\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx\) [132]

Optimal result

Integrand size = 15, antiderivative size = 187 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=-\frac {\left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 B\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))} \]

[Out]

-(2*A*a^3-3*A*a*b^2+4*B*a^2*b-B*b^3)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(7/2)-1/3*(A*b-B*a)*
cosh(x)/(a^2+b^2)/(a+b*sinh(x))^3-1/6*(5*A*a*b-2*B*a^2+3*B*b^2)*cosh(x)/(a^2+b^2)^2/(a+b*sinh(x))^2-1/6*(11*A*
a^2*b-4*A*b^3-2*B*a^3+13*B*a*b^2)*cosh(x)/(a^2+b^2)^3/(a+b*sinh(x))

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2833, 12, 2739, 632, 212} \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (2 a^3 A+4 a^2 b B-3 a A b^2-b^3 B\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))} \]

[In]

Int[(A + B*Sinh[x])/(a + b*Sinh[x])^4,x]

[Out]

-(((2*a^3*A - 3*a*A*b^2 + 4*a^2*b*B - b^3*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) -
((A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^3) - ((5*a*A*b - 2*a^2*B + 3*b^2*B)*Cosh[x])/(6*(a^2 + b^
2)^2*(a + b*Sinh[x])^2) - ((11*a^2*A*b - 4*A*b^3 - 2*a^3*B + 13*a*b^2*B)*Cosh[x])/(6*(a^2 + b^2)^3*(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)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\int \frac {-3 (a A+b B)+2 (A b-a B) \sinh (x)}{(a+b \sinh (x))^3} \, dx}{3 \left (a^2+b^2\right )} \\ & = -\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}+\frac {\int \frac {2 \left (3 a^2 A-2 A b^2+5 a b B\right )-\left (5 a A b-2 a^2 B+3 b^2 B\right ) \sinh (x)}{(a+b \sinh (x))^2} \, dx}{6 \left (a^2+b^2\right )^2} \\ & = -\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))}-\frac {\int -\frac {3 \left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 B\right )}{a+b \sinh (x)} \, dx}{6 \left (a^2+b^2\right )^3} \\ & = -\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 B\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{2 \left (a^2+b^2\right )^3} \\ & = -\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))}+\frac {\left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 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 )^3} \\ & = -\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))}-\frac {\left (2 \left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 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 )^3} \\ & = -\frac {\left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 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 )^{7/2}}-\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\frac {\frac {6 \left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 B\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {2 \left (a^2+b^2\right )^2 (-A b+a B) \cosh (x)}{(a+b \sinh (x))^3}+\frac {\left (a^2+b^2\right ) \left (-5 a A b+2 a^2 B-3 b^2 B\right ) \cosh (x)}{(a+b \sinh (x))^2}+\frac {\left (-11 a^2 A b+4 A b^3+2 a^3 B-13 a b^2 B\right ) \cosh (x)}{a+b \sinh (x)}}{6 \left (a^2+b^2\right )^3} \]

[In]

Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^4,x]

[Out]

((6*(2*a^3*A - 3*a*A*b^2 + 4*a^2*b*B - b^3*B)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + (
2*(a^2 + b^2)^2*(-(A*b) + a*B)*Cosh[x])/(a + b*Sinh[x])^3 + ((a^2 + b^2)*(-5*a*A*b + 2*a^2*B - 3*b^2*B)*Cosh[x
])/(a + b*Sinh[x])^2 + ((-11*a^2*A*b + 4*A*b^3 + 2*a^3*B - 13*a*b^2*B)*Cosh[x])/(a + b*Sinh[x]))/(6*(a^2 + b^2
)^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(632\) vs. \(2(175)=350\).

