### 3.751 $$\int \frac{1}{(a+a \cosh (x)+c \sinh (x))^3} \, dx$$

Optimal. Leaf size=89 $\frac{\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{2 c^5}-\frac{3 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^4 (a \cosh (x)+a+c \sinh (x))}-\frac{a \sinh (x)+c \cosh (x)}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}$

[Out]

((3*a^2 - c^2)*Log[a + c*Tanh[x/2]])/(2*c^5) - (c*Cosh[x] + a*Sinh[x])/(2*c^2*(a + a*Cosh[x] + c*Sinh[x])^2) -
(3*(a*c*Cosh[x] + a^2*Sinh[x]))/(2*c^4*(a + a*Cosh[x] + c*Sinh[x]))

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Rubi [A]  time = 0.0936418, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {3129, 3153, 3124, 31} $\frac{\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{2 c^5}-\frac{3 \left (a^2 \sinh (x)+a c \cosh (x)\right )}{2 c^4 (a \cosh (x)+a+c \sinh (x))}-\frac{a \sinh (x)+c \cosh (x)}{2 c^2 (a \cosh (x)+a+c \sinh (x))^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + a*Cosh[x] + c*Sinh[x])^(-3),x]

[Out]

((3*a^2 - c^2)*Log[a + c*Tanh[x/2]])/(2*c^5) - (c*Cosh[x] + a*Sinh[x])/(2*c^2*(a + a*Cosh[x] + c*Sinh[x])^2) -
(3*(a*c*Cosh[x] + a^2*Sinh[x]))/(2*c^4*(a + a*Cosh[x] + c*Sinh[x]))

Rule 3129

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[((-(c*Cos[d
+ e*x]) + b*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1))/(e*(n + 1)*(a^2 - b^2 - c^2)), x] +
Dist[1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c*(n + 2)*Sin[d + e*x])*(a + b*C
os[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n
, -1] && NeQ[n, -3/2]

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{(a+a \cosh (x)+c \sinh (x))^3} \, dx &=-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{\int \frac{-2 a+a \cosh (x)+c \sinh (x)}{(a+a \cosh (x)+c \sinh (x))^2} \, dx}{2 c^2}\\ &=-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}+\frac{\left (3 a^2-c^2\right ) \int \frac{1}{a+a \cosh (x)+c \sinh (x)} \, dx}{2 c^4}\\ &=-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}+\frac{\left (3 a^2-c^2\right ) \operatorname{Subst}\left (\int \frac{1}{2 a+2 c x} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{c^4}\\ &=\frac{\left (3 a^2-c^2\right ) \log \left (a+c \tanh \left (\frac{x}{2}\right )\right )}{2 c^5}-\frac{c \cosh (x)+a \sinh (x)}{2 c^2 (a+a \cosh (x)+c \sinh (x))^2}-\frac{3 \left (a c \cosh (x)+a^2 \sinh (x)\right )}{2 c^4 (a+a \cosh (x)+c \sinh (x))}\\ \end{align*}

Mathematica [A]  time = 0.494929, size = 148, normalized size = 1.66 $\frac{4 \left (c^2-3 a^2\right ) \log \left (\cosh \left (\frac{x}{2}\right )\right )+\frac{6 c \left (c^2-a^2\right ) \sinh \left (\frac{x}{2}\right )}{a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )}+4 \left (3 a^2-c^2\right ) \log \left (a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )\right )+\frac{c^2 (a-c) (a+c)}{\left (a \cosh \left (\frac{x}{2}\right )+c \sinh \left (\frac{x}{2}\right )\right )^2}-6 a c \tanh \left (\frac{x}{2}\right )+c^2 \left (-\text{sech}^2\left (\frac{x}{2}\right )\right )}{8 c^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + a*Cosh[x] + c*Sinh[x])^(-3),x]

[Out]

(4*(-3*a^2 + c^2)*Log[Cosh[x/2]] + 4*(3*a^2 - c^2)*Log[a*Cosh[x/2] + c*Sinh[x/2]] - c^2*Sech[x/2]^2 + ((a - c)
*c^2*(a + c))/(a*Cosh[x/2] + c*Sinh[x/2])^2 + (6*c*(-a^2 + c^2)*Sinh[x/2])/(a*Cosh[x/2] + c*Sinh[x/2]) - 6*a*c
*Tanh[x/2])/(8*c^5)

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Maple [A]  time = 0.08, size = 138, normalized size = 1.6 \begin{align*}{\frac{1}{8\,{c}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{3\,a}{4\,{c}^{4}}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{{a}^{4}}{8\,{c}^{5}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{2}}{4\,{c}^{3}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{1}{8\,c} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{3}}{{c}^{5}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{a}{{c}^{3}} \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{3\,{a}^{2}}{2\,{c}^{5}}\ln \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{2\,{c}^{3}}\ln \left ( a+c\tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+a*cosh(x)+c*sinh(x))^3,x)

[Out]

