\(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx\) [254]

Optimal result

Integrand size = 31, antiderivative size = 113 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=-\frac {2 (a A+c C) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{3/2} e}-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4461, 2833, 12, 2739, 632, 210, 2747, 32} \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=-\frac {2 (a A+c C) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{3/2}}-\frac {(A c-a C) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}-\frac {B}{c e (a+c \sinh (d+e x))} \]

[In]

Int[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^2,x]

[Out]

(-2*(a*A + c*C)*ArcTanh[(c - a*Tanh[(d + e*x)/2])/Sqrt[a^2 + c^2]])/((a^2 + c^2)^(3/2)*e) - B/(c*e*(a + c*Sinh
[d + e*x])) - ((A*c - a*C)*Cosh[d + e*x])/((a^2 + c^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*} \text {integral}& = B \int \frac {\cosh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx+\int \frac {A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx \\ & = -\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}-\frac {\int \frac {-a A-c C}{a+c \sinh (d+e x)} \, dx}{a^2+c^2}+\frac {B \text {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,c \sinh (d+e x)\right )}{c e} \\ & = -\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}+\frac {(a A+c C) \int \frac {1}{a+c \sinh (d+e x)} \, dx}{a^2+c^2} \\ & = -\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}-\frac {(2 i (a A+c C)) \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 ) e} \\ & = -\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}+\frac {(4 i (a A+c C)) \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 ) e} \\ & = -\frac {2 (a A+c C) \text {arctanh}\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 )^{3/2} e}-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=\frac {\frac {2 (a A+c C) \arctan \left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}-\frac {B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)}{c (a+c \sinh (d+e x))}}{\left (a^2+c^2\right ) e} \]

[In]

Integrate[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^2,x]

[Out]

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

Maple [A] (verified)

Time = 3.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (A \,c^{2}-B \,a^{2}-B \,c^{2}-C a c \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{2}+c^{2}\right )}-\frac {A c -C a}{a^{2}+c^{2}}\right )}{a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a}+\frac {2 \left (A a +c C \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}}}}{e}\)	\(151\)
default	\(\frac {-\frac {2 \left (-\frac {\left (A \,c^{2}-B \,a^{2}-B \,c^{2}-C a c \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{2}+c^{2}\right )}-\frac {A c -C a}{a^{2}+c^{2}}\right )}{a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a}+\frac {2 \left (A a +c C \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}}}}{e}\)	\(151\)
parts	\(\frac {-\frac {2 \left (-\frac {c \left (A c -C a \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{2}+c^{2}\right )}-\frac {A c -C a}{a^{2}+c^{2}}\right )}{a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a}+\frac {2 \left (A a +c C \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}}}}{e}-\frac {B}{c e \left (a +c \sinh \left (e x +d \right )\right )}\)	\(159\)
risch	\(\frac {2 A a c \,{\mathrm e}^{e x +d}-2 B \,a^{2} {\mathrm e}^{e x +d}-2 B \,c^{2} {\mathrm e}^{e x +d}-2 C \,a^{2} {\mathrm e}^{e x +d}-2 A \,c^{2}+2 C a c}{c e \left (a^{2}+c^{2}\right ) \left (c \,{\mathrm e}^{2 e x +2 d}+2 a \,{\mathrm e}^{e x +d}-c \right )}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a -a^{4}-2 a^{2} c^{2}-c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) A a}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a -a^{4}-2 a^{2} c^{2}-c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) c C}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a +a^{4}+2 a^{2} c^{2}+c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) A a}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a +a^{4}+2 a^{2} c^{2}+c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) c C}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}\)	\(360\)

[In]

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (109) = 218\).