Time = 0.83 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.39

method	result	size
default	\(-\frac {2 \left (-\frac {b \left (9 A \,a^{4} b +6 a^{2} A \,b^{3}+2 A \,b^{5}-4 B \,a^{5}+a^{3} B \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (6 A \,a^{6} b -27 A \,a^{4} b^{3}-12 A \,a^{2} b^{5}-4 A \,b^{7}-2 B \,a^{7}+14 B \,a^{5} b^{2}-11 B \,a^{3} b^{4}-2 B a \,b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{4}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{2}}+\frac {b \left (54 A \,a^{6} b -21 A \,a^{4} b^{3}-4 A \,a^{2} b^{5}-4 A \,b^{7}-18 B \,a^{7}+42 B \,a^{5} b^{2}-17 B \,a^{3} b^{4}-2 B a \,b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (6 A \,a^{6} b -20 A \,a^{4} b^{3}-3 A \,a^{2} b^{5}-2 A \,b^{7}-2 B \,a^{7}+10 B \,a^{5} b^{2}-14 B \,a^{3} b^{4}-B a \,b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (27 A \,a^{4} b +4 a^{2} A \,b^{3}+2 A \,b^{5}-8 B \,a^{5}+19 a^{3} B \,b^{2}+2 B a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 A \,a^{4} b +5 a^{2} A \,b^{3}+2 A \,b^{5}-6 B \,a^{5}+10 a^{3} B \,b^{2}+B a \,b^{4}}{6 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )^{3}}+\frac {\left (2 a^{3} A -3 A a \,b^{2}+4 B \,a^{2} b -B \,b^{3}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \sqrt {a^{2}+b^{2}}}\)	\(633\)
risch	\(\frac {-8 B \,a^{6} {\mathrm e}^{3 x}-12 A \,b^{6} {\mathrm e}^{2 x}-13 B a \,b^{5}-3 B \,b^{6} {\mathrm e}^{5 x}-11 A \,a^{2} b^{4}+2 B \,a^{3} b^{3}+3 B \,b^{6} {\mathrm e}^{x}-78 B \,a^{2} b^{4} {\mathrm e}^{3 x}-102 A \,a^{4} b^{2} {\mathrm e}^{2 x}+36 A \,a^{2} b^{4} {\mathrm e}^{2 x}+24 B \,a^{5} b \,{\mathrm e}^{2 x}-102 B \,a^{3} b^{3} {\mathrm e}^{2 x}+60 A \,a^{3} b^{3} {\mathrm e}^{x}-15 A a \,b^{5} {\mathrm e}^{x}-12 B \,a^{4} b^{2} {\mathrm e}^{x}+66 B \,a^{2} b^{4} {\mathrm e}^{x}+44 A \,a^{5} b \,{\mathrm e}^{3 x}-82 A \,a^{3} b^{3} {\mathrm e}^{3 x}+6 A \,a^{3} b^{3} {\mathrm e}^{5 x}-9 A a \,b^{5} {\mathrm e}^{5 x}+12 B \,a^{2} b^{4} {\mathrm e}^{5 x}+24 B a \,b^{5} {\mathrm e}^{2 x}+30 A \,a^{4} b^{2} {\mathrm e}^{4 x}-45 A \,a^{2} b^{4} {\mathrm e}^{4 x}+24 A a \,b^{5} {\mathrm e}^{3 x}+64 B \,a^{4} b^{2} {\mathrm e}^{3 x}+4 A \,b^{6}+60 B \,a^{3} b^{3} {\mathrm e}^{4 x}-15 B a \,b^{5} {\mathrm e}^{4 x}}{3 b \left (a^{2}+b^{2}\right )^{3} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )^{3}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{3} A}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) A a \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {2 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,a^{2} b}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,b^{3}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{3} A}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) A a \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {2 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,a^{2} b}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,b^{3}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\)	\(965\)

[In]

int((A+B*sinh(x))/(a+b*sinh(x))^4,x,method=_RETURNVERBOSE)

[Out]

-2*(-1/2*b*(9*A*a^4*b+6*A*a^2*b^3+2*A*b^5-4*B*a^5+B*a^3*b^2)/a/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*x)^5-1/2
*(6*A*a^6*b-27*A*a^4*b^3-12*A*a^2*b^5-4*A*b^7-2*B*a^7+14*B*a^5*b^2-11*B*a^3*b^4-2*B*a*b^6)/(a^6+3*a^4*b^2+3*a^
2*b^4+b^6)/a^2*tanh(1/2*x)^4+1/3/a^3*b*(54*A*a^6*b-21*A*a^4*b^3-4*A*a^2*b^5-4*A*b^7-18*B*a^7+42*B*a^5*b^2-17*B
*a^3*b^4-2*B*a*b^6)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*x)^3+1/a^2*(6*A*a^6*b-20*A*a^4*b^3-3*A*a^2*b^5-2*A*
b^7-2*B*a^7+10*B*a^5*b^2-14*B*a^3*b^4-B*a*b^6)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*x)^2-1/2/a*b*(27*A*a^4*b
+4*A*a^2*b^3+2*A*b^5-8*B*a^5+19*B*a^3*b^2+2*B*a*b^4)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*x)-1/6*(18*A*a^4*b
+5*A*a^2*b^3+2*A*b^5-6*B*a^5+10*B*a^3*b^2+B*a*b^4)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6))/(tanh(1/2*x)^2*a-2*b*tanh(1/
2*x)-a)^3+(2*A*a^3-3*A*a*b^2+4*B*a^2*b-B*b^3)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*t
anh(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. 3870 vs. \(2 (177) = 354\).

