### 3.799 $$\int \frac{A+B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx$$

Optimal. Leaf size=121 $-\frac{2 \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right ) (a A-b B+c C)}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac{\sinh (x) (A b-a B)+\cosh (x) (A c-a C)-b C+B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}$

[Out]

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

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Rubi [A]  time = 0.150261, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.174, Rules used = {3153, 3124, 618, 206} $-\frac{2 \tanh ^{-1}\left (\frac{c-(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2+c^2}}\right ) (a A-b B+c C)}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac{\sinh (x) (A b-a B)+\cosh (x) (A c-a C)-b C+B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3153

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)
*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B - c*C)/(a^2 -
b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[
a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]

Rule 3124

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(-1), x_Symbol] :> Module[{f = Free
Factors[Tan[(d + e*x)/2], x]}, Dist[(2*f)/e, Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d +
e*x)/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]

Rule 618

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 206

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

Rubi steps

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

Mathematica [A]  time = 0.372806, size = 143, normalized size = 1.18 $\frac{a^2 C+\sinh (x) \left (-a b B+a c C+A \left (b^2-c^2\right )\right )-a A c+b (B c-b C)}{b \left (-a^2+b^2-c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac{2 (a A-b B+c C) \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )+c}{\sqrt{-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [B]  time = 0.081, size = 376, normalized size = 3.1 \begin{align*} -2\,{\frac{1}{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}- \left ( \tanh \left ( x/2 \right ) \right ) ^{2}b-2\,c\tanh \left ( x/2 \right ) -a-b} \left ( -{\frac{ \left ( aAb-A{b}^{2}+A{c}^{2}-{a}^{2}B+abB-B{c}^{2}-acC+Ccb \right ) \tanh \left ( x/2 \right ) }{{a}^{3}-{a}^{2}b-a{b}^{2}+a{c}^{2}+{b}^{3}-b{c}^{2}}}-{\frac{aAc-bBc-{a}^{2}C+C{b}^{2}}{{a}^{3}-{a}^{2}b-a{b}^{2}+a{c}^{2}+{b}^{3}-b{c}^{2}}} \right ) }-2\,{\frac{aA}{ \left ({a}^{2}-{b}^{2}+{c}^{2} \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) }+2\,{\frac{Bb}{ \left ({a}^{2}-{b}^{2}+{c}^{2} \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{Cc}{ \left ({a}^{2}-{b}^{2}+{c}^{2} \right ) \sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tanh \left ( x/2 \right ) -2\,c}{\sqrt{-{a}^{2}+{b}^{2}-{c}^{2}}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(-(A*a*b-A*b^2+A*c^2-B*a^2+B*a*b-B*c^2-C*a*c+C*b*c)/(a^3-a^2*b-a*b^2+a*c^2+b^3-b*c^2)*tanh(1/2*x)-(A*a*c-B*
