### 3.801 $$\int \frac{b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx$$

Optimal. Leaf size=22 $\frac{b \sinh (x)+c \cosh (x)}{a+b \cosh (x)+c \sinh (x)}$

[Out]

(c*Cosh[x] + b*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])

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Rubi [A]  time = 0.0829528, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.031, Rules used = {3150} $\frac{b \sinh (x)+c \cosh (x)}{a+b \cosh (x)+c \sinh (x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c*Cosh[x] + b*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])

Rule 3150

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] /; FreeQ[{a, b, c, d, e, A, B,
C}, x] && NeQ[a^2 - b^2 - c^2, 0] && EqQ[a*A - b*B - c*C, 0]

Rubi steps

\begin{align*} \int \frac{b^2-c^2+a b \cosh (x)+a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2} \, dx &=\frac{c \cosh (x)+b \sinh (x)}{a+b \cosh (x)+c \sinh (x)}\\ \end{align*}

Mathematica [A]  time = 0.0831417, size = 34, normalized size = 1.55 $\frac{-a c+b^2 \sinh (x)-c^2 \sinh (x)}{b (a+b \cosh (x)+c \sinh (x))}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-(a*c) + b^2*Sinh[x] - c^2*Sinh[x])/(b*(a + b*Cosh[x] + c*Sinh[x]))

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Maple [B]  time = 0.082, size = 73, normalized size = 3.3 \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 ( ab-{b}^{2}+{c}^{2} \right ) \tanh \left ( x/2 \right ) }{a-b}}-{\frac{ac}{a-b}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2-c^2+a*b*cosh(x)+a*c*sinh(x))/(a+b*cosh(x)+c*sinh(x))^2,x)

[Out]

2*(-(a*b-b^2+c^2)/(a-b)*tanh(1/2*x)-a*c/(a-b))/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-2*c*tanh(1/2*x)-a-b)

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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((b^2-c^2+a*b*cosh(x)+a*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 = 2.3252, size = 178, normalized size = 8.09 \begin{align*} -\frac{2 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b - c\right )}}{{\left (b + c\right )} \cosh \left (x\right )^{2} +{\left (b + c\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \,{\left ({\left (b + c\right )} \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) + b - c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*cosh(x) + a*sinh(x) + b - c)/((b + c)*cosh(x)^2 + (b + c)*sinh(x)^2 + 2*a*cosh(x) + 2*((b + c)*cosh(x) +
a)*sinh(x) + b - c)

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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((b**2-c**2+a*b*cosh(x)+a*c*sinh(x))/(a+b*cosh(x)+c*sinh(x))**2,x)

[Out]

Timed out

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Giac [A]  time = 1.1841, size = 47, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (a e^{x} + b - c\right )}}{b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*e^x + b - c)/(b*e^(2*x) + c*e^(2*x) + 2*a*e^x + b - c)