Time = 0.42 (sec) , antiderivative size = 3870, normalized size of antiderivative = 20.70 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^4,x, algorithm="fricas")

[Out]

-1/6*(4*B*a^5*b^3 - 22*A*a^4*b^4 - 22*B*a^3*b^5 - 14*A*a^2*b^6 - 26*B*a*b^7 + 8*A*b^8 + 6*(2*A*a^5*b^3 + 4*B*a
^4*b^4 - A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 - B*b^8)*cosh(x)^5 + 6*(2*A*a^5*b^3 + 4*B*a^4*b^4 - A*a^3*b^5 + 3
*B*a^2*b^6 - 3*A*a*b^7 - B*b^8)*sinh(x)^5 + 30*(2*A*a^6*b^2 + 4*B*a^5*b^3 - A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*
b^6 - B*a*b^7)*cosh(x)^4 + 30*(2*A*a^6*b^2 + 4*B*a^5*b^3 - A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 - B*a*b^7 + (
2*A*a^5*b^3 + 4*B*a^4*b^4 - A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 - B*b^8)*cosh(x))*sinh(x)^4 - 4*(4*B*a^8 - 22*
A*a^7*b - 28*B*a^6*b^2 + 19*A*a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 + 39*B*a^2*b^6 - 12*A*a*b^7)*cosh(x)^3 - 4*
(4*B*a^8 - 22*A*a^7*b - 28*B*a^6*b^2 + 19*A*a^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 + 39*B*a^2*b^6 - 12*A*a*b^7 -
15*(2*A*a^5*b^3 + 4*B*a^4*b^4 - A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 - B*b^8)*cosh(x)^2 - 30*(2*A*a^6*b^2 + 4*
B*a^5*b^3 - A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 - B*a*b^7)*cosh(x))*sinh(x)^3 + 12*(4*B*a^7*b - 17*A*a^6*b^2
- 13*B*a^5*b^3 - 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 + 4*B*a*b^7 - 2*A*b^8)*cosh(x)^2 + 12*(4*B*a^7*b -
17*A*a^6*b^2 - 13*B*a^5*b^3 - 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 + 4*B*a*b^7 - 2*A*b^8 + 5*(2*A*a^5*b^
3 + 4*B*a^4*b^4 - A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 - B*b^8)*cosh(x)^3 + 15*(2*A*a^6*b^2 + 4*B*a^5*b^3 - A*a
^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 - B*a*b^7)*cosh(x)^2 - (4*B*a^8 - 22*A*a^7*b - 28*B*a^6*b^2 + 19*A*a^5*b^3
+ 7*B*a^4*b^4 + 29*A*a^3*b^5 + 39*B*a^2*b^6 - 12*A*a*b^7)*cosh(x))*sinh(x)^2 + 3*(2*A*a^3*b^4 + 4*B*a^2*b^5 -
3*A*a*b^6 - B*b^7 - (2*A*a^3*b^4 + 4*B*a^2*b^5 - 3*A*a*b^6 - B*b^7)*cosh(x)^6 - (2*A*a^3*b^4 + 4*B*a^2*b^5 - 3
*A*a*b^6 - B*b^7)*sinh(x)^6 - 6*(2*A*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^5 - 6*(2*A*a^4*b^3
+ 4*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 + 4*B*a^2*b^5 - 3*A*a*b^6 - B*b^7)*cosh(x))*sinh(x)^5 -
3*(8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 + B*b^7)*cosh(x)^4 - 3*(8*A*a^5*b^2 + 1
6*B*a^4*b^3 - 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 + B*b^7 + 5*(2*A*a^3*b^4 + 4*B*a^2*b^5 - 3*A*a*b^6 - B*b^
7)*cosh(x)^2 + 10*(2*A*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6)*cosh(x))*sinh(x)^4 - 4*(4*A*a^6*b + 8*B*
a^5*b^2 - 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 + 3*B*a*b^6)*cosh(x)^3 - 4*(4*A*a^6*b + 8*B*a^5*b^2 - 12*A
*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 + 3*B*a*b^6 + 5*(2*A*a^3*b^4 + 4*B*a^2*b^5 - 3*A*a*b^6 - B*b^7)*cosh(x)^
3 + 15*(2*A*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6)*cosh(x)^2 + 3*(8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^
3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 + B*b^7)*cosh(x))*sinh(x)^3 + 3*(8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^3*b^4 - 8
*B*a^2*b^5 + 3*A*a*b^6 + B*b^7)*cosh(x)^2 + 3*(8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a
*b^6 + B*b^7 - 5*(2*A*a^3*b^4 + 4*B*a^2*b^5 - 3*A*a*b^6 - B*b^7)*cosh(x)^4 - 20*(2*A*a^4*b^3 + 4*B*a^3*b^4 - 3
*A*a^2*b^5 - B*a*b^6)*cosh(x)^3 - 6*(8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 + B*b
^7)*cosh(x)^2 - 4*(4*A*a^6*b + 8*B*a^5*b^2 - 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 + 3*B*a*b^6)*cosh(x))*s