b*c-C*a^2+C*b^2)/(a^3-a^2*b-a*b^2+a*c^2+b^3-b*c^2))/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-2*c*tanh(1/2*x)-a-b)-2/(a
^2-b^2+c^2)/(-a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/2))*a*A+2/(a^2-b^2+c^2
)/(-a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/2))*B*b-2/(a^2-b^2+c^2)/(-a^2+b^
2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/2))*C*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.82662, size = 5477, normalized size = 45.26 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(2*B*a^3*b - 2*A*a^2*b^2 - 2*B*a*b^3 + 2*A*b^4 - 2*C*a*c^3 + 2*A*c^4 + 2*(A*a^2 + B*a*b - 2*A*b^2)*c^2 - (A*
a*b^2 - B*b^3 + C*b^2*c - C*c^3 - (A*a - B*b)*c^2 + (A*a*b^2 - B*b^3 + C*c^3 + (A*a - (B - 2*C)*b)*c^2 + (2*A*
a*b - (2*B - C)*b^2)*c)*cosh(x)^2 + (A*a*b^2 - B*b^3 + C*c^3 + (A*a - (B - 2*C)*b)*c^2 + (2*A*a*b - (2*B - C)*
b^2)*c)*sinh(x)^2 + 2*(A*a^2*b - B*a*b^2 + C*a*c^2 + (A*a^2 - (B - C)*a*b)*c)*cosh(x) + 2*(A*a^2*b - B*a*b^2 +
C*a*c^2 + (A*a^2 - (B - C)*a*b)*c + (A*a*b^2 - B*b^3 + C*c^3 + (A*a - (B - 2*C)*b)*c^2 + (2*A*a*b - (2*B - C)
*b^2)*c)*cosh(x))*sinh(x))*sqrt(a^2 - b^2 + c^2)*log(((b^2 + 2*b*c + c^2)*cosh(x)^2 + (b^2 + 2*b*c + c^2)*sinh
(x)^2 + 2*a^2 - b^2 + c^2 + 2*(a*b + a*c)*cosh(x) + 2*(a*b + a*c + (b^2 + 2*b*c + c^2)*cosh(x))*sinh(x) - 2*sq
rt(a^2 - b^2 + c^2)*((b + c)*cosh(x) + (b + c)*sinh(x) + a))/((b + c)*cosh(x)^2 + (b + c)*sinh(x)^2 + 2*a*cosh
(x) + 2*((b + c)*cosh(x) + a)*sinh(x) + b - c)) - 2*(C*a^3 - C*a*b^2)*c + 2*((B + C)*a^4 - A*a^3*b - (B + 2*C)
*a^2*b^2 + A*a*b^3 + C*b^4 + B*c^4 - (A*a - (B - C)*b)*c^3 + ((2*B + C)*a^2 - A*a*b - (B + C)*b^2)*c^2 - (A*a^
3 - (B - C)*a^2*b - A*a*b^2 + (B - C)*b^3)*c)*cosh(x) + 2*((B + C)*a^4 - A*a^3*b - (B + 2*C)*a^2*b^2 + A*a*b^3
+ C*b^4 + B*c^4 - (A*a - (B - C)*b)*c^3 + ((2*B + C)*a^2 - A*a*b - (B + C)*b^2)*c^2 - (A*a^3 - (B - C)*a^2*b
- A*a*b^2 + (B - C)*b^3)*c)*sinh(x))/(a^4*b^2 - 2*a^2*b^4 + b^6 - c^6 - (2*a^2 - 3*b^2)*c^4 - (a^4 - 4*a^2*b^2
+ 3*b^4)*c^2 + (a^4*b^2 - 2*a^2*b^4 + b^6 + 2*b*c^5 + c^6 + (2*a^2 - b^2)*c^4 + 4*(a^2*b - b^3)*c^3 + (a^4 -
b^4)*c^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*c)*cosh(x)^2 + (a^4*b^2 - 2*a^2*b^4 + b^6 + 2*b*c^5 + c^6 + (2*a^2 - b^
2)*c^4 + 4*(a^2*b - b^3)*c^3 + (a^4 - b^4)*c^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*c)*sinh(x)^2 + 2*(a^5*b - 2*a^3*b
^3 + a*b^5 + a*b*c^4 + a*c^5 + 2*(a^3 - a*b^2)*c^3 + 2*(a^3*b - a*b^3)*c^2 + (a^5 - 2*a^3*b^2 + a*b^4)*c)*cosh
(x) + 2*(a^5*b - 2*a^3*b^3 + a*b^5 + a*b*c^4 + a*c^5 + 2*(a^3 - a*b^2)*c^3 + 2*(a^3*b - a*b^3)*c^2 + (a^5 - 2*
a^3*b^2 + a*b^4)*c + (a^4*b^2 - 2*a^2*b^4 + b^6 + 2*b*c^5 + c^6 + (2*a^2 - b^2)*c^4 + 4*(a^2*b - b^3)*c^3 + (a
