3.30 \(\int \frac {\sinh ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)

Optimal. Leaf size=79 \[ \frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{5/2} d \sqrt {a-b}}-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d} \]

[Out]

-(a+b)*cosh(d*x+c)/b^2/d+1/3*cosh(d*x+c)^3/b/d+a^2*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))/b^(5/2)/d/(a-b)^(1/
2)

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Rubi [A]  time = 0.11, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3186, 390, 205} \[ \frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{b^{5/2} d \sqrt {a-b}}-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]

[Out]

(a^2*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(5/2)*d) - ((a + b)*Cosh[c + d*x])/(b^2*d) +
Cosh[c + d*x]^3/(3*b*d)

Rule 205

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

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 3186

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 ^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a+b}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a-b+b x^2\right )}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{b^2 d}\\ &=\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{\sqrt {a-b} b^{5/2} d}-\frac {(a+b) \cosh (c+d x)}{b^2 d}+\frac {\cosh ^3(c+d x)}{3 b d}\\ \end {align*}

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Mathematica [C]  time = 0.43, size = 134, normalized size = 1.70 \[ \frac {\frac {12 a^2 \left (\tan ^{-1}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+\tan ^{-1}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )\right )}{\sqrt {a-b}}-3 \sqrt {b} (4 a+3 b) \cosh (c+d x)+b^{3/2} \cosh (3 (c+d x))}{12 b^{5/2} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^5/(a + b*Sinh[c + d*x]^2),x]

[Out]

((12*a^2*(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] - 3*Sqrt[b]*(4*a + 3*b)*Cosh[c + d*x] + b^(3/2)*Cosh[3*(c + d*x)])/(12*b^(
5/2)*d)

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fricas [B]  time = 0.66, size = 1668, normalized size = 21.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/24*((a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (a*b^2 - b^3)*sinh(d*x
+ c)^6 - 3*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c)^4 - 3*(4*a^2*b - a*b^2 - 3*b^3 - 5*(a*b^2 - b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 4*(5*(a*b^2 - b^3)*cosh(d*x + c)^3 - 3*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(
d*x + c)^3 + a*b^2 - b^3 - 3*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c)^2 + 3*(5*(a*b^2 - b^3)*cosh(d*x + c)^4 -
4*a^2*b + a*b^2 + 3*b^3 - 6*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 12*(a^2*cosh(d*x + c)
^3 + 3*a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*sinh(d*x + c)^3)*sqrt(-a*
b + b^2)*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*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 - 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)) + 6*((a*b^2 - b^3)*cosh(d*x + c)^5 - 2*(4
*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c)^3 - (4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a*b^3 - b^
4)*d*cosh(d*x + c)^3 + 3*(a*b^3 - b^4)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(
d*x + c)^2 + (a*b^3 - b^4)*d*sinh(d*x + c)^3), 1/24*((a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(a*b^2 - b^3)*cosh(d*x
+ c)*sinh(d*x + c)^5 + (a*b^2 - b^3)*sinh(d*x + c)^6 - 3*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c)^4 - 3*(4*a^2*
b - a*b^2 - 3*b^3 - 5*(a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a*b^2 - b^3)*cosh(d*x + c)^3 - 3*
(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + a*b^2 - b^3 - 3*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x
+ c)^2 + 3*(5*(a*b^2 - b^3)*cosh(d*x + c)^4 - 4*a^2*b + a*b^2 + 3*b^3 - 6*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*s
inh(d*x + c)^2 + a^2*sinh(d*x + c)^3)*sqrt(a*b - b^2)*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)) - 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*
x + c)^2 + a^2*sinh(d*x + c)^3)*sqrt(a*b - b^2)*arctan(-1/2*sqrt(a*b - b^2)*(cosh(d*x + c) + sinh(d*x + c))/(a
- b)) + 6*((a*b^2 - b^3)*cosh(d*x + c)^5 - 2*(4*a^2*b - a*b^2 - 3*b^3)*cosh(d*x + c)^3 - (4*a^2*b - a*b^2 - 3
*b^3)*cosh(d*x + c))*sinh(d*x + c))/((a*b^3 - b^4)*d*cosh(d*x + c)^3 + 3*(a*b^3 - b^4)*d*cosh(d*x + c)^2*sinh(
d*x + c) + 3*(a*b^3 - b^4)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a*b^3 - b^4)*d*sinh(d*x + c)^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5/(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,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[31,78]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[85,31]Warning, need to choose a branch for the root of a p
olynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[46,18]Warning, need to choo
se a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,
b]=[-27,57]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.
The choice was done assuming [a,b]=[22,73]Warning, need to choose a branch for the root of a polynomial with p
arameters. This might be wrong.The choice was done assuming [a,b]=[-10,75]Warning, need to choose a branch for
the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-1,84]Warni
ng, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was d
one assuming [a,b]=[-91,-60]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-33,-40]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-18,-85]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumin
g [a,b]=[1,-81]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wr
ong.The choice was done assuming [a,b]=[70,33]Warning, need to choose a branch for the root of a polynomial wi
th parameters. This might be wrong.The choice was done assuming [a,b]=[14,-81]Warning, need to choose a branch
for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[39,-90]
Undef/Unsigned Inf encountered in limitEvaluation time: 2.08Limit: Max order reached or unable to make series
expansion Error: Bad Argument Value

