3.3.55 \(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx\) [255]

Optimal. Leaf size=180 \[ -\frac {\left (2 a^2 A-A c^2+3 a c C\right ) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{5/2} e}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))} \]

[Out]

-(2*A*a^2-A*c^2+3*C*a*c)*arctanh((c-a*tanh(1/2*e*x+1/2*d))/(a^2+c^2)^(1/2))/(a^2+c^2)^(5/2)/e-1/2*B/c/e/(a+c*s
inh(e*x+d))^2-1/2*(A*c-C*a)*cosh(e*x+d)/(a^2+c^2)/e/(a+c*sinh(e*x+d))^2-1/2*(3*A*a*c-C*a^2+2*C*c^2)*cosh(e*x+d
)/(a^2+c^2)^2/e/(a+c*sinh(e*x+d))

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4461, 2833, 12, 2739, 632, 210, 2747, 32} \begin {gather*} -\frac {\left (2 a^2 A+3 a c C-A c^2\right ) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{5/2}}-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right )^2 (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^3,x]

[Out]

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

Rule 4461

Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] :> With[{e = FreeFactors[Sin[c*(a +
b*x)], x]}, Int[ActivateTrig[u*v], x] + Dist[d, Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x], x] /; FunctionOfQ[
Sin[c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] &&  !FreeQ[v, x] && IntegerQ[(n - 1)/2] && NonsumQ[u] &&
(EqQ[F, Cos] || EqQ[F, cos])

Rubi steps

\begin {align*} \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx &=B \int \frac {\cosh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx+\int \frac {A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx\\ &=-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\int \frac {-2 (a A+c C)+(A c-a C) \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx}{2 \left (a^2+c^2\right )}+\frac {B \text {Subst}\left (\int \frac {1}{(a+x)^3} \, dx,x,c \sinh (d+e x)\right )}{c e}\\ &=-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}+\frac {\int \frac {2 a^2 A-A c^2+3 a c C}{a+c \sinh (d+e x)} \, dx}{2 \left (a^2+c^2\right )^2}\\ &=-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}+\frac {\left (2 a^2 A-A c^2+3 a c C\right ) \int \frac {1}{a+c \sinh (d+e x)} \, dx}{2 \left (a^2+c^2\right )^2}\\ &=-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}-\frac {\left (i \left (2 a^2 A-A c^2+3 a c C\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i c x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right )^2 e}\\ &=-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}+\frac {\left (2 i \left (2 a^2 A-A c^2+3 a c C\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+c^2\right )-x^2} \, dx,x,-2 i c+2 a \tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right )^2 e}\\ &=-\frac {\left (2 a^2 A-A c^2+3 a c C\right ) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{5/2} e}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.48, size = 170, normalized size = 0.94 \begin {gather*} \frac {\frac {2 \left (2 a^2 A-A c^2+3 a c C\right ) \text {ArcTan}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}-\frac {\left (a^2+c^2\right ) \left (B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)\right )}{c (a+c \sinh (d+e x))^2}+\frac {\left (-3 a A c+a^2 C-2 c^2 C\right ) \cosh (d+e x)}{a+c \sinh (d+e x)}}{2 \left (a^2+c^2\right )^2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^3,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(415\) vs. \(2(169)=338\).
time = 6.01, size = 416, normalized size = 2.31

