 2*B*a^2*b^4 - 6*A*a*b^5 - 4*B*b^6 - 2*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*a^2*b^4
+ 3*B*a*b^5 - A*b^6)*cosh(x)^3 - 2*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*a^2*b^4 + 3*B*a*b^5 - A*b^6)*sinh(x)^3 + 2*
(2*B*a^6 - 6*A*a^5*b - 3*B*a^4*b^2 - 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 + 2*B*b^6)*cosh(x)^2 + 2*(2*B*a^6 -
6*A*a^5*b - 3*B*a^4*b^2 - 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 + 2*B*b^6 - 3*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*
a^2*b^4 + 3*B*a*b^5 - A*b^6)*cosh(x))*sinh(x)^2 + (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5 + (2*A*a^2*b^3 + 3*B*a*b^4
- A*b^5)*cosh(x)^4 + (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*sinh(x)^4 + 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*cos
h(x)^3 + 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4 + (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cosh(x))*sinh(x)^3 + 2*(4*
A*a^4*b + 6*B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x)^2 + 2*(4*A*a^4*b + 6*B*a^3*b^2 - 4*A*a^2*b^3
- 3*B*a*b^4 + A*b^5 + 3*(2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cosh(x)^2 + 6*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*
cosh(x))*sinh(x)^2 - 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*cosh(x) - 4*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4
- (2*A*a^2*b^3 + 3*B*a*b^4 - A*b^5)*cosh(x)^3 - 3*(2*A*a^3*b^2 + 3*B*a^2*b^3 - A*a*b^4)*cosh(x)^2 - (4*A*a^4*b
+ 6*B*a^3*b^2 - 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*s
inh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh
(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 2*(4*B*a^5*b - 10*A*a^4
*b^2 - B*a^3*b^3 - 11*A*a^2*b^4 - 5*B*a*b^5 - A*b^6)*cosh(x) - 2*(4*B*a^5*b - 10*A*a^4*b^2 - B*a^3*b^3 - 11*A*
a^2*b^4 - 5*B*a*b^5 - A*b^6 + 3*(2*A*a^4*b^2 + 3*B*a^3*b^3 + A*a^2*b^4 + 3*B*a*b^5 - A*b^6)*cosh(x)^2 - 2*(2*B
*a^6 - 6*A*a^5*b - 3*B*a^4*b^2 - 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 + 2*B*b^6)*cosh(x))*sinh(x))/(a^6*b^3 +
3*a^4*b^5 + 3*a^2*b^7 + b^9 + (a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*cosh(x)^4 + (a^6*b^3 + 3*a^4*b^5 + 3*a^
2*b^7 + b^9)*sinh(x)^4 + 4*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x)^3 + 4*(a^7*b^2 + 3*a^5*b^4 + 3*a^
3*b^6 + a*b^8 + (a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*cosh(x))*sinh(x)^3 + 2*(2*a^8*b + 5*a^6*b^3 + 3*a^4*b^
5 - a^2*b^7 - b^9)*cosh(x)^2 + 2*(2*a^8*b + 5*a^6*b^3 + 3*a^4*b^5 - a^2*b^7 - b^9 + 3*(a^6*b^3 + 3*a^4*b^5 + 3
*a^2*b^7 + b^9)*cosh(x)^2 + 6*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^2 - 4*(a^7*b^2 + 3*a^
5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x) - 4*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8 - (a^6*b^3 + 3*a^4*b^5 + 3*a^2
*b^7 + b^9)*cosh(x)^3 - 3*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*cosh(x)^2 - (2*a^8*b + 5*a^6*b^3 + 3*a^4*b
^5 - a^2*b^7 - b^9)*cosh(x))*sinh(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (119) = 238\).