inh(x)^2 - 6*(2*A*a^4*b^3 + 4*B*a^3*b^4 - 3*A*a^2*b^5 - B*a*b^6)*cosh(x) - 6*(2*A*a^4*b^3 + 4*B*a^3*b^4 - 3*A*
a^2*b^5 - B*a*b^6 + (2*A*a^3*b^4 + 4*B*a^2*b^5 - 3*A*a*b^6 - B*b^7)*cosh(x)^5 + 5*(2*A*a^4*b^3 + 4*B*a^3*b^4 -
3*A*a^2*b^5 - B*a*b^6)*cosh(x)^4 + 2*(8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 + B
*b^7)*cosh(x)^3 + 2*(4*A*a^6*b + 8*B*a^5*b^2 - 12*A*a^4*b^3 - 14*B*a^3*b^4 + 9*A*a^2*b^5 + 3*B*a*b^6)*cosh(x)^
2 - (8*A*a^5*b^2 + 16*B*a^4*b^3 - 14*A*a^3*b^4 - 8*B*a^2*b^5 + 3*A*a*b^6 + B*b^7)*cosh(x))*sinh(x))*sqrt(a^2 +
b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) + 2*sqr
t(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)) - 6*(4*B*a^6*b^2 - 20*A*a^5*b^3 - 18*B*a^4*b^4 - 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 - B*b^8)*cosh(
x) - 6*(4*B*a^6*b^2 - 20*A*a^5*b^3 - 18*B*a^4*b^4 - 15*A*a^3*b^5 - 23*B*a^2*b^6 + 5*A*a*b^7 - B*b^8 - 5*(2*A*a
^5*b^3 + 4*B*a^4*b^4 - A*a^3*b^5 + 3*B*a^2*b^6 - 3*A*a*b^7 - B*b^8)*cosh(x)^4 - 20*(2*A*a^6*b^2 + 4*B*a^5*b^3
- A*a^4*b^4 + 3*B*a^3*b^5 - 3*A*a^2*b^6 - B*a*b^7)*cosh(x)^3 + 2*(4*B*a^8 - 22*A*a^7*b - 28*B*a^6*b^2 + 19*A*a
^5*b^3 + 7*B*a^4*b^4 + 29*A*a^3*b^5 + 39*B*a^2*b^6 - 12*A*a*b^7)*cosh(x)^2 - 4*(4*B*a^7*b - 17*A*a^6*b^2 - 13*
B*a^5*b^3 - 11*A*a^4*b^4 - 13*B*a^3*b^5 + 4*A*a^2*b^6 + 4*B*a*b^7 - 2*A*b^8)*cosh(x))*sinh(x))/(a^8*b^4 + 4*a^
6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12 - (a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*cosh(x)^6 - (a^8
*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*sinh(x)^6 - 6*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 +
a*b^11)*cosh(x)^5 - 6*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11 + (a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^
8 + 4*a^2*b^10 + b^12)*cosh(x))*sinh(x)^5 - 3*(4*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^12)*cosh(
x)^4 - 3*(4*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^12 + 5*(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^
2*b^10 + b^12)*cosh(x)^2 + 10*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*cosh(x))*sinh(x)^4 - 4*(2
*a^11*b + 5*a^9*b^3 - 10*a^5*b^7 - 10*a^3*b^9 - 3*a*b^11)*cosh(x)^3 - 4*(2*a^11*b + 5*a^9*b^3 - 10*a^5*b^7 - 1
0*a^3*b^9 - 3*a*b^11 + 5*(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*cosh(x)^3 + 15*(a^9*b^3 + 4*a^7
*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*cosh(x)^2 + 3*(4*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^12
)*cosh(x))*sinh(x)^3 + 3*(4*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^12)*cosh(x)^2 + 3*(4*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^12 - 5*(a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*cosh(x
)^4 - 20*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*cosh(x)^3 - 6*(4*a^10*b^2 + 15*a^8*b^4 + 20*a^
6*b^6 + 10*a^4*b^8 - b^12)*cosh(x)^2 - 4*(2*a^11*b + 5*a^9*b^3 - 10*a^5*b^7 - 10*a^3*b^9 - 3*a*b^11)*cosh(x))*
sinh(x)^2 - 6*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*cosh(x) - 6*(a^9*b^3 + 4*a^7*b^5 + 6*a^5*
b^7 + 4*a^3*b^9 + a*b^11 + (a^8*b^4 + 4*a^6*b^6 + 6*a^4*b^8 + 4*a^2*b^10 + b^12)*cosh(x)^5 + 5*(a^9*b^3 + 4*a^
7*b^5 + 6*a^5*b^7 + 4*a^3*b^9 + a*b^11)*cosh(x)^4 + 2*(4*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^1
2)*cosh(x)^3 + 2*(2*a^11*b + 5*a^9*b^3 - 10*a^5*b^7 - 10*a^3*b^9 - 3*a*b^11)*cosh(x)^2 - (4*a^10*b^2 + 15*a^8*
b^4 + 20*a^6*b^6 + 10*a^4*b^8 - b^12)*cosh(x))*sinh(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\text {Timed out} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))**4,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (177) = 354\).