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (5 A \,a^{2} c^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-3 C \,a^{3} c \right ) \left (\tanh ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}-\frac {\left (4 A \,a^{4} c -7 A \,a^{2} c^{3}-2 A \,c^{5}+2 B \,a^{4} c +4 B \,a^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+5 C \,a^{3} c^{2}-2 C a \,c^{4}\right ) \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a^{2}}+\frac {\left (11 A \,a^{2} c^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-5 C \,a^{3} c +4 C a \,c^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a}+\frac {4 A \,a^{2} c +A \,c^{3}-2 C \,a^{3}+C a \,c^{2}}{2 a^{4}+4 a^{2} c^{2}+2 c^{4}}\right )}{\left (a \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,c^{2}+3 C a c \right ) \arctanh \left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) \sqrt {a^{2}+c^{2}}}}{e}\)	\(416\)
default	\(\frac {-\frac {2 \left (-\frac {\left (5 A \,a^{2} c^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-3 C \,a^{3} c \right ) \left (\tanh ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}-\frac {\left (4 A \,a^{4} c -7 A \,a^{2} c^{3}-2 A \,c^{5}+2 B \,a^{4} c +4 B \,a^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+5 C \,a^{3} c^{2}-2 C a \,c^{4}\right ) \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a^{2}}+\frac {\left (11 A \,a^{2} c^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-5 C \,a^{3} c +4 C a \,c^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a}+\frac {4 A \,a^{2} c +A \,c^{3}-2 C \,a^{3}+C a \,c^{2}}{2 a^{4}+4 a^{2} c^{2}+2 c^{4}}\right )}{\left (a \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,c^{2}+3 C a c \right ) \arctanh \left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) \sqrt {a^{2}+c^{2}}}}{e}\)	\(416\)
risch	\(\frac {2 A \,a^{2} c^{2} {\mathrm e}^{3 e x +3 d}-A \,c^{4} {\mathrm e}^{3 e x +3 d}+3 C a \,c^{3} {\mathrm e}^{3 e x +3 d}+6 A \,a^{3} c \,{\mathrm e}^{2 e x +2 d}-3 A a \,c^{3} {\mathrm e}^{2 e x +2 d}-2 B \,a^{4} {\mathrm e}^{2 e x +2 d}-4 B \,a^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 B \,c^{4} {\mathrm e}^{2 e x +2 d}-2 C \,a^{4} {\mathrm e}^{2 e x +2 d}+5 C \,a^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 C \,c^{4} {\mathrm e}^{2 e x +2 d}-10 A \,a^{2} c^{2} {\mathrm e}^{e x +d}-A \,c^{4} {\mathrm e}^{e x +d}+4 C \,a^{3} c \,{\mathrm e}^{e x +d}-5 C a \,c^{3} {\mathrm e}^{e x +d}+3 A a \,c^{3}-C \,a^{2} c^{2}+2 C \,c^{4}}{c e \left (a^{2}+c^{2}\right )^{2} \left (c \,{\mathrm e}^{2 e x +2 d}+2 a \,{\mathrm e}^{e x +d}-c \right )^{2}}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) A \,c^{2}}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}+\frac {3 \ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) C a c}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) A \,c^{2}}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}-\frac {3 \ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) C a c}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}\)	\(744\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x,method=_RETURNVERBOSE)

[Out]