1/8/c^3*tanh(1/2*x)^2-3/4/c^4*a*tanh(1/2*x)-1/8/c^5/(a+c*tanh(1/2*x))^2*a^4+1/4/c^3/(a+c*tanh(1/2*x))^2*a^2-1/
8/c/(a+c*tanh(1/2*x))^2+a^3/c^5/(a+c*tanh(1/2*x))-a/c^3/(a+c*tanh(1/2*x))+3/2/c^5*ln(a+c*tanh(1/2*x))*a^2-1/2/
c^3*ln(a+c*tanh(1/2*x))

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Maxima [B]  time = 1.12245, size = 335, normalized size = 3.76 \begin{align*} -\frac{3 \, a^{3} + 6 \, a^{2} c + 3 \, a c^{2} +{\left (9 \, a^{3} + 9 \, a^{2} c + a c^{2} + c^{3}\right )} e^{\left (-x\right )} + 3 \,{\left (3 \, a^{3} - a c^{2}\right )} e^{\left (-2 \, x\right )} +{\left (3 \, a^{3} - 3 \, a^{2} c - a c^{2} + c^{3}\right )} e^{\left (-3 \, x\right )}}{a^{2} c^{4} + 2 \, a c^{5} + c^{6} + 4 \,{\left (a^{2} c^{4} + a c^{5}\right )} e^{\left (-x\right )} + 2 \,{\left (3 \, a^{2} c^{4} - c^{6}\right )} e^{\left (-2 \, x\right )} + 4 \,{\left (a^{2} c^{4} - a c^{5}\right )} e^{\left (-3 \, x\right )} +{\left (a^{2} c^{4} - 2 \, a c^{5} + c^{6}\right )} e^{\left (-4 \, x\right )}} + \frac{{\left (3 \, a^{2} - c^{2}\right )} \log \left (-{\left (a - c\right )} e^{\left (-x\right )} - a - c\right )}{2 \, c^{5}} - \frac{{\left (3 \, a^{2} - c^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, c^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cosh(x)+c*sinh(x))^3,x, algorithm="maxima")

[Out]

-(3*a^3 + 6*a^2*c + 3*a*c^2 + (9*a^3 + 9*a^2*c + a*c^2 + c^3)*e^(-x) + 3*(3*a^3 - a*c^2)*e^(-2*x) + (3*a^3 - 3
*a^2*c - a*c^2 + c^3)*e^(-3*x))/(a^2*c^4 + 2*a*c^5 + c^6 + 4*(a^2*c^4 + a*c^5)*e^(-x) + 2*(3*a^2*c^4 - c^6)*e^
(-2*x) + 4*(a^2*c^4 - a*c^5)*e^(-3*x) + (a^2*c^4 - 2*a*c^5 + c^6)*e^(-4*x)) + 1/2*(3*a^2 - c^2)*log(-(a - c)*e
^(-x) - a - c)/c^5 - 1/2*(3*a^2 - c^2)*log(e^(-x) + 1)/c^5

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Fricas [B]  time = 2.82751, size = 3359, normalized size = 37.74 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cosh(x)+c*sinh(x))^3,x, algorithm="fricas")

[Out]