Time = 0.31 (sec) , antiderivative size = 982, normalized size of antiderivative = 5.25 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^4,x, algorithm="maxima")

[Out]

1/6*(3*(2*a^2 - 3*b^2)*a*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) - 2*(11*a^2*b^3 - 4*b^5 + 15*(4*a^3*b^2 - a*b^4)*e^(-x) + 6*(17*a^4*b
- 6*a^2*b^3 + 2*b^5)*e^(-2*x) + 2*(22*a^5 - 41*a^3*b^2 + 12*a*b^4)*e^(-3*x) - 15*(2*a^4*b - 3*a^2*b^3)*e^(-4*x
) + 3*(2*a^3*b^2 - 3*a*b^4)*e^(-5*x))/(a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9 + 6*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*
b^6 + a*b^8)*e^(-x) + 3*(4*a^8*b + 11*a^6*b^3 + 9*a^4*b^5 + a^2*b^7 - b^9)*e^(-2*x) + 4*(2*a^9 + 3*a^7*b^2 - 3
*a^5*b^4 - 7*a^3*b^6 - 3*a*b^8)*e^(-3*x) - 3*(4*a^8*b + 11*a^6*b^3 + 9*a^4*b^5 + a^2*b^7 - b^9)*e^(-4*x) + 6*(
a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*e^(-5*x) - (a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*e^(-6*x)))*A + 1/6
*B*(3*(4*a^2*b - b^3)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(2*a^3*b^3 - 13*a*b^5 + 3*(4*a^4*b^2 - 22*a^2*b^4 - b^6)*e^(-x) + 6*(
4*a^5*b - 17*a^3*b^3 + 4*a*b^5)*e^(-2*x) + 2*(4*a^6 - 32*a^4*b^2 + 39*a^2*b^4)*e^(-3*x) + 15*(4*a^3*b^3 - a*b^
5)*e^(-4*x) - 3*(4*a^2*b^4 - b^6)*e^(-5*x))/(a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10 + 6*(a^7*b^3 + 3*a^5*b^5 +
3*a^3*b^7 + a*b^9)*e^(-x) + 3*(4*a^8*b^2 + 11*a^6*b^4 + 9*a^4*b^6 + a^2*b^8 - b^10)*e^(-2*x) + 4*(2*a^9*b + 3
*a^7*b^3 - 3*a^5*b^5 - 7*a^3*b^7 - 3*a*b^9)*e^(-3*x) - 3*(4*a^8*b^2 + 11*a^6*b^4 + 9*a^4*b^6 + a^2*b^8 - b^10)
*e^(-4*x) + 6*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*e^(-5*x) - (a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10)*e^
(-6*x)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (177) = 354\).