1/e*(-2*(-1/2*(5*A*a^2*c^2+2*A*c^4-2*B*a^4-4*B*a^2*c^2-2*B*c^4-3*C*a^3*c)/a/(a^4+2*a^2*c^2+c^4)*tanh(1/2*e*x+1
/2*d)^3-1/2*(4*A*a^4*c-7*A*a^2*c^3-2*A*c^5+2*B*a^4*c+4*B*a^2*c^3+2*B*c^5-2*C*a^5+5*C*a^3*c^2-2*C*a*c^4)/(a^4+2
*a^2*c^2+c^4)/a^2*tanh(1/2*e*x+1/2*d)^2+1/2*(11*A*a^2*c^2+2*A*c^4-2*B*a^4-4*B*a^2*c^2-2*B*c^4-5*C*a^3*c+4*C*a*
c^3)/(a^4+2*a^2*c^2+c^4)/a*tanh(1/2*e*x+1/2*d)+1/2*(4*A*a^2*c+A*c^3-2*C*a^3+C*a*c^2)/(a^4+2*a^2*c^2+c^4))/(a*t
anh(1/2*e*x+1/2*d)^2-2*c*tanh(1/2*e*x+1/2*d)-a)^2+(2*A*a^2-A*c^2+3*C*a*c)/(a^4+2*a^2*c^2+c^4)/(a^2+c^2)^(1/2)*
arctanh(1/2*(2*a*tanh(1/2*e*x+1/2*d)-2*c)/(a^2+c^2)^(1/2)))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (172) = 344\).
time = 0.52, size = 743, normalized size = 4.13 \begin {gather*} \frac {1}{2} \, {\left (\frac {{\left (2 \, a^{2} - c^{2}\right )} e^{\left (-1\right )} \log \left (\frac {c e^{\left (-x e - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-x e - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )} \sqrt {a^{2} + c^{2}}} - \frac {2 \, {\left (3 \, a c^{2} + {\left (10 \, a^{2} c + c^{3}\right )} e^{\left (-x e - d\right )} + 3 \, {\left (2 \, a^{3} - a c^{2}\right )} e^{\left (-2 \, x e - 2 \, d\right )} - {\left (2 \, a^{2} c - c^{3}\right )} e^{\left (-3 \, x e - 3 \, d\right )}\right )} e^{\left (-1\right )}}{a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6} + 4 \, {\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} e^{\left (-x e - d\right )} + 2 \, {\left (2 \, a^{6} + 3 \, a^{4} c^{2} - c^{6}\right )} e^{\left (-2 \, x e - 2 \, d\right )} - 4 \, {\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} e^{\left (-3 \, x e - 3 \, d\right )} + {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} e^{\left (-4 \, x e - 4 \, d\right )}}\right )} A + \frac {1}{2} \, {\left (\frac {3 \, a c e^{\left (-1\right )} \log \left (\frac {c e^{\left (-x e - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-x e - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )} \sqrt {a^{2} + c^{2}}} + \frac {2 \, {\left (3 \, a c^{3} e^{\left (-3 \, x e - 3 \, d\right )} + a^{2} c^{2} - 2 \, c^{4} + {\left (4 \, a^{3} c - 5 \, a c^{3}\right )} e^{\left (-x e - d\right )} + {\left (2 \, a^{4} - 5 \, a^{2} c^{2} + 2 \, c^{4}\right )} e^{\left (-2 \, x e - 2 \, d\right )}\right )} e^{\left (-1\right )}}{a^{4} c^{3} + 2 \, a^{2} c^{5} + c^{7} + 4 \, {\left (a^{5} c^{2} + 2 \, a^{3} c^{4} + a c^{6}\right )} e^{\left (-x e - d\right )} + 2 \, {\left (2 \, a^{6} c + 3 \, a^{4} c^{3} - c^{7}\right )} e^{\left (-2 \, x e - 2 \, d\right )} - 4 \, {\left (a^{5} c^{2} + 2 \, a^{3} c^{4} + a c^{6}\right )} e^{\left (-3 \, x e - 3 \, d\right )} + {\left (a^{4} c^{3} + 2 \, a^{2} c^{5} + c^{7}\right )} e^{\left (-4 \, x e - 4 \, d\right )}}\right )} C - \frac {2 \, B e^{\left (-2 \, x e - 2 \, d - 1\right )}}{4 \, a c^{2} e^{\left (-x e - d\right )} - 4 \, a c^{2} e^{\left (-3 \, x e - 3 \, d\right )} + c^{3} e^{\left (-4 \, x e - 4 \, d\right )} + c^{3} + 2 \, {\left (2 \, a^{2} c - c^{3}\right )} e^{\left (-2 \, x e - 2 \, d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x, algorithm="maxima")

[Out]

