3.1.32 \(\int \frac {\sinh ^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [32]

Optimal. Leaf size=56 \[ -\frac {a \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} b^{3/2} d}+\frac {\cosh (c+d x)}{b d} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3265, 396, 211} \begin {gather*} \frac {\cosh (c+d x)}{b d}-\frac {a \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{3/2} d \sqrt {a-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]

[Out]

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

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.17, size = 107, normalized size = 1.91 \begin {gather*} \frac {-\frac {a \left (\text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{\sqrt {a-b}}+\sqrt {b} \cosh (c+d x)}{b^{3/2} d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]

[Out]

(-((a*(ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] + ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x
)/2])/Sqrt[a - b]]))/Sqrt[a - b]) + Sqrt[b]*Cosh[c + d*x])/(b^(3/2)*d)

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Maple [A]
time = 0.95, size = 93, normalized size = 1.66

method	result	size
derivativedivides	\(\frac {\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b \sqrt {a b -b^{2}}}}{d}\)	\(93\)
default	\(\frac {\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {a \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{b \sqrt {a b -b^{2}}}}{d}\)	\(93\)
risch	\(\frac {{\mathrm e}^{d x +c}}{2 b d}+\frac {{\mathrm e}^{-d x -c}}{2 b d}-\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d b}+\frac {a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a -b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a b +b^{2}}}+1\right )}{2 \sqrt {-a b +b^{2}}\, d b}\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/b/(tanh(1/2*d*x+1/2*c)+1)-1/b/(tanh(1/2*d*x+1/2*c)-1)-a/b/(a*b-b^2)^(1/2)*arctan(1/4*(2*a*tanh(1/2*d*x+
1/2*c)^2-2*a+4*b)/(a*b-b^2)^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/2*(e^(2*d*x + 2*c) + 1)*e^(-d*x - c)/(b*d) - 1/8*integrate(16*(a*e^(3*d*x + 3*c) - a*e^(d*x + c))/(b^2*e^(4*
d*x + 4*c) + b^2 + 2*(2*a*b*e^(2*c) - b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (48) = 96\).
time = 0.45, size = 746, normalized size = 13.32 \begin {gather*} \left [\frac {{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} - \sqrt {-a b + b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \log \left (\frac {b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} - 2 \, {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} - 2 \, a + 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} - {\left (2 \, a - 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right )\right )} \sqrt {-a b + b^{2}} + b}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) + a b - b^{2}}{2 \, {\left ({\left (a b^{2} - b^{3}\right )} d \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d \sinh \left (d x + c\right )\right )}}, \frac {{\left (a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} - 2 \, \sqrt {a b - b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \arctan \left (-\frac {b \cosh \left (d x + c\right )^{3} + 3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, b \cosh \left (d x + c\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a b - b^{2}}}\right ) + 2 \, \sqrt {a b - b^{2}} {\left (a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )\right )} \arctan \left (-\frac {\sqrt {a b - b^{2}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}}{2 \, {\left (a - b\right )}}\right ) + a b - b^{2}}{2 \, {\left ({\left (a b^{2} - b^{3}\right )} d \cosh \left (d x + c\right ) + {\left (a b^{2} - b^{3}\right )} d \sinh \left (d x + c\right )\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/2*((a*b - b^2)*cosh(d*x + c)^2 + 2*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b - b^2)*sinh(d*x + c)^2 -
sqrt(-a*b + b^2)*(a*cosh(d*x + c) + a*sinh(d*x + c))*log((b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^
3 + b*sinh(d*x + c)^4 - 2*(2*a - 3*b)*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - 2*a + 3*b)*sinh(d*x + c)^2 +
4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*
x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 + 1)*sinh(d*x + c) + cosh(d*x + c))*sqrt(-a*b + b^2) + b)/(b*c
osh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*(2*a - b)*cosh(d*x + c)^2 + 2*(3*b*
cosh(d*x + c)^2 + 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + (2*a - b)*cosh(d*x + c))*sinh(d*x + c) + b
)) + a*b - b^2)/((a*b^2 - b^3)*d*cosh(d*x + c) + (a*b^2 - b^3)*d*sinh(d*x + c)), 1/2*((a*b - b^2)*cosh(d*x + c
)^2 + 2*(a*b - b^2)*cosh(d*x + c)*sinh(d*x + c) + (a*b - b^2)*sinh(d*x + c)^2 - 2*sqrt(a*b - b^2)*(a*cosh(d*x
+ c) + a*sinh(d*x + c))*arctan(-1/2*(b*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)*sinh(d*x + c)^2 + b*sinh(d*x + c)^3
+ (4*a - 3*b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2 + 4*a - 3*b)*sinh(d*x + c))/sqrt(a*b - b^2)) + 2*sqrt(a*b
- b^2)*(a*cosh(d*x + c) + a*sinh(d*x + c))*arctan(-1/2*sqrt(a*b - b^2)*(cosh(d*x + c) + sinh(d*x + c))/(a - b)
) + a*b - b^2)/((a*b^2 - b^3)*d*cosh(d*x + c) + (a*b^2 - b^3)*d*sinh(d*x + c))]

