3.3.37 \(\int \frac {\sinh ^3(c+d x)}{a+b \cosh (c+d x)} \, dx\) [237]

Optimal. Leaf size=61 \[ -\frac {a \cosh (c+d x)}{b^2 d}+\frac {\cosh ^2(c+d x)}{2 b d}+\frac {\left (a^2-b^2\right ) \log (a+b \cosh (c+d x))}{b^3 d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 711} \begin {gather*} \frac {\left (a^2-b^2\right ) \log (a+b \cosh (c+d x))}{b^3 d}-\frac {a \cosh (c+d x)}{b^2 d}+\frac {\cosh ^2(c+d x)}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 711

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

Rule 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.08, size = 55, normalized size = 0.90 \begin {gather*} \frac {-4 a b \cosh (c+d x)+b^2 \cosh (2 (c+d x))+4 \left (a^2-b^2\right ) \log (a+b \cosh (c+d x))}{4 b^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(204\) vs. \(2(59)=118\).
time = 0.99, size = 205, normalized size = 3.36

method	result	size
risch	\(-\frac {x \,a^{2}}{b^{3}}+\frac {x}{b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {a \,{\mathrm e}^{d x +c}}{2 b^{2} d}-\frac {{\mathrm e}^{-d x -c} a}{2 b^{2} d}+\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}-\frac {2 a^{2} c}{b^{3} d}+\frac {2 c}{b d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}+1\right ) a^{2}}{b^{3} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}+1\right )}{b d}\)	\(170\)
derivativedivides	\(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}+\frac {\left (a^{3}-a^{2} b -a \,b^{2}+b^{3}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{b^{3} \left (a -b \right )}}{d}\)	\(205\)
default	\(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-b -2 a}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{3}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a +b}{2 b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{3}}+\frac {\left (a^{3}-a^{2} b -a \,b^{2}+b^{3}\right ) \ln \left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{b^{3} \left (a -b \right )}}{d}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (59) = 118\).
time = 0.28, size = 130, normalized size = 2.13 \begin {gather*} -\frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{8 \, b^{2} d} + \frac {{\left (a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{3} d} - \frac {4 \, a e^{\left (-d x - c\right )} - b e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, b^{2} d} + \frac {{\left (a^{2} - 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^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + (a^2 - b^2)*(d*x + c)/(b^3*d) - 1/8*(4*a*e^(-d*x - c) -
b*e^(-2*d*x - 2*c))/(b^2*d) + (a^2 - b^2)*log(2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) + b)/(b^3*d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (59) = 118\).
time = 0.51, size = 340, normalized size = 5.57 \begin {gather*} \frac {b^{2} \cosh \left (d x + c\right )^{4} + b^{2} \sinh \left (d x + c\right )^{4} - 8 \, {\left (a^{2} - b^{2}\right )} d x \cosh \left (d x + c\right )^{2} - 4 \, a b \cosh \left (d x + c\right )^{3} + 4 \, {\left (b^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )^{3} - 4 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - 4 \, {\left (a^{2} - b^{2}\right )} d x - 6 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 8 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - 4 \, {\left (a^{2} - b^{2}\right )} d x \cosh \left (d x + c\right ) - 3 \, a b \cosh \left (d x + c\right )^{2} - a b\right )} \sinh \left (d x + c\right )}{8 \, {\left (b^{3} d \cosh \left (d x + c\right )^{2} + 2 \, b^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{3} d \sinh \left (d x + c\right )^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b^2*cosh(d*x + c)^4 + b^2*sinh(d*x + c)^4 - 8*(a^2 - b^2)*d*x*cosh(d*x + c)^2 - 4*a*b*cosh(d*x + c)^3 + 4
*(b^2*cosh(d*x + c) - a*b)*sinh(d*x + c)^3 - 4*a*b*cosh(d*x + c) + 2*(3*b^2*cosh(d*x + c)^2 - 4*(a^2 - b^2)*d*
x - 6*a*b*cosh(d*x + c))*sinh(d*x + c)^2 + b^2 + 8*((a^2 - b^2)*cosh(d*x + c)^2 + 2*(a^2 - b^2)*cosh(d*x + c)*
sinh(d*x + c) + (a^2 - b^2)*sinh(d*x + c)^2)*log(2*(b*cosh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + 4*
(b^2*cosh(d*x + c)^3 - 4*(a^2 - b^2)*d*x*cosh(d*x + c) - 3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c))/(b^3*d*co
sh(d*x + c)^2 + 2*b^3*d*cosh(d*x + c)*sinh(d*x + c) + b^3*d*sinh(d*x + c)^2)

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

[Out]

Timed out

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Giac [A]
time = 0.43, size = 88, normalized size = 1.44 \begin {gather*} \frac {\frac {b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{b^{2}} + \frac {8 \, {\left (a^{2} - 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^{3}}}{8 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((b*(e^(d*x + c) + e^(-d*x - c))^2 - 4*a*(e^(d*x + c) + e^(-d*x - c)))/b^2 + 8*(a^2 - b^2)*log(abs(b*(e^(d
*x + c) + e^(-d*x - c)) + 2*a))/b^3)/d

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Mupad [B]
time = 1.06, size = 122, normalized size = 2.00 \begin {gather*} \frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}-\frac {x\,\left (a^2-b^2\right )}{b^3}+\frac {\ln \left (b+2\,a\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a^2-b^2\right )}{b^3\,d}-\frac {a\,{\mathrm {e}}^{-c-d\,x}}{2\,b^2\,d}-\frac {a\,{\mathrm {e}}^{c+d\,x}}{2\,b^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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