1/2*((2*a^2 - c^2)*e^(-1)*log((c*e^(-x*e - d) - a - sqrt(a^2 + c^2))/(c*e^(-x*e - d) - a + sqrt(a^2 + c^2)))/(
(a^4 + 2*a^2*c^2 + c^4)*sqrt(a^2 + c^2)) - 2*(3*a*c^2 + (10*a^2*c + c^3)*e^(-x*e - d) + 3*(2*a^3 - a*c^2)*e^(-
2*x*e - 2*d) - (2*a^2*c - c^3)*e^(-3*x*e - 3*d))*e^(-1)/(a^4*c^2 + 2*a^2*c^4 + c^6 + 4*(a^5*c + 2*a^3*c^3 + a*
c^5)*e^(-x*e - d) + 2*(2*a^6 + 3*a^4*c^2 - c^6)*e^(-2*x*e - 2*d) - 4*(a^5*c + 2*a^3*c^3 + a*c^5)*e^(-3*x*e - 3
*d) + (a^4*c^2 + 2*a^2*c^4 + c^6)*e^(-4*x*e - 4*d)))*A + 1/2*(3*a*c*e^(-1)*log((c*e^(-x*e - d) - a - sqrt(a^2
+ c^2))/(c*e^(-x*e - d) - a + sqrt(a^2 + c^2)))/((a^4 + 2*a^2*c^2 + c^4)*sqrt(a^2 + c^2)) + 2*(3*a*c^3*e^(-3*x
*e - 3*d) + a^2*c^2 - 2*c^4 + (4*a^3*c - 5*a*c^3)*e^(-x*e - d) + (2*a^4 - 5*a^2*c^2 + 2*c^4)*e^(-2*x*e - 2*d))
*e^(-1)/(a^4*c^3 + 2*a^2*c^5 + c^7 + 4*(a^5*c^2 + 2*a^3*c^4 + a*c^6)*e^(-x*e - d) + 2*(2*a^6*c + 3*a^4*c^3 - c
^7)*e^(-2*x*e - 2*d) - 4*(a^5*c^2 + 2*a^3*c^4 + a*c^6)*e^(-3*x*e - 3*d) + (a^4*c^3 + 2*a^2*c^5 + c^7)*e^(-4*x*
e - 4*d)))*C - 2*B*e^(-2*x*e - 2*d - 1)/(4*a*c^2*e^(-x*e - d) - 4*a*c^2*e^(-3*x*e - 3*d) + c^3*e^(-4*x*e - 4*d
) + c^3 + 2*(2*a^2*c - c^3)*e^(-2*x*e - 2*d))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2652 vs. \(2 (172) = 344\).
time = 0.42, size = 2652, normalized size = 14.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x, algorithm="fricas")

[Out]