^4 - b^4)*c^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*c)*cosh(x))*sinh(x)), -2*(B*a^3*b - A*a^2*b^2 - B*a*b^3 + A*b^4 -
C*a*c^3 + A*c^4 + (A*a^2 + B*a*b - 2*A*b^2)*c^2 - (A*a*b^2 - B*b^3 + C*b^2*c - C*c^3 - (A*a - B*b)*c^2 + (A*a*
b^2 - B*b^3 + C*c^3 + (A*a - (B - 2*C)*b)*c^2 + (2*A*a*b - (2*B - C)*b^2)*c)*cosh(x)^2 + (A*a*b^2 - B*b^3 + C*
c^3 + (A*a - (B - 2*C)*b)*c^2 + (2*A*a*b - (2*B - C)*b^2)*c)*sinh(x)^2 + 2*(A*a^2*b - B*a*b^2 + C*a*c^2 + (A*a
^2 - (B - C)*a*b)*c)*cosh(x) + 2*(A*a^2*b - B*a*b^2 + C*a*c^2 + (A*a^2 - (B - C)*a*b)*c + (A*a*b^2 - B*b^3 + C
*c^3 + (A*a - (B - 2*C)*b)*c^2 + (2*A*a*b - (2*B - C)*b^2)*c)*cosh(x))*sinh(x))*sqrt(-a^2 + b^2 - c^2)*arctan(
sqrt(-a^2 + b^2 - c^2)*((b + c)*cosh(x) + (b + c)*sinh(x) + a)/(a^2 - b^2 + c^2)) - (C*a^3 - C*a*b^2)*c + ((B
+ C)*a^4 - A*a^3*b - (B + 2*C)*a^2*b^2 + A*a*b^3 + C*b^4 + B*c^4 - (A*a - (B - C)*b)*c^3 + ((2*B + C)*a^2 - A*
a*b - (B + C)*b^2)*c^2 - (A*a^3 - (B - C)*a^2*b - A*a*b^2 + (B - C)*b^3)*c)*cosh(x) + ((B + C)*a^4 - A*a^3*b -
(B + 2*C)*a^2*b^2 + A*a*b^3 + C*b^4 + B*c^4 - (A*a - (B - C)*b)*c^3 + ((2*B + C)*a^2 - A*a*b - (B + C)*b^2)*c
^2 - (A*a^3 - (B - C)*a^2*b - A*a*b^2 + (B - C)*b^3)*c)*sinh(x))/(a^4*b^2 - 2*a^2*b^4 + b^6 - c^6 - (2*a^2 - 3
*b^2)*c^4 - (a^4 - 4*a^2*b^2 + 3*b^4)*c^2 + (a^4*b^2 - 2*a^2*b^4 + b^6 + 2*b*c^5 + c^6 + (2*a^2 - b^2)*c^4 + 4
*(a^2*b - b^3)*c^3 + (a^4 - b^4)*c^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*c)*cosh(x)^2 + (a^4*b^2 - 2*a^2*b^4 + b^6 +
2*b*c^5 + c^6 + (2*a^2 - b^2)*c^4 + 4*(a^2*b - b^3)*c^3 + (a^4 - b^4)*c^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*c)*si
nh(x)^2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5 + a*b*c^4 + a*c^5 + 2*(a^3 - a*b^2)*c^3 + 2*(a^3*b - a*b^3)*c^2 + (a^5
- 2*a^3*b^2 + a*b^4)*c)*cosh(x) + 2*(a^5*b - 2*a^3*b^3 + a*b^5 + a*b*c^4 + a*c^5 + 2*(a^3 - a*b^2)*c^3 + 2*(a^
3*b - a*b^3)*c^2 + (a^5 - 2*a^3*b^2 + a*b^4)*c + (a^4*b^2 - 2*a^2*b^4 + b^6 + 2*b*c^5 + c^6 + (2*a^2 - b^2)*c^
4 + 4*(a^2*b - b^3)*c^3 + (a^4 - b^4)*c^2 + 2*(a^4*b - 2*a^2*b^3 + b^5)*c)*cosh(x))*sinh(x))]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.19316, size = 279, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (A a - B b + C c\right )} \arctan \left (\frac{b e^{x} + c e^{x} + a}{\sqrt{-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{2} - b^{2} + c^{2}\right )} \sqrt{-a^{2} + b^{2} - c^{2}}} - \frac{2 \,{\left (B a^{2} e^{x} + C a^{2} e^{x} - A a b e^{x} - C b^{2} e^{x} - A a c e^{x} + B b c e^{x} - C b c e^{x} + B c^{2} e^{x} + B a b - A b^{2} - C a c + A c^{2}\right )}}{{\left (a^{2} b - b^{3} + a^{2} c - b^{2} c + b c^{2} + c^{3}\right )}{\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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