\(\int \frac {1}{(a+b \sinh (c+d x))^4} \, dx\) [104]

Optimal result

Integrand size = 12, antiderivative size = 174 \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=-\frac {a \left (2 a^2-3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2} d}-\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \sinh (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2743, 2833, 12, 2739, 632, 210} \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=-\frac {a \left (2 a^2-3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{d \left (a^2+b^2\right )^{7/2}}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 d \left (a^2+b^2\right )^3 (a+b \sinh (c+d x))}-\frac {5 a b \cosh (c+d x)}{6 d \left (a^2+b^2\right )^2 (a+b \sinh (c+d x))^2}-\frac {b \cosh (c+d x)}{3 d \left (a^2+b^2\right ) (a+b \sinh (c+d x))^3} \]

[In]

Int[(a + b*Sinh[c + d*x])^(-4),x]

[Out]

-((a*(2*a^2 - 3*b^2)*ArcTanh[(b - a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/((a^2 + b^2)^(7/2)*d)) - (b*Cosh[c +
d*x])/(3*(a^2 + b^2)*d*(a + b*Sinh[c + d*x])^3) - (5*a*b*Cosh[c + d*x])/(6*(a^2 + b^2)^2*d*(a + b*Sinh[c + d*x
])^2) - (b*(11*a^2 - 4*b^2)*Cosh[c + d*x])/(6*(a^2 + b^2)^3*d*(a + b*Sinh[c + d*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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

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 {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {\int \frac {-3 a+2 b \sinh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx}{3 \left (a^2+b^2\right )} \\ & = -\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}+\frac {\int \frac {2 \left (3 a^2-2 b^2\right )-5 a b \sinh (c+d x)}{(a+b \sinh (c+d x))^2} \, dx}{6 \left (a^2+b^2\right )^2} \\ & = -\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \sinh (c+d x))}-\frac {\int -\frac {3 a \left (2 a^2-3 b^2\right )}{a+b \sinh (c+d x)} \, dx}{6 \left (a^2+b^2\right )^3} \\ & = -\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \sinh (c+d x))}+\frac {\left (a \left (2 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{2 \left (a^2+b^2\right )^3} \\ & = -\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \sinh (c+d x))}-\frac {\left (i a \left (2 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2+b^2\right )^3 d} \\ & = -\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \sinh (c+d x))}+\frac {\left (2 i a \left (2 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{\left (a^2+b^2\right )^3 d} \\ & = -\frac {a \left (2 a^2-3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2} d}-\frac {b \cosh (c+d x)}{3 \left (a^2+b^2\right ) d (a+b \sinh (c+d x))^3}-\frac {5 a b \cosh (c+d x)}{6 \left (a^2+b^2\right )^2 d (a+b \sinh (c+d x))^2}-\frac {b \left (11 a^2-4 b^2\right ) \cosh (c+d x)}{6 \left (a^2+b^2\right )^3 d (a+b \sinh (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=\frac {\frac {6 a \left (2 a^2-3 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {b \cosh (c+d x) \left (-18 a^4-5 a^2 b^2-2 b^4+3 a b \left (-9 a^2+b^2\right ) \sinh (c+d x)+\left (-11 a^2 b^2+4 b^4\right ) \sinh ^2(c+d x)\right )}{(a+b \sinh (c+d x))^3}}{6 \left (a^2+b^2\right )^3 d} \]

[In]

Integrate[(a + b*Sinh[c + d*x])^(-4),x]

[Out]

((6*a*(2*a^2 - 3*b^2)*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + (b*Cosh[c + d*x]*
(-18*a^4 - 5*a^2*b^2 - 2*b^4 + 3*a*b*(-9*a^2 + b^2)*Sinh[c + d*x] + (-11*a^2*b^2 + 4*b^4)*Sinh[c + d*x]^2))/(a
+ b*Sinh[c + d*x])^3)/(6*(a^2 + b^2)^3*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(163)=326\).