1/2*(6*a^3*c - 12*a^2*c^2 + 6*a*c^3 + 2*(3*a^3*c + 3*a^2*c^2 - a*c^3 - c^4)*cosh(x)^3 + 2*(3*a^3*c + 3*a^2*c^2
- a*c^3 - c^4)*sinh(x)^3 + 6*(3*a^3*c - a*c^3)*cosh(x)^2 + 6*(3*a^3*c - a*c^3 + (3*a^3*c + 3*a^2*c^2 - a*c^3
- c^4)*cosh(x))*sinh(x)^2 + 2*(9*a^3*c - 9*a^2*c^2 + a*c^3 - c^4)*cosh(x) + ((3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*
a*c^3 - c^4)*cosh(x)^4 + (3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*sinh(x)^4 + 3*a^4 - 6*a^3*c + 2*a^2*c^2
+ 2*a*c^3 - c^4 + 4*(3*a^4 + 3*a^3*c - a^2*c^2 - a*c^3)*cosh(x)^3 + 4*(3*a^4 + 3*a^3*c - a^2*c^2 - a*c^3 + (3
*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x))*sinh(x)^3 + 2*(9*a^4 - 6*a^2*c^2 + c^4)*cosh(x)^2 + 2*(9*
a^4 - 6*a^2*c^2 + c^4 + 3*(3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x)^2 + 6*(3*a^4 + 3*a^3*c - a^2*c
^2 - a*c^3)*cosh(x))*sinh(x)^2 + 4*(3*a^4 - 3*a^3*c - a^2*c^2 + a*c^3)*cosh(x) + 4*(3*a^4 - 3*a^3*c - a^2*c^2
+ a*c^3 + (3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x)^3 + 3*(3*a^4 + 3*a^3*c - a^2*c^2 - a*c^3)*cosh
(x)^2 + (9*a^4 - 6*a^2*c^2 + c^4)*cosh(x))*sinh(x))*log((a + c)*cosh(x) + (a + c)*sinh(x) + a - c) - ((3*a^4 +
6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x)^4 + (3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*sinh(x)^4 + 3*
a^4 - 6*a^3*c + 2*a^2*c^2 + 2*a*c^3 - c^4 + 4*(3*a^4 + 3*a^3*c - a^2*c^2 - a*c^3)*cosh(x)^3 + 4*(3*a^4 + 3*a^3
*c - a^2*c^2 - a*c^3 + (3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x))*sinh(x)^3 + 2*(9*a^4 - 6*a^2*c^2
+ c^4)*cosh(x)^2 + 2*(9*a^4 - 6*a^2*c^2 + c^4 + 3*(3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x)^2 + 6
*(3*a^4 + 3*a^3*c - a^2*c^2 - a*c^3)*cosh(x))*sinh(x)^2 + 4*(3*a^4 - 3*a^3*c - a^2*c^2 + a*c^3)*cosh(x) + 4*(3
*a^4 - 3*a^3*c - a^2*c^2 + a*c^3 + (3*a^4 + 6*a^3*c + 2*a^2*c^2 - 2*a*c^3 - c^4)*cosh(x)^3 + 3*(3*a^4 + 3*a^3*
c - a^2*c^2 - a*c^3)*cosh(x)^2 + (9*a^4 - 6*a^2*c^2 + c^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 2*(9
*a^3*c - 9*a^2*c^2 + a*c^3 - c^4 + 3*(3*a^3*c + 3*a^2*c^2 - a*c^3 - c^4)*cosh(x)^2 + 6*(3*a^3*c - a*c^3)*cosh(
x))*sinh(x))/(a^2*c^5 - 2*a*c^6 + c^7 + (a^2*c^5 + 2*a*c^6 + c^7)*cosh(x)^4 + (a^2*c^5 + 2*a*c^6 + c^7)*sinh(x
)^4 + 4*(a^2*c^5 + a*c^6)*cosh(x)^3 + 4*(a^2*c^5 + a*c^6 + (a^2*c^5 + 2*a*c^6 + c^7)*cosh(x))*sinh(x)^3 + 2*(3
*a^2*c^5 - c^7)*cosh(x)^2 + 2*(3*a^2*c^5 - c^7 + 3*(a^2*c^5 + 2*a*c^6 + c^7)*cosh(x)^2 + 6*(a^2*c^5 + a*c^6)*c
osh(x))*sinh(x)^2 + 4*(a^2*c^5 - a*c^6)*cosh(x) + 4*(a^2*c^5 - a*c^6 + (a^2*c^5 + 2*a*c^6 + c^7)*cosh(x)^3 + 3
*(a^2*c^5 + a*c^6)*cosh(x)^2 + (3*a^2*c^5 - c^7)*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(1/(a+a*cosh(x)+c*sinh(x))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.17161, size = 277, normalized size = 3.11 \begin{align*} \frac{{\left (3 \, a^{3} + 3 \, a^{2} c - a c^{2} - c^{3}\right )} \log \left ({\left | a e^{x} + c e^{x} + a - c \right |}\right )}{2 \,{\left (a c^{5} + c^{6}\right )}} - \frac{{\left (3 \, a^{2} - c^{2}\right )} \log \left (e^{x} + 1\right )}{2 \, c^{5}} + \frac{3 \, a^{3} e^{\left (3 \, x\right )} + 3 \, a^{2} c e^{\left (3 \, x\right )} - a c^{2} e^{\left (3 \, x\right )} - c^{3} e^{\left (3 \, x\right )} + 9 \, a^{3} e^{\left (2 \, x\right )} - 3 \, a c^{2} e^{\left (2 \, x\right )} + 9 \, a^{3} e^{x} - 9 \, a^{2} c e^{x} + a c^{2} e^{x} - c^{3} e^{x} + 3 \, a^{3} - 6 \, a^{2} c + 3 \, a c^{2}}{{\left (a e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + a - c\right )}^{2} c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+a*cosh(x)+c*sinh(x))^3,x, algorithm="giac")

[Out]

1/2*(3*a^3 + 3*a^2*c - a*c^2 - c^3)*log(abs(a*e^x + c*e^x + a - c))/(a*c^5 + c^6) - 1/2*(3*a^2 - c^2)*log(e^x
+ 1)/c^5 + (3*a^3*e^(3*x) + 3*a^2*c*e^(3*x) - a*c^2*e^(3*x) - c^3*e^(3*x) + 9*a^3*e^(2*x) - 3*a*c^2*e^(2*x) +
9*a^3*e^x - 9*a^2*c*e^x + a*c^2*e^x - c^3*e^x + 3*a^3 - 6*a^2*c + 3*a*c^2)/((a*e^(2*x) + c*e^(2*x) + 2*a*e^x +
a - c)^2*c^4)