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maple [B]  time = 0.07, size = 227, normalized size = 2.87 \[ -\frac {1}{3 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a}{d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a^{2} \arctan \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{d \,b^{2} \sqrt {a b -b^{2}}}+\frac {1}{3 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {a}{d \,b^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {1}{2 d b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (3 \, {\left (4 \, a e^{\left (4 \, c\right )} + 3 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 3 \, {\left (4 \, a e^{\left (2 \, c\right )} + 3 \, b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - b e^{\left (6 \, d x + 6 \, c\right )} - b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, b^{2} d} + \frac {1}{32} \, \int \frac {64 \, {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - a^{2} e^{\left (d x + c\right )}\right )}}{b^{3} e^{\left (4 \, d x + 4 \, c\right )} + b^{3} + 2 \, {\left (2 \, a b^{2} e^{\left (2 \, c\right )} - b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^5/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

-1/24*(3*(4*a*e^(4*c) + 3*b*e^(4*c))*e^(4*d*x) + 3*(4*a*e^(2*c) + 3*b*e^(2*c))*e^(2*d*x) - b*e^(6*d*x + 6*c) -
b)*e^(-3*d*x - 3*c)/(b^2*d) + 1/32*integrate(64*(a^2*e^(3*d*x + 3*c) - a^2*e^(d*x + c))/(b^3*e^(4*d*x + 4*c)
+ b^3 + 2*(2*a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 1.38, size = 348, normalized size = 4.41 \[ \frac {\left (2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {b^5\,d^2\,\left (a-b\right )}}{2\,b^2\,d\,\left (a-b\right )\,\sqrt {a^4}}\right )+2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,a^2}{b^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^4}}-\frac {4\,\left (2\,a^3\,b^3\,d\,\sqrt {a^4}-2\,a^4\,b^2\,d\,\sqrt {a^4}\right )}{a^5\,b^6\,\left (a-b\right )\,\sqrt {a\,b^5\,d^2-b^6\,d^2}\,\sqrt {b^5\,d^2\,\left (a-b\right )}}\right )+\frac {2\,a^2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{b^8\,d\,{\left (a-b\right )}^2\,\sqrt {a^4}}\right )\,\left (\frac {b^7\,\sqrt {a\,b^5\,d^2-b^6\,d^2}}{4}-\frac {a\,b^6\,\sqrt {a\,b^5\,d^2-b^6\,d^2}}{4}\right )\right )\right )\,\sqrt {a^4}}{2\,\sqrt {a\,b^5\,d^2-b^6\,d^2}}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,b\,d}+\frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,b\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (4\,a+3\,b\right )}{8\,b^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (4\,a+3\,b\right )}{8\,b^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(c + d*x)^5/(a + b*sinh(c + d*x)^2),x)

[Out]

((2*atan((a^2*exp(d*x)*exp(c)*(b^5*d^2*(a - b))^(1/2))/(2*b^2*d*(a - b)*(a^4)^(1/2))) + 2*atan((exp(d*x)*exp(c
)*((2*a^2)/(b^8*d*(a - b)^2*(a^4)^(1/2)) - (4*(2*a^3*b^3*d*(a^4)^(1/2) - 2*a^4*b^2*d*(a^4)^(1/2)))/(a^5*b^6*(a
- b)*(a*b^5*d^2 - b^6*d^2)^(1/2)*(b^5*d^2*(a - b))^(1/2))) + (2*a^2*exp(3*c)*exp(3*d*x))/(b^8*d*(a - b)^2*(a^
4)^(1/2)))*((b^7*(a*b^5*d^2 - b^6*d^2)^(1/2))/4 - (a*b^6*(a*b^5*d^2 - b^6*d^2)^(1/2))/4)))*(a^4)^(1/2))/(2*(a*
b^5*d^2 - b^6*d^2)^(1/2)) + exp(- 3*c - 3*d*x)/(24*b*d) + exp(3*c + 3*d*x)/(24*b*d) - (exp(c + d*x)*(4*a + 3*b
))/(8*b^2*d) - (exp(- c - d*x)*(4*a + 3*b))/(8*b^2*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)**5/(a+b*sinh(d*x+c)**2),x)

[Out]

Timed out

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