Time = 0.27 (sec) , antiderivative size = 570, normalized size of antiderivative = 5.04 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=\frac {2 \, C a^{3} c - 2 \, A a^{2} c^{2} + 2 \, C a c^{3} - 2 \, A c^{4} - {\left (A a c^{2} + C c^{3} - {\left (A a c^{2} + C c^{3}\right )} \cosh \left (e x + d\right )^{2} - {\left (A a c^{2} + C c^{3}\right )} \sinh \left (e x + d\right )^{2} - 2 \, {\left (A a^{2} c + C a c^{2}\right )} \cosh \left (e x + d\right ) - 2 \, {\left (A a^{2} c + C a c^{2} + {\left (A a c^{2} + C c^{3}\right )} \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right )\right )} \sqrt {a^{2} + c^{2}} \log \left (\frac {c^{2} \cosh \left (e x + d\right )^{2} + c^{2} \sinh \left (e x + d\right )^{2} + 2 \, a c \cosh \left (e x + d\right ) + 2 \, a^{2} + c^{2} + 2 \, {\left (c^{2} \cosh \left (e x + d\right ) + a c\right )} \sinh \left (e x + d\right ) - 2 \, \sqrt {a^{2} + c^{2}} {\left (c \cosh \left (e x + d\right ) + c \sinh \left (e x + d\right ) + a\right )}}{c \cosh \left (e x + d\right )^{2} + c \sinh \left (e x + d\right )^{2} + 2 \, a \cosh \left (e x + d\right ) + 2 \, {\left (c \cosh \left (e x + d\right ) + a\right )} \sinh \left (e x + d\right ) - c}\right ) - 2 \, {\left ({\left (B + C\right )} a^{4} - A a^{3} c + {\left (2 \, B + C\right )} a^{2} c^{2} - A a c^{3} + B c^{4}\right )} \cosh \left (e x + d\right ) - 2 \, {\left ({\left (B + C\right )} a^{4} - A a^{3} c + {\left (2 \, B + C\right )} a^{2} c^{2} - A a c^{3} + B c^{4}\right )} \sinh \left (e x + d\right )}{{\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} e \cosh \left (e x + d\right )^{2} + {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} e \sinh \left (e x + d\right )^{2} + 2 \, {\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} e \cosh \left (e x + d\right ) - {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} e + 2 \, {\left ({\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} e \cosh \left (e x + d\right ) + {\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} e\right )} \sinh \left (e x + d\right )} \]

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^2,x, algorithm="fricas")

[Out]

(2*C*a^3*c - 2*A*a^2*c^2 + 2*C*a*c^3 - 2*A*c^4 - (A*a*c^2 + C*c^3 - (A*a*c^2 + C*c^3)*cosh(e*x + d)^2 - (A*a*c
^2 + C*c^3)*sinh(e*x + d)^2 - 2*(A*a^2*c + C*a*c^2)*cosh(e*x + d) - 2*(A*a^2*c + C*a*c^2 + (A*a*c^2 + C*c^3)*c
osh(e*x + d))*sinh(e*x + d))*sqrt(a^2 + c^2)*log((c^2*cosh(e*x + d)^2 + c^2*sinh(e*x + d)^2 + 2*a*c*cosh(e*x +
d) + 2*a^2 + c^2 + 2*(c^2*cosh(e*x + d) + a*c)*sinh(e*x + d) - 2*sqrt(a^2 + c^2)*(c*cosh(e*x + d) + c*sinh(e*
x + d) + a))/(c*cosh(e*x + d)^2 + c*sinh(e*x + d)^2 + 2*a*cosh(e*x + d) + 2*(c*cosh(e*x + d) + a)*sinh(e*x + d
) - c)) - 2*((B + C)*a^4 - A*a^3*c + (2*B + C)*a^2*c^2 - A*a*c^3 + B*c^4)*cosh(e*x + d) - 2*((B + C)*a^4 - A*a
^3*c + (2*B + C)*a^2*c^2 - A*a*c^3 + B*c^4)*sinh(e*x + d))/((a^4*c^2 + 2*a^2*c^4 + c^6)*e*cosh(e*x + d)^2 + (a
^4*c^2 + 2*a^2*c^4 + c^6)*e*sinh(e*x + d)^2 + 2*(a^5*c + 2*a^3*c^3 + a*c^5)*e*cosh(e*x + d) - (a^4*c^2 + 2*a^2
*c^4 + c^6)*e + 2*((a^4*c^2 + 2*a^2*c^4 + c^6)*e*cosh(e*x + d) + (a^5*c + 2*a^3*c^3 + a*c^5)*e)*sinh(e*x + d))

Sympy [F(-1)]

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

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (109) = 218\).

