3.4.46 \(\int \frac {\cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [346]

Optimal. Leaf size=85 \[ -\frac {a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \]

[Out]

-a*(a^2+b^2)*ln(a+b*sinh(d*x+c))/b^4/d+(a^2+b^2)*sinh(d*x+c)/b^3/d-1/2*a*sinh(d*x+c)^2/b^2/d+1/3*sinh(d*x+c)^3
/b/d

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Rubi [A]
time = 0.09, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2916, 12, 786} \begin {gather*} -\frac {a \left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{b^4 d}+\frac {\left (a^2+b^2\right ) \sinh (c+d x)}{b^3 d}-\frac {a \sinh ^2(c+d x)}{2 b^2 d}+\frac {\sinh ^3(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[c + d*x]^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-((a*(a^2 + b^2)*Log[a + b*Sinh[c + d*x]])/(b^4*d)) + ((a^2 + b^2)*Sinh[c + d*x])/(b^3*d) - (a*Sinh[c + d*x]^2
)/(2*b^2*d) + Sinh[c + d*x]^3/(3*b*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 786

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rule 2916

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.23, size = 90, normalized size = 1.06 \begin {gather*} \frac {-3 a b^2 \cosh (2 (c+d x))-12 a^3 \log (a+b \sinh (c+d x))-12 a b^2 \log (a+b \sinh (c+d x))+3 b \left (4 a^2+3 b^2\right ) \sinh (c+d x)+b^3 \sinh (3 (c+d x))}{12 b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[c + d*x]^3*Sinh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(-3*a*b^2*Cosh[2*(c + d*x)] - 12*a^3*Log[a + b*Sinh[c + d*x]] - 12*a*b^2*Log[a + b*Sinh[c + d*x]] + 3*b*(4*a^2
+ 3*b^2)*Sinh[c + d*x] + b^3*Sinh[3*(c + d*x)])/(12*b^4*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(81)=162\).
time = 0.89, size = 246, normalized size = 2.89