-1/2*(2*C*a^4*c^2 - 6*A*a^3*c^3 - 2*C*a^2*c^4 - 6*A*a*c^5 - 4*C*c^6 - 2*(2*A*a^4*c^2 + 3*C*a^3*c^3 + A*a^2*c^4
+ 3*C*a*c^5 - A*c^6)*cosh(x*cosh(1) + x*sinh(1) + d)^3 - 2*(2*A*a^4*c^2 + 3*C*a^3*c^3 + A*a^2*c^4 + 3*C*a*c^5
- A*c^6)*sinh(x*cosh(1) + x*sinh(1) + d)^3 + 2*(2*(B + C)*a^6 - 6*A*a^5*c + 3*(2*B - C)*a^4*c^2 - 3*A*a^3*c^3
+ 3*(2*B - C)*a^2*c^4 + 3*A*a*c^5 + 2*(B + C)*c^6)*cosh(x*cosh(1) + x*sinh(1) + d)^2 + 2*(2*(B + C)*a^6 - 6*A
*a^5*c + 3*(2*B - C)*a^4*c^2 - 3*A*a^3*c^3 + 3*(2*B - C)*a^2*c^4 + 3*A*a*c^5 + 2*(B + C)*c^6 - 3*(2*A*a^4*c^2
+ 3*C*a^3*c^3 + A*a^2*c^4 + 3*C*a*c^5 - A*c^6)*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d
)^2 + (2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5 + (2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5)*cosh(x*cosh(1) + x*sinh(1) + d)^4 +
(2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5)*sinh(x*cosh(1) + x*sinh(1) + d)^4 + 4*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A*a*c^4)
*cosh(x*cosh(1) + x*sinh(1) + d)^3 + 4*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A*a*c^4 + (2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5
)*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d)^3 + 2*(4*A*a^4*c + 6*C*a^3*c^2 - 4*A*a^2*c^
3 - 3*C*a*c^4 + A*c^5)*cosh(x*cosh(1) + x*sinh(1) + d)^2 + 2*(4*A*a^4*c + 6*C*a^3*c^2 - 4*A*a^2*c^3 - 3*C*a*c^
4 + A*c^5 + 3*(2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5)*cosh(x*cosh(1) + x*sinh(1) + d)^2 + 6*(2*A*a^3*c^2 + 3*C*a^2*c
^3 - A*a*c^4)*cosh(x*cosh(1) + x*sinh(1) + d))*sinh(x*cosh(1) + x*sinh(1) + d)^2 - 4*(2*A*a^3*c^2 + 3*C*a^2*c^
3 - A*a*c^4)*cosh(x*cosh(1) + x*sinh(1) + d) - 4*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A*a*c^4 - (2*A*a^2*c^3 + 3*C*a*c
^4 - A*c^5)*cosh(x*cosh(1) + x*sinh(1) + d)^3 - 3*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A*a*c^4)*cosh(x*cosh(1) + x*sin
h(1) + d)^2 - (4*A*a^4*c + 6*C*a^3*c^2 - 4*A*a^2*c^3 - 3*C*a*c^4 + A*c^5)*cosh(x*cosh(1) + x*sinh(1) + d))*sin
h(x*cosh(1) + x*sinh(1) + d))*sqrt(a^2 + c^2)*log((c^2*cosh(x*cosh(1) + x*sinh(1) + d)^2 + c^2*sinh(x*cosh(1)
+ x*sinh(1) + d)^2 + 2*a*c*cosh(x*cosh(1) + x*sinh(1) + d) + 2*a^2 + c^2 + 2*(c^2*cosh(x*cosh(1) + x*sinh(1) +
d) + a*c)*sinh(x*cosh(1) + x*sinh(1) + d) + 2*sqrt(a^2 + c^2)*(c*cosh(x*cosh(1) + x*sinh(1) + d) + c*sinh(x*c
osh(1) + x*sinh(1) + d) + a))/(c*cosh(x*cosh(1) + x*sinh(1) + d)^2 + c*sinh(x*cosh(1) + x*sinh(1) + d)^2 + 2*a
*cosh(x*cosh(1) + x*sinh(1) + d) + 2*(c*cosh(x*cosh(1) + x*sinh(1) + d) + a)*sinh(x*cosh(1) + x*sinh(1) + d) -
c)) - 2*(4*C*a^5*c - 10*A*a^4*c^2 - C*a^3*c^3 - 11*A*a^2*c^4 - 5*C*a*c^5 - A*c^6)*cosh(x*cosh(1) + x*sinh(1)
+ d) - 2*(4*C*a^5*c - 10*A*a^4*c^2 - C*a^3*c^3 - 11*A*a^2*c^4 - 5*C*a*c^5 - A*c^6 + 3*(2*A*a^4*c^2 + 3*C*a^3*c
^3 + A*a^2*c^4 + 3*C*a*c^5 - A*c^6)*cosh(x*cosh(1) + x*sinh(1) + d)^2 - 2*(2*(B + C)*a^6 - 6*A*a^5*c + 3*(2*B
- C)*a^4*c^2 - 3*A*a^3*c^3 + 3*(2*B - C)*a^2*c^4 + 3*A*a*c^5 + 2*(B + C)*c^6)*cosh(x*cosh(1) + x*sinh(1) + d))
*sinh(x*cosh(1) + x*sinh(1) + d))/(((a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*cosh(1) + (a^6*c^3 + 3*a^4*c^5 + 3
*a^2*c^7 + c^9)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^4 + ((a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*cosh(1)
+ (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*sinh(1))*sinh(x*cosh(1) + x*sinh(1) + d)^4 + 4*((a^7*c^2 + 3*a^5*c^4
+ 3*a^3*c^6 + a*c^8)*cosh(1) + (a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*sinh(1))*cosh(x*cosh(1) + x*sinh(1)
+ d)^3 + 4*((a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*cosh(1) + ((a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*cosh(
1) + (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d) + (a^7*c^2 + 3*a^5*c^4 +
3*a^3*c^6 + a*c^8)*sinh(1))*sinh(x*cosh(1) + x*sinh(1) + d)^3 + 2*((2*a^8*c + 5*a^6*c^3 + 3*a^4*c^5 - a^2*c^7
- c^9)*cosh(1) + (2*a^8*c + 5*a^6*c^3 + 3*a^4*c^5 - a^2*c^7 - c^9)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2
+ 2*(3*((a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*cosh(1) + (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*sinh(1))*co
sh(x*cosh(1) + x*sinh(1) + d)^2 + (2*a^8*c + 5*a^6*c^3 + 3*a^4*c^5 - a^2*c^7 - c^9)*cosh(1) + 6*((a^7*c^2 + 3*
a^5*c^4 + 3*a^3*c^6 + a*c^8)*cosh(1) + (a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*sinh(1))*cosh(x*cosh(1) + x*s
inh(1) + d) + (2*a^8*c + 5*a^6*c^3 + 3*a^4*c^5 - a^2*c^7 - c^9)*sinh(1))*sinh(x*cosh(1) + x*sinh(1) + d)^2 + (
a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*cosh(1) - 4*((a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*cosh(1) + (a^7*c
^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d) + (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^
7 + c^9)*sinh(1) + 4*(((a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^9)*cosh(1) + (a^6*c^3 + 3*a^4*c^5 + 3*a^2*c^7 + c^
9)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^3 + 3*((a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*cosh(1) + (a^7*c^
2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d)^2 - (a^7*c^2 + 3*a^5*c^4 + 3*a^3*c
^6 + a*c^8)*cosh(1) + ((2*a^8*c + 5*a^6*c^3 + 3*a^4*c^5 - a^2*c^7 - c^9)*cosh(1) + (2*a^8*c + 5*a^6*c^3 + 3*a^
4*c^5 - a^2*c^7 - c^9)*sinh(1))*cosh(x*cosh(1) + x*sinh(1) + d) - (a^7*c^2 + 3*a^5*c^4 + 3*a^3*c^6 + a*c^8)*si
nh(1))*sinh(x*cosh(1) + x*sinh(1) + d))