Time = 0.29 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.55 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\frac {{\left (2 \, A a^{3} + 4 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {6 \, A a^{3} b^{3} e^{\left (5 \, x\right )} + 12 \, B a^{2} b^{4} e^{\left (5 \, x\right )} - 9 \, A a b^{5} e^{\left (5 \, x\right )} - 3 \, B b^{6} e^{\left (5 \, x\right )} + 30 \, A a^{4} b^{2} e^{\left (4 \, x\right )} + 60 \, B a^{3} b^{3} e^{\left (4 \, x\right )} - 45 \, A a^{2} b^{4} e^{\left (4 \, x\right )} - 15 \, B a b^{5} e^{\left (4 \, x\right )} - 8 \, B a^{6} e^{\left (3 \, x\right )} + 44 \, A a^{5} b e^{\left (3 \, x\right )} + 64 \, B a^{4} b^{2} e^{\left (3 \, x\right )} - 82 \, A a^{3} b^{3} e^{\left (3 \, x\right )} - 78 \, B a^{2} b^{4} e^{\left (3 \, x\right )} + 24 \, A a b^{5} e^{\left (3 \, x\right )} + 24 \, B a^{5} b e^{\left (2 \, x\right )} - 102 \, A a^{4} b^{2} e^{\left (2 \, x\right )} - 102 \, B a^{3} b^{3} e^{\left (2 \, x\right )} + 36 \, A a^{2} b^{4} e^{\left (2 \, x\right )} + 24 \, B a b^{5} e^{\left (2 \, x\right )} - 12 \, A b^{6} e^{\left (2 \, x\right )} - 12 \, B a^{4} b^{2} e^{x} + 60 \, A a^{3} b^{3} e^{x} + 66 \, B a^{2} b^{4} e^{x} - 15 \, A a b^{5} e^{x} + 3 \, B b^{6} e^{x} + 2 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4} - 13 \, B a b^{5} + 4 \, A b^{6}}{3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}^{3}} \]

[In]

integrate((A+B*sinh(x))/(a+b*sinh(x))^4,x, algorithm="giac")

[Out]

1/2*(2*A*a^3 + 4*B*a^2*b - 3*A*a*b^2 - B*b^3)*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^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 1/3*(6*A*a^3*b^3*e^(5*x) + 12*B*a^2
*b^4*e^(5*x) - 9*A*a*b^5*e^(5*x) - 3*B*b^6*e^(5*x) + 30*A*a^4*b^2*e^(4*x) + 60*B*a^3*b^3*e^(4*x) - 45*A*a^2*b^
4*e^(4*x) - 15*B*a*b^5*e^(4*x) - 8*B*a^6*e^(3*x) + 44*A*a^5*b*e^(3*x) + 64*B*a^4*b^2*e^(3*x) - 82*A*a^3*b^3*e^
(3*x) - 78*B*a^2*b^4*e^(3*x) + 24*A*a*b^5*e^(3*x) + 24*B*a^5*b*e^(2*x) - 102*A*a^4*b^2*e^(2*x) - 102*B*a^3*b^3
*e^(2*x) + 36*A*a^2*b^4*e^(2*x) + 24*B*a*b^5*e^(2*x) - 12*A*b^6*e^(2*x) - 12*B*a^4*b^2*e^x + 60*A*a^3*b^3*e^x
+ 66*B*a^2*b^4*e^x - 15*A*a*b^5*e^x + 3*B*b^6*e^x + 2*B*a^3*b^3 - 11*A*a^2*b^4 - 13*B*a*b^5 + 4*A*b^6)/((a^6*b
+ 3*a^4*b^3 + 3*a^2*b^5 + b^7)*(b*e^(2*x) + 2*a*e^x - b)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \]

[In]

int((A + B*sinh(x))/(a + b*sinh(x))^4,x)

[Out]

int((A + B*sinh(x))/(a + b*sinh(x))^4, x)