method	result	size
risch	\(\frac {a^{3} x}{b^{4}}+\frac {a x}{b^{2}}+\frac {{\mathrm e}^{3 d x +3 c}}{24 b d}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 b^{2} d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 b^{3} d}+\frac {3 \,{\mathrm e}^{d x +c}}{8 b d}-\frac {{\mathrm e}^{-d x -c} a^{2}}{2 b^{3} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 b d}-\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 b^{2} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 b d}+\frac {2 a^{3} c}{b^{4} d}+\frac {2 a c}{b^{2} d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{4} d}-\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{b^{2} d}\)	\(244\)
derivativedivides	\(\frac {-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-b +a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a^{2}-a b +2 b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {2 a \left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{4}}}{d}\)	\(246\)
default	\(\frac {-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}-\frac {1}{3 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-b +a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a^{2}-a b +2 b^{2}}{2 b^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {a \left (a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {2 a \left (\frac {a^{2}}{2}+\frac {b^{2}}{2}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{b^{4}}}{d}\)	\(246\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/3/b/(tanh(1/2*d*x+1/2*c)-1)^3-1/2*(a+b)/b^2/(tanh(1/2*d*x+1/2*c)-1)^2-1/2*(2*a^2+a*b+2*b^2)/b^3/(tanh(
1/2*d*x+1/2*c)-1)+a*(a^2+b^2)/b^4*ln(tanh(1/2*d*x+1/2*c)-1)-1/3/b/(tanh(1/2*d*x+1/2*c)+1)^3-1/2*(-b+a)/b^2/(ta
nh(1/2*d*x+1/2*c)+1)^2-1/2*(2*a^2-a*b+2*b^2)/b^3/(tanh(1/2*d*x+1/2*c)+1)+a*(a^2+b^2)/b^4*ln(tanh(1/2*d*x+1/2*c
)+1)-2/b^4*a*(1/2*a^2+1/2*b^2)*ln(a*tanh(1/2*d*x+1/2*c)^2-2*b*tanh(1/2*d*x+1/2*c)-a))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (81) = 162\).
time = 0.27, size = 183, normalized size = 2.15 \begin {gather*} -\frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{24 \, b^{3} d} - \frac {{\left (a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{b^{4} d} - \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{24 \, b^{3} d} - \frac {{\left (a^{3} + a b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + 3*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) - (a^3 + a*b^2)*
(d*x + c)/(b^4*d) - 1/24*(3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + 3*b^2)*e^(-d*x - c))/(b^3
*d) - (a^3 + a*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^4*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (81) = 162\).
time = 0.39, size = 652, normalized size = 7.67 \begin {gather*} \frac {b^{3} \cosh \left (d x + c\right )^{6} + b^{3} \sinh \left (d x + c\right )^{6} - 3 \, a b^{2} \cosh \left (d x + c\right )^{5} + 24 \, {\left (a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{3} + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right ) - a b^{2}\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{2} - 5 \, a b^{2} \cosh \left (d x + c\right ) + 4 \, a^{2} b + 3 \, b^{3}\right )} \sinh \left (d x + c\right )^{4} - 3 \, a b^{2} \cosh \left (d x + c\right ) + 2 \, {\left (10 \, b^{3} \cosh \left (d x + c\right )^{3} - 15 \, a b^{2} \cosh \left (d x + c\right )^{2} + 12 \, {\left (a^{3} + a b^{2}\right )} d x + 6 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - b^{3} - 3 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} - 10 \, a b^{2} \cosh \left (d x + c\right )^{3} + 24 \, {\left (a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right ) - 4 \, a^{2} b - 3 \, b^{3} + 6 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} - 24 \, {\left ({\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (a^{3} + a b^{2}\right )} \sinh \left (d x + c\right )^{3}\right )} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 3 \, {\left (2 \, b^{3} \cosh \left (d x + c\right )^{5} - 5 \, a b^{2} \cosh \left (d x + c\right )^{4} + 24 \, {\left (a^{3} + a b^{2}\right )} d x \cosh \left (d x + c\right )^{2} + 4 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - a b^{2} - 2 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (b^{4} d \cosh \left (d x + c\right )^{3} + 3 \, b^{4} d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, b^{4} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^3*cosh(d*x + c)^6 + b^3*sinh(d*x + c)^6 - 3*a*b^2*cosh(d*x + c)^5 + 24*(a^3 + a*b^2)*d*x*cosh(d*x + c)
^3 + 3*(2*b^3*cosh(d*x + c) - a*b^2)*sinh(d*x + c)^5 + 3*(4*a^2*b + 3*b^3)*cosh(d*x + c)^4 + 3*(5*b^3*cosh(d*x
+ c)^2 - 5*a*b^2*cosh(d*x + c) + 4*a^2*b + 3*b^3)*sinh(d*x + c)^4 - 3*a*b^2*cosh(d*x + c) + 2*(10*b^3*cosh(d*
x + c)^3 - 15*a*b^2*cosh(d*x + c)^2 + 12*(a^3 + a*b^2)*d*x + 6*(4*a^2*b + 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^
3 - b^3 - 3*(4*a^2*b + 3*b^3)*cosh(d*x + c)^2 + 3*(5*b^3*cosh(d*x + c)^4 - 10*a*b^2*cosh(d*x + c)^3 + 24*(a^3
+ a*b^2)*d*x*cosh(d*x + c) - 4*a^2*b - 3*b^3 + 6*(4*a^2*b + 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 24*((a^3
+ a*b^2)*cosh(d*x + c)^3 + 3*(a^3 + a*b^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^3 + a*b^2)*cosh(d*x + c)*sinh
(d*x + c)^2 + (a^3 + a*b^2)*sinh(d*x + c)^3)*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + 3*
(2*b^3*cosh(d*x + c)^5 - 5*a*b^2*cosh(d*x + c)^4 + 24*(a^3 + a*b^2)*d*x*cosh(d*x + c)^2 + 4*(4*a^2*b + 3*b^3)*
cosh(d*x + c)^3 - a*b^2 - 2*(4*a^2*b + 3*b^3)*cosh(d*x + c))*sinh(d*x + c))/(b^4*d*cosh(d*x + c)^3 + 3*b^4*d*c
osh(d*x + c)^2*sinh(d*x + c) + 3*b^4*d*cosh(d*x + c)*sinh(d*x + c)^2 + b^4*d*sinh(d*x + c)^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [A]
time = 0.45, size = 145, normalized size = 1.71 \begin {gather*} \frac {\frac {b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 3 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 12 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{b^{3}} - \frac {24 \, {\left (a^{3} + a b^{2}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b^{4}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((b^2*(e^(d*x + c) - e^(-d*x - c))^3 - 3*a*b*(e^(d*x + c) - e^(-d*x - c))^2 + 12*a^2*(e^(d*x + c) - e^(-d
*x - c)) + 12*b^2*(e^(d*x + c) - e^(-d*x - c)))/b^3 - 24*(a^3 + a*b^2)*log(abs(b*(e^(d*x + c) - e^(-d*x - c))
+ 2*a))/b^4)/d

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Mupad [B]
time = 0.43, size = 180, normalized size = 2.12 \begin {gather*} \frac {x\,\left (a^3+a\,b^2\right )}{b^4}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {a\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b^2\,d}-\frac {a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a^2+3\,b^2\right )}{8\,b^3\,d}-\frac {\ln \left (2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^3+a\,b^2\right )}{b^4\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a^2+3\,b^2\right )}{8\,b^3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(a*b^2 + a^3))/b^4 - exp(- 3*c - 3*d*x)/(24*b*d) + exp(3*c + 3*d*x)/(24*b*d) - (a*exp(- 2*c - 2*d*x))/(8*b^
2*d) - (a*exp(2*c + 2*d*x))/(8*b^2*d) - (exp(- c - d*x)*(4*a^2 + 3*b^2))/(8*b^3*d) - (log(2*a*exp(d*x)*exp(c)
- b + b*exp(2*c)*exp(2*d*x))*(a*b^2 + a^3))/(b^4*d) + (exp(c + d*x)*(4*a^2 + 3*b^2))/(8*b^3*d)

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