Time = 0.29 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.20 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\frac {1}{2} \, {\left (\frac {3 \, a b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a b^{3} e^{\left (-3 \, x\right )} + a^{2} b^{2} - 2 \, b^{4} + {\left (4 \, a^{3} b - 5 \, a b^{3}\right )} e^{\left (-x\right )} + {\left (2 \, a^{4} - 5 \, a^{2} b^{2} + 2 \, b^{4}\right )} e^{\left (-2 \, x\right )}\right )}}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + 4 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (2 \, a^{6} b + 3 \, a^{4} b^{3} - b^{7}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-3 \, x\right )} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-4 \, x\right )}}\right )} B + \frac {1}{2} \, A {\left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (3 \, a b^{2} + {\left (10 \, a^{2} b + b^{3}\right )} e^{\left (-x\right )} + 3 \, {\left (2 \, a^{3} - a b^{2}\right )} e^{\left (-2 \, x\right )} - {\left (2 \, a^{2} b - b^{3}\right )} e^{\left (-3 \, x\right )}\right )}}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} e^{\left (-x\right )} + 2 \, {\left (2 \, a^{6} + 3 \, a^{4} b^{2} - b^{6}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} e^{\left (-3 \, x\right )} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-4 \, x\right )}}\right )} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="maxima")

[Out]

1/2*(3*a*b*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqr
t(a^2 + b^2)) + 2*(3*a*b^3*e^(-3*x) + a^2*b^2 - 2*b^4 + (4*a^3*b - 5*a*b^3)*e^(-x) + (2*a^4 - 5*a^2*b^2 + 2*b^
4)*e^(-2*x))/(a^4*b^3 + 2*a^2*b^5 + b^7 + 4*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*e^(-x) + 2*(2*a^6*b + 3*a^4*b^3 - b^
7)*e^(-2*x) - 4*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*e^(-3*x) + (a^4*b^3 + 2*a^2*b^5 + b^7)*e^(-4*x)))*B + 1/2*A*((2*
a^2 - b^2)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqr
t(a^2 + b^2)) - 2*(3*a*b^2 + (10*a^2*b + b^3)*e^(-x) + 3*(2*a^3 - a*b^2)*e^(-2*x) - (2*a^2*b - b^3)*e^(-3*x))/
(a^4*b^2 + 2*a^2*b^4 + b^6 + 4*(a^5*b + 2*a^3*b^3 + a*b^5)*e^(-x) + 2*(2*a^6 + 3*a^4*b^2 - b^6)*e^(-2*x) - 4*(
a^5*b + 2*a^3*b^3 + a*b^5)*e^(-3*x) + (a^4*b^2 + 2*a^2*b^4 + b^6)*e^(-4*x)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (119) = 238\).

Time = 0.28 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.18 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=-\frac {{\left (2 \, A a^{2} + 3 \, B a b - A b^{2}\right )} \log \left (\frac {{\left | -2 \, b e^{x} - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | -2 \, b e^{x} - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, A a^{2} b^{2} e^{\left (3 \, x\right )} + 3 \, B a b^{3} e^{\left (3 \, x\right )} - A b^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} + 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} - 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} + 4 \, B a^{3} b e^{x} - 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} + 2 \, B b^{4}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}^{2}} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="giac")

[Out]

-1/2*(2*A*a^2 + 3*B*a*b - A*b^2)*log(abs(-2*b*e^x - 2*a - 2*sqrt(a^2 + b^2))/abs(-2*b*e^x - 2*a + 2*sqrt(a^2 +
b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)) + (2*A*a^2*b^2*e^(3*x) + 3*B*a*b^3*e^(3*x) - A*b^4*e^(3*x) -
2*B*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) + 5*B*a^2*b^2*e^(2*x) - 3*A*a*b^3*e^(2*x) - 2*B*b^4*e^(2*x) + 4*B*a^3*b*e
^x - 10*A*a^2*b^2*e^x - 5*B*a*b^3*e^x - A*b^4*e^x - B*a^2*b^2 + 3*A*a*b^3 + 2*B*b^4)/((a^4*b + 2*a^2*b^3 + b^5
)*(b*e^(2*x) + 2*a*e^x - b)^2)

Mupad [F(-1)]

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

[In]

int((A + B*sinh(x))/(a + b*sinh(x))^3,x)

[Out]

int((A + B*sinh(x))/(a + b*sinh(x))^3, x)