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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*sinh(d*x+c)**2),x)

[Out]

Timed out

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 1.17, size = 293, normalized size = 5.23 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}}{2\,b\,d}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^3\,d^2\,\left (a-b\right )}}{2\,b\,d\,\left (a-b\right )\,{\left (a^2\right )}^{3/2}}\right )+2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^3}{b^5\,d\,{\left (a-b\right )}^2\,{\left (a^2\right )}^{3/2}}-\frac {4\,\left (2\,b^2\,d\,{\left (a^2\right )}^{3/2}-2\,a\,b\,d\,{\left (a^2\right )}^{3/2}\right )}{a^3\,b^4\,\left (a-b\right )\,\sqrt {a\,b^3\,d^2-b^4\,d^2}\,\sqrt {b^3\,d^2\,\left (a-b\right )}}\right )+\frac {2\,a^3\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{b^5\,d\,{\left (a-b\right )}^2\,{\left (a^2\right )}^{3/2}}\right )\,\left (\frac {b^5\,\sqrt {a\,b^3\,d^2-b^4\,d^2}}{4}-\frac {a\,b^4\,\sqrt {a\,b^3\,d^2-b^4\,d^2}}{4}\right )\right )\right )\,\sqrt {a^2}}{2\,\sqrt {a\,b^3\,d^2-b^4\,d^2}}+\frac {{\mathrm {e}}^{-c-d\,x}}{2\,b\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^3/(a + b*sinh(c + d*x)^2),x)

[Out]

exp(c + d*x)/(2*b*d) - ((2*atan((a^3*exp(d*x)*exp(c)*(b^3*d^2*(a - b))^(1/2))/(2*b*d*(a - b)*(a^2)^(3/2))) + 2
*atan((exp(d*x)*exp(c)*((2*a^3)/(b^5*d*(a - b)^2*(a^2)^(3/2)) - (4*(2*b^2*d*(a^2)^(3/2) - 2*a*b*d*(a^2)^(3/2))
)/(a^3*b^4*(a - b)*(a*b^3*d^2 - b^4*d^2)^(1/2)*(b^3*d^2*(a - b))^(1/2))) + (2*a^3*exp(3*c)*exp(3*d*x))/(b^5*d*
(a - b)^2*(a^2)^(3/2)))*((b^5*(a*b^3*d^2 - b^4*d^2)^(1/2))/4 - (a*b^4*(a*b^3*d^2 - b^4*d^2)^(1/2))/4)))*(a^2)^
(1/2))/(2*(a*b^3*d^2 - b^4*d^2)^(1/2)) + exp(- c - d*x)/(2*b*d)

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