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (170) = 340\).
time = 0.45, size = 405, normalized size = 2.25 \begin {gather*} -\frac {\frac {{\left (2 \, A a^{2} + 3 \, C a c - A c^{2}\right )} \log \left (\frac {{\left | -2 \, c e^{\left (e x + d\right )} - 2 \, a - 2 \, \sqrt {a^{2} + c^{2}} \right |}}{{\left | -2 \, c e^{\left (e x + d\right )} - 2 \, a + 2 \, \sqrt {a^{2} + c^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )} \sqrt {a^{2} + c^{2}}} - \frac {2 \, {\left (2 \, A a^{2} c^{2} e^{\left (3 \, e x + 3 \, d\right )} + 3 \, C a c^{3} e^{\left (3 \, e x + 3 \, d\right )} - A c^{4} e^{\left (3 \, e x + 3 \, d\right )} - 2 \, B a^{4} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, C a^{4} e^{\left (2 \, e x + 2 \, d\right )} + 6 \, A a^{3} c e^{\left (2 \, e x + 2 \, d\right )} - 4 \, B a^{2} c^{2} e^{\left (2 \, e x + 2 \, d\right )} + 5 \, C a^{2} c^{2} e^{\left (2 \, e x + 2 \, d\right )} - 3 \, A a c^{3} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, B c^{4} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, C c^{4} e^{\left (2 \, e x + 2 \, d\right )} + 4 \, C a^{3} c e^{\left (e x + d\right )} - 10 \, A a^{2} c^{2} e^{\left (e x + d\right )} - 5 \, C a c^{3} e^{\left (e x + d\right )} - A c^{4} e^{\left (e x + d\right )} - C a^{2} c^{2} + 3 \, A a c^{3} + 2 \, C c^{4}\right )}}{{\left (a^{4} c + 2 \, a^{2} c^{3} + c^{5}\right )} {\left (c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c\right )}^{2}}}{2 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+c\,\mathrm {sinh}\left (d+e\,x\right )\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^3,x)

[Out]

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^3, x)

________________________________________________________________________________________