Time = 1.44 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.84

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {b^{2} \left (9 a^{4}+6 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (6 a^{6}-27 a^{4} b^{2}-12 a^{2} b^{4}-4 b^{6}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{2} \left (54 a^{6}-21 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b \left (6 a^{6}-20 a^{4} b^{2}-3 a^{2} b^{4}-2 b^{6}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{2} \left (27 a^{4}+4 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (18 a^{4}+5 a^{2} b^{2}+2 b^{4}\right )}{6 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}+\frac {a \left (2 a^{2}-3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \sqrt {a^{2}+b^{2}}}}{d}\)	\(494\)
default	\(\frac {-\frac {2 \left (-\frac {b^{2} \left (9 a^{4}+6 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (6 a^{6}-27 a^{4} b^{2}-12 a^{2} b^{4}-4 b^{6}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{2} \left (54 a^{6}-21 a^{4} b^{2}-4 a^{2} b^{4}-4 b^{6}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b \left (6 a^{6}-20 a^{4} b^{2}-3 a^{2} b^{4}-2 b^{6}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{2} \left (27 a^{4}+4 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (18 a^{4}+5 a^{2} b^{2}+2 b^{4}\right )}{6 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )^{3}}+\frac {a \left (2 a^{2}-3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \sqrt {a^{2}+b^{2}}}}{d}\)	\(494\)
risch	\(\frac {6 a^{3} b^{2} {\mathrm e}^{5 d x +5 c}-9 a \,b^{4} {\mathrm e}^{5 d x +5 c}+30 a^{4} b \,{\mathrm e}^{4 d x +4 c}-45 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}+44 a^{5} {\mathrm e}^{3 d x +3 c}-82 a^{3} b^{2} {\mathrm e}^{3 d x +3 c}+24 a \,b^{4} {\mathrm e}^{3 d x +3 c}-102 a^{4} b \,{\mathrm e}^{2 d x +2 c}+36 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}-12 b^{5} {\mathrm e}^{2 d x +2 c}+60 a^{3} b^{2} {\mathrm e}^{d x +c}-15 a \,b^{4} {\mathrm e}^{d x +c}-11 a^{2} b^{3}+4 b^{5}}{3 d \left (a^{2}+b^{2}\right )^{3} \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {3 a \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}+\frac {3 a \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}} d}\)	\(567\)

[In]

int(1/(a+b*sinh(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*(-1/2*b^2*(9*a^4+6*a^2*b^2+2*b^4)/a/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*d*x+1/2*c)^5-1/2*b*(6*a^6-2
7*a^4*b^2-12*a^2*b^4-4*b^6)/a^2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*d*x+1/2*c)^4+1/3/a^3*b^2*(54*a^6-21*a^4
*b^2-4*a^2*b^4-4*b^6)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*d*x+1/2*c)^3+1/a^2*b*(6*a^6-20*a^4*b^2-3*a^2*b^4-
2*b^6)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*d*x+1/2*c)^2-1/2/a*b^2*(27*a^4+4*a^2*b^2+2*b^4)/(a^6+3*a^4*b^2+3
*a^2*b^4+b^6)*tanh(1/2*d*x+1/2*c)-1/6*b*(18*a^4+5*a^2*b^2+2*b^4)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6))/(tanh(1/2*d*x+
1/2*c)^2*a-2*b*tanh(1/2*d*x+1/2*c)-a)^3+a*(2*a^2-3*b^2)/(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*tanh(1/2*d*x+1/2*c)-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. 2934 vs. \(2 (165) = 330\).

