∫ sinh(x)_____ (a cosh(x)+b sinh(x))3 dx [702]

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3166, 37} \begin {gather*} \frac {\tanh ^2(x)}{2 a (a+b \tanh (x))^2} \end {gather*}

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +

1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ

Int[sin[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symb

ol] :> Dist[1/d, Subst[Int[x^m*((a + b*x)^n/(1 + x^2)^((m + n + 2)/2)), x], x, Tan[c + d*x]], x] /; FreeQ[{a,

b, c, d}, x] && IntegerQ[n] && IntegerQ[(m + n)/2] && NeQ[n, -1] && !(GtQ[n, 0] && GtQ[m, 1])

\begin {align*} \int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^3} \, dx &=-\text {Subst}\left (\int \frac {x}{(a-i b x)^3} \, dx,x,i \tanh (x)\right )\\ &=\frac {\tanh ^2(x)}{2 a (a+b \tanh (x))^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(19)=38\).
time = 0.09, size = 54, normalized size = 2.84 \begin {gather*} -\frac {a^2-b^2+b^2 \cosh (2 x)+a b \sinh (2 x)}{2 a (a-b) (a+b) (a \cosh (x)+b \sinh (x))^2} \end {gather*}

-1/2*(a^2 - b^2 + b^2*Cosh[2*x] + a*b*Sinh[2*x])/(a*(a - b)*(a + b)*(a*Cosh[x] + b*Sinh[x])^2)

________________________________________________________________________________________


method	result	size

default	\(\frac {2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{a \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )^{2}}\)	\(31\)
risch	\(-\frac {2 \left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}-b \right )}{\left ({\mathrm e}^{2 x} a +b \,{\mathrm e}^{2 x}+a -b \right )^{2} \left (a +b \right )^{2}}\)	\(43\)

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (17) = 34\).
time = 0.27, size = 167, normalized size = 8.79 \begin {gather*} -\frac {2 \, {\left (a - b\right )} e^{\left (-2 \, x\right )}}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )} + {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, x\right )}} - \frac {2 \, b}{a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, x\right )} + {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, x\right )}} \end {gather*}

-2*(a - b)*e^(-2*x)/(a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*x) + (a^4 - 4*a^3*b + 6*a

^2*b^2 - 4*a*b^3 + b^4)*e^(-4*x)) - 2*b/(a^4 - 2*a^2*b^2 + b^4 + 2*(a^4 - 2*a^3*b + 2*a*b^3 - b^4)*e^(-2*x) +

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (17) = 34\).
time = 0.33, size = 216, normalized size = 11.37 \begin {gather*} -\frac {2 \, {\left (a \cosh \left (x\right ) + {\left (a + 2 \, b\right )} \sinh \left (x\right )\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cosh \left (x\right )^{3} + 3 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \sinh \left (x\right )^{3} + {\left (3 \, a^{4} + 4 \, a^{3} b - 2 \, a^{2} b^{2} - 4 \, a b^{3} - b^{4}\right )} \cosh \left (x\right ) + {\left (a^{4} + 4 \, a^{3} b + 2 \, a^{2} b^{2} - 4 \, a b^{3} - 3 \, b^{4} + 3 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )} \end {gather*}

-2*(a*cosh(x) + (a + 2*b)*sinh(x))/((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(x)^3 + 3*(a^4 + 4*a^3*b +

6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(x)*sinh(x)^2 + (a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sinh(x)^3 + (3*a^4

+ 4*a^3*b - 2*a^2*b^2 - 4*a*b^3 - b^4)*cosh(x) + (a^4 + 4*a^3*b + 2*a^2*b^2 - 4*a*b^3 - 3*b^4 + 3*(a^4 + 4*a^

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (17) = 34\).
time = 0.42, size = 50, normalized size = 2.63 \begin {gather*} -\frac {2 \, {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} - b\right )}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}^{2}} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 1.60, size = 42, normalized size = 2.21 \begin {gather*} \frac {2\,b-{\mathrm {e}}^{2\,x}\,\left (2\,a+2\,b\right )}{{\left (a+b\right )}^2\,{\left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}^2} \end {gather*}

________________________________________________________________________________________

3.8.2 \(\int \frac {\sinh (x)}{(a \cosh (x)+b \sinh (x))^3} \, dx\) [702]