Time = 0.35 (sec) , antiderivative size = 339, normalized size of antiderivative = 3.00 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=A {\left (\frac {a \log \left (\frac {c e^{\left (-e x - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-e x - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{2} + c^{2}\right )}^{\frac {3}{2}} e} - \frac {2 \, {\left (a e^{\left (-e x - d\right )} + c\right )}}{{\left (a^{2} c + c^{3} + 2 \, {\left (a^{3} + a c^{2}\right )} e^{\left (-e x - d\right )} - {\left (a^{2} c + c^{3}\right )} e^{\left (-2 \, e x - 2 \, d\right )}\right )} e}\right )} + C {\left (\frac {c \log \left (\frac {c e^{\left (-e x - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-e x - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{2} + c^{2}\right )}^{\frac {3}{2}} e} + \frac {2 \, {\left (a^{2} e^{\left (-e x - d\right )} + a c\right )}}{{\left (a^{2} c^{2} + c^{4} + 2 \, {\left (a^{3} c + a c^{3}\right )} e^{\left (-e x - d\right )} - {\left (a^{2} c^{2} + c^{4}\right )} e^{\left (-2 \, e x - 2 \, d\right )}\right )} e}\right )} - \frac {2 \, B e^{\left (-e x - d\right )}}{{\left (2 \, a c e^{\left (-e x - d\right )} - c^{2} e^{\left (-2 \, e x - 2 \, d\right )} + c^{2}\right )} e} \]

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.50 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=\frac {\frac {{\left (A a + C c\right )} \log \left (\frac {{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a - 2 \, \sqrt {a^{2} + c^{2}} \right |}}{{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a + 2 \, \sqrt {a^{2} + c^{2}} \right |}}\right )}{{\left (a^{2} + c^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (B a^{2} e^{\left (e x + d\right )} + C a^{2} e^{\left (e x + d\right )} - A a c e^{\left (e x + d\right )} + B c^{2} e^{\left (e x + d\right )} - C a c + A c^{2}\right )}}{{\left (a^{2} c + c^{3}\right )} {\left (c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c\right )}}}{e} \]

[In]

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^2,x, algorithm="giac")

[Out]

((A*a + C*c)*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^2 + c^2)^(3/2) - 2*(B*a^2*e^(e*x + d) + C*a^2*e^(e*x + d) - A*a*c*e^(e*x + d) + B*c^2*e^(e*x + d) - C*a*c
+ A*c^2)/((a^2*c + c^3)*(c*e^(2*e*x + 2*d) + 2*a*e^(e*x + d) - c)))/e

Mupad [B] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.47 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx=\frac {\ln \left (\frac {2\,\left (A\,a+C\,c\right )\,\left (c-a\,{\mathrm {e}}^{d+e\,x}\right )}{c\,{\left (a^2+c^2\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (A\,a+C\,c\right )}{c\,\left (a^2+c^2\right )}\right )\,\left (A\,a+C\,c\right )}{e\,{\left (a^2+c^2\right )}^{3/2}}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (A\,a+C\,c\right )}{c\,\left (a^2+c^2\right )}-\frac {2\,\left (A\,a+C\,c\right )\,\left (c-a\,{\mathrm {e}}^{d+e\,x}\right )}{c\,{\left (a^2+c^2\right )}^{3/2}}\right )\,\left (A\,a+C\,c\right )}{e\,{\left (a^2+c^2\right )}^{3/2}}-\frac {\frac {2\,\left (A\,c^3-C\,a\,c^2\right )}{c\,e\,\left (a^2\,c+c^3\right )}+\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (B\,c^4+B\,a^2\,c^2+C\,a^2\,c^2-A\,a\,c^3\right )}{c^2\,e\,\left (a^2\,c+c^3\right )}}{2\,a\,{\mathrm {e}}^{d+e\,x}-c+c\,{\mathrm {e}}^{2\,d+2\,e\,x}} \]

[In]

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^2,x)

[Out]

(log((2*(A*a + C*c)*(c - a*exp(d + e*x)))/(c*(a^2 + c^2)^(3/2)) - (2*exp(d + e*x)*(A*a + C*c))/(c*(a^2 + c^2))
)*(A*a + C*c))/(e*(a^2 + c^2)^(3/2)) - (log(- (2*exp(d + e*x)*(A*a + C*c))/(c*(a^2 + c^2)) - (2*(A*a + C*c)*(c
- a*exp(d + e*x)))/(c*(a^2 + c^2)^(3/2)))*(A*a + C*c))/(e*(a^2 + c^2)^(3/2)) - ((2*(A*c^3 - C*a*c^2))/(c*e*(a
^2*c + c^3)) + (2*exp(d + e*x)*(B*c^4 + B*a^2*c^2 + C*a^2*c^2 - A*a*c^3))/(c^2*e*(a^2*c + c^3)))/(2*a*exp(d +
e*x) - c + c*exp(2*d + 2*e*x))