Time = 0.35 (sec) , antiderivative size = 2934, normalized size of antiderivative = 16.86 \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*sinh(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/6*(22*a^4*b^3 + 14*a^2*b^5 - 8*b^7 - 6*(2*a^5*b^2 - a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^5 - 6*(2*a^5*b^2 - a^3
*b^4 - 3*a*b^6)*sinh(d*x + c)^5 - 30*(2*a^6*b - a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^4 - 30*(2*a^6*b - a^4*b^3 -
3*a^2*b^5 + (2*a^5*b^2 - a^3*b^4 - 3*a*b^6)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(22*a^7 - 19*a^5*b^2 - 29*a^3*
b^4 + 12*a*b^6)*cosh(d*x + c)^3 - 4*(22*a^7 - 19*a^5*b^2 - 29*a^3*b^4 + 12*a*b^6 + 15*(2*a^5*b^2 - a^3*b^4 - 3
*a*b^6)*cosh(d*x + c)^2 + 30*(2*a^6*b - a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 12*(17*a^6*b + 1
1*a^4*b^3 - 4*a^2*b^5 + 2*b^7)*cosh(d*x + c)^2 + 12*(17*a^6*b + 11*a^4*b^3 - 4*a^2*b^5 + 2*b^7 - 5*(2*a^5*b^2
- a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^3 - 15*(2*a^6*b - a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^2 - (22*a^7 - 19*a^5*b
^2 - 29*a^3*b^4 + 12*a*b^6)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((2*a^3*b^3 - 3*a*b^5)*cosh(d*x + c)^6 + (2*a^3
*b^3 - 3*a*b^5)*sinh(d*x + c)^6 - 2*a^3*b^3 + 3*a*b^5 + 6*(2*a^4*b^2 - 3*a^2*b^4)*cosh(d*x + c)^5 + 6*(2*a^4*b
^2 - 3*a^2*b^4 + (2*a^3*b^3 - 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5)*cos
h(d*x + c)^4 + 3*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5 + 5*(2*a^3*b^3 - 3*a*b^5)*cosh(d*x + c)^2 + 10*(2*a^4*b^2 - 3
*a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*(4*a^6 - 12*a^4*b^2 + 9*a^2*b^4)*cosh(d*x + c)^3 + 4*(4*a^6 - 12*
a^4*b^2 + 9*a^2*b^4 + 5*(2*a^3*b^3 - 3*a*b^5)*cosh(d*x + c)^3 + 15*(2*a^4*b^2 - 3*a^2*b^4)*cosh(d*x + c)^2 + 3
*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5)*cosh(d*x
+ c)^2 - 3*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5 - 5*(2*a^3*b^3 - 3*a*b^5)*cosh(d*x + c)^4 - 20*(2*a^4*b^2 - 3*a^2*
b^4)*cosh(d*x + c)^3 - 6*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5)*cosh(d*x + c)^2 - 4*(4*a^6 - 12*a^4*b^2 + 9*a^2*b^4)
*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(2*a^4*b^2 - 3*a^2*b^4)*cosh(d*x + c) + 6*(2*a^4*b^2 - 3*a^2*b^4 + (2*a^3*
b^3 - 3*a*b^5)*cosh(d*x + c)^5 + 5*(2*a^4*b^2 - 3*a^2*b^4)*cosh(d*x + c)^4 + 2*(8*a^5*b - 14*a^3*b^3 + 3*a*b^5
)*cosh(d*x + c)^3 + 2*(4*a^6 - 12*a^4*b^2 + 9*a^2*b^4)*cosh(d*x + c)^2 - (8*a^5*b - 14*a^3*b^3 + 3*a*b^5)*cosh
(d*x + c))*sinh(d*x + c))*sqrt(a^2 + b^2)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c)
+ 2*a^2 + b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) + 2*sqrt(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x +
c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) -
b)) - 30*(4*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(d*x + c) - 6*(20*a^5*b^2 + 15*a^3*b^4 - 5*a*b^6 + 5*(2*a^5*b^2
- a^3*b^4 - 3*a*b^6)*cosh(d*x + c)^4 + 20*(2*a^6*b - a^4*b^3 - 3*a^2*b^5)*cosh(d*x + c)^3 + 2*(22*a^7 - 19*a^5
*b^2 - 29*a^3*b^4 + 12*a*b^6)*cosh(d*x + c)^2 - 4*(17*a^6*b + 11*a^4*b^3 - 4*a^2*b^5 + 2*b^7)*cosh(d*x + c))*s
inh(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^6 + (a^8*b^3 + 4*a^6*b^5 +
6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*sinh(d*x + c)^6 + 6*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*
cosh(d*x + c)^5 + 3*(4*a^10*b + 15*a^8*b^3 + 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d*cosh(d*x + c)^4 + 6*((a^8*b^3 +
4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*cosh(d*x + c) + (a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*
b^10)*d)*sinh(d*x + c)^5 + 4*(2*a^11 + 5*a^9*b^2 - 10*a^5*b^6 - 10*a^3*b^8 - 3*a*b^10)*d*cosh(d*x + c)^3 + 3*(
5*(a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^2 + 10*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6
+ 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c) + (4*a^10*b + 15*a^8*b^3 + 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d)*sinh(d*x
+ c)^4 - 3*(4*a^10*b + 15*a^8*b^3 + 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d*cosh(d*x + c)^2 + 4*(5*(a^8*b^3 + 4*a^6*
b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^3 + 15*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^
10)*d*cosh(d*x + c)^2 + 3*(4*a^10*b + 15*a^8*b^3 + 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d*cosh(d*x + c) + (2*a^11 +
5*a^9*b^2 - 10*a^5*b^6 - 10*a^3*b^8 - 3*a*b^10)*d)*sinh(d*x + c)^3 + 6*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a
^3*b^8 + a*b^10)*d*cosh(d*x + c) + 3*(5*(a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*cosh(d*x + c)^4
+ 20*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^3 + 6*(4*a^10*b + 15*a^8*b^3 + 20
*a^6*b^5 + 10*a^4*b^7 - b^11)*d*cosh(d*x + c)^2 + 4*(2*a^11 + 5*a^9*b^2 - 10*a^5*b^6 - 10*a^3*b^8 - 3*a*b^10)*
d*cosh(d*x + c) - (4*a^10*b + 15*a^8*b^3 + 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d)*sinh(d*x + c)^2 - (a^8*b^3 + 4*a
^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d + 6*((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*cosh(d*x
+ c)^5 + 5*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*cosh(d*x + c)^4 + 2*(4*a^10*b + 15*a^8*b^3
+ 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d*cosh(d*x + c)^3 + 2*(2*a^11 + 5*a^9*b^2 - 10*a^5*b^6 - 10*a^3*b^8 - 3*a*b
^10)*d*cosh(d*x + c)^2 - (4*a^10*b + 15*a^8*b^3 + 20*a^6*b^5 + 10*a^4*b^7 - b^11)*d*cosh(d*x + c) + (a^9*b^2 +
4*a^7*b^4 + 6*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d)*sinh(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*sinh(d*x+c))**4,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (165) = 330\).

Time = 0.30 (sec) , antiderivative size = 551, normalized size of antiderivative = 3.17 \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=\frac {{\left (2 \, a^{2} - 3 \, b^{2}\right )} a \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {11 \, a^{2} b^{3} - 4 \, b^{5} + 15 \, {\left (4 \, a^{3} b^{2} - a b^{4}\right )} e^{\left (-d x - c\right )} + 6 \, {\left (17 \, a^{4} b - 6 \, a^{2} b^{3} + 2 \, b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (22 \, a^{5} - 41 \, a^{3} b^{2} + 12 \, a b^{4}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 15 \, {\left (2 \, a^{4} b - 3 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, {\left (2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{3 \, {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9} + 6 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} e^{\left (-d x - c\right )} + 3 \, {\left (4 \, a^{8} b + 11 \, a^{6} b^{3} + 9 \, a^{4} b^{5} + a^{2} b^{7} - b^{9}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 4 \, {\left (2 \, a^{9} + 3 \, a^{7} b^{2} - 3 \, a^{5} b^{4} - 7 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - 3 \, {\left (4 \, a^{8} b + 11 \, a^{6} b^{3} + 9 \, a^{4} b^{5} + a^{2} b^{7} - b^{9}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 6 \, {\left (a^{7} b^{2} + 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} + a b^{8}\right )} e^{\left (-5 \, d x - 5 \, c\right )} - {\left (a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )} d} \]

[In]

integrate(1/(a+b*sinh(d*x+c))^4,x, algorithm="maxima")

[Out]

1/2*(2*a^2 - 3*b^2)*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^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)*d) - 1/3*(11*a^2*b^3 - 4*b^5 + 15*(4*a^3*b^2 - a*b^4)*e^(-d*x
- c) + 6*(17*a^4*b - 6*a^2*b^3 + 2*b^5)*e^(-2*d*x - 2*c) + 2*(22*a^5 - 41*a^3*b^2 + 12*a*b^4)*e^(-3*d*x - 3*c)
- 15*(2*a^4*b - 3*a^2*b^3)*e^(-4*d*x - 4*c) + 3*(2*a^3*b^2 - 3*a*b^4)*e^(-5*d*x - 5*c))/((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^(-d*x - c) + 3*(4*a^8*b + 11*a^6*b^3 + 9*a^
4*b^5 + a^2*b^7 - b^9)*e^(-2*d*x - 2*c) + 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*d*x -
3*c) - 3*(4*a^8*b + 11*a^6*b^3 + 9*a^4*b^5 + a^2*b^7 - b^9)*e^(-4*d*x - 4*c) + 6*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*
b^6 + a*b^8)*e^(-5*d*x - 5*c) - (a^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9)*e^(-6*d*x - 6*c))*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (165) = 330\).

Time = 0.30 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.05 \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (6 \, a^{3} b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 9 \, a b^{4} e^{\left (5 \, d x + 5 \, c\right )} + 30 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} - 45 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 44 \, a^{5} e^{\left (3 \, d x + 3 \, c\right )} - 82 \, a^{3} b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b^{4} e^{\left (3 \, d x + 3 \, c\right )} - 102 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 36 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 12 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 60 \, a^{3} b^{2} e^{\left (d x + c\right )} - 15 \, a b^{4} e^{\left (d x + c\right )} - 11 \, a^{2} b^{3} + 4 \, b^{5}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - b\right )}^{3}}}{6 \, d} \]

[In]

integrate(1/(a+b*sinh(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(3*(2*a^3 - 3*a*b^2)*log(abs(2*b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^(d*x + c) + 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)) + 2*(6*a^3*b^2*e^(5*d*x + 5*c) - 9*a*b^4*e
^(5*d*x + 5*c) + 30*a^4*b*e^(4*d*x + 4*c) - 45*a^2*b^3*e^(4*d*x + 4*c) + 44*a^5*e^(3*d*x + 3*c) - 82*a^3*b^2*e
^(3*d*x + 3*c) + 24*a*b^4*e^(3*d*x + 3*c) - 102*a^4*b*e^(2*d*x + 2*c) + 36*a^2*b^3*e^(2*d*x + 2*c) - 12*b^5*e^
(2*d*x + 2*c) + 60*a^3*b^2*e^(d*x + c) - 15*a*b^4*e^(d*x + c) - 11*a^2*b^3 + 4*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*
b^4 + b^6)*(b*e^(2*d*x + 2*c) + 2*a*e^(d*x + c) - b)^3))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \sinh (c+d x))^4} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int(1/(a + b*sinh(c + d*x))^4,x)

[Out]

int(1/(a + b*sinh(c + d*x))^4, x)