3.1.40 \(\int \frac {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\) [40]

Optimal. Leaf size=130 \[ -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a^3 \sqrt {a-b} d}-\frac {\left (3 a^2+4 a b+8 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{8 a^3 d}+\frac {(3 a+4 b) \coth (c+d x) \text {csch}(c+d x)}{8 a^2 d}-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d} \]

[Out]

-1/8*(3*a^2+4*a*b+8*b^2)*arctanh(cosh(d*x+c))/a^3/d+1/8*(3*a+4*b)*coth(d*x+c)*csch(d*x+c)/a^2/d-1/4*coth(d*x+c
)*csch(d*x+c)^3/a/d-b^(5/2)*arctan(cosh(d*x+c)*b^(1/2)/(a-b)^(1/2))/a^3/d/(a-b)^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3265, 425, 541, 536, 212, 211} \begin {gather*} -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a^3 d \sqrt {a-b}}+\frac {(3 a+4 b) \coth (c+d x) \text {csch}(c+d x)}{8 a^2 d}-\frac {\left (3 a^2+4 a b+8 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{8 a^3 d}-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^(5/2)*ArcTan[(Sqrt[b]*Cosh[c + d*x])/Sqrt[a - b]])/(a^3*Sqrt[a - b]*d)) - ((3*a^2 + 4*a*b + 8*b^2)*ArcTan
h[Cosh[c + d*x]])/(8*a^3*d) + ((3*a + 4*b)*Coth[c + d*x]*Csch[c + d*x])/(8*a^2*d) - (Coth[c + d*x]*Csch[c + d*
x]^3)/(4*a*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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

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 {\text {csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d}-\frac {\text {Subst}\left (\int \frac {3 a+b+3 b x^2}{\left (1-x^2\right )^2 \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{4 a d}\\ &=\frac {(3 a+4 b) \coth (c+d x) \text {csch}(c+d x)}{8 a^2 d}-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d}-\frac {\text {Subst}\left (\int \frac {3 a^2+a b+4 b^2+b (3 a+4 b) x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\cosh (c+d x)\right )}{8 a^2 d}\\ &=\frac {(3 a+4 b) \coth (c+d x) \text {csch}(c+d x)}{8 a^2 d}-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\cosh (c+d x)\right )}{a^3 d}-\frac {\left (3 a^2+4 a b+8 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{8 a^3 d}\\ &=-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \cosh (c+d x)}{\sqrt {a-b}}\right )}{a^3 \sqrt {a-b} d}-\frac {\left (3 a^2+4 a b+8 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{8 a^3 d}+\frac {(3 a+4 b) \coth (c+d x) \text {csch}(c+d x)}{8 a^2 d}-\frac {\coth (c+d x) \text {csch}^3(c+d x)}{4 a d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 5.25, size = 295, normalized size = 2.27 \begin {gather*} -\frac {(2 a-b+b \cosh (2 (c+d x))) \text {csch}^2(c+d x) \left (64 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b}-i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )+64 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b}+i \sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a-b}}\right )-2 a \sqrt {a-b} (3 a+4 b) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+a^2 \sqrt {a-b} \text {csch}^4\left (\frac {1}{2} (c+d x)\right )-8 \sqrt {a-b} \left (3 a^2+4 a b+8 b^2\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-2 a \sqrt {a-b} (3 a+4 b) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-a^2 \sqrt {a-b} \text {sech}^4\left (\frac {1}{2} (c+d x)\right )\right )}{128 a^3 \sqrt {a-b} d \left (b+a \text {csch}^2(c+d x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/128*((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*(64*b^(5/2)*ArcTan[(Sqrt[b] - I*Sqrt[a]*Tanh[(c + d*x)
/2])/Sqrt[a - b]] + 64*b^(5/2)*ArcTan[(Sqrt[b] + I*Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[a - b]] - 2*a*Sqrt[a - b]*(
3*a + 4*b)*Csch[(c + d*x)/2]^2 + a^2*Sqrt[a - b]*Csch[(c + d*x)/2]^4 - 8*Sqrt[a - b]*(3*a^2 + 4*a*b + 8*b^2)*L
og[Tanh[(c + d*x)/2]] - 2*a*Sqrt[a - b]*(3*a + 4*b)*Sech[(c + d*x)/2]^2 - a^2*Sqrt[a - b]*Sech[(c + d*x)/2]^4)
)/(a^3*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2))

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Maple [A]
time = 5.52, size = 156, normalized size = 1.20

method	result	size
derivativedivides	\(\frac {\frac {\left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a -4 b \right )^{2}}{64 a^{3}}-\frac {-4 a -4 b}{32 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{64 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\left (6 a^{2}+8 a b +16 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{3}}-\frac {b^{3} \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a^{3} \sqrt {a b -b^{2}}}}{d}\)	\(156\)
default	\(\frac {\frac {\left (a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 a -4 b \right )^{2}}{64 a^{3}}-\frac {-4 a -4 b}{32 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{64 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {\left (6 a^{2}+8 a b +16 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{3}}-\frac {b^{3} \arctan \left (\frac {2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a +4 b}{4 \sqrt {a b -b^{2}}}\right )}{a^{3} \sqrt {a b -b^{2}}}}{d}\)	\(156\)
risch	\(\frac {{\mathrm e}^{d x +c} \left (3 a \,{\mathrm e}^{6 d x +6 c}+4 b \,{\mathrm e}^{6 d x +6 c}-11 a \,{\mathrm e}^{4 d x +4 c}-4 b \,{\mathrm e}^{4 d x +4 c}-11 a \,{\mathrm e}^{2 d x +2 c}-4 b \,{\mathrm e}^{2 d x +2 c}+3 a +4 b \right )}{4 d \,a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right )}{8 d a}+\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a^{2} d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right )}{8 d a}-\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a^{2} d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{a^{3} d}+\frac {\sqrt {-b \left (a -b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d \,a^{3}}-\frac {\sqrt {-b \left (a -b \right )}\, b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-b \left (a -b \right )}\, {\mathrm e}^{d x +c}}{b}+1\right )}{2 \left (a -b \right ) d \,a^{3}}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/64*(a*tanh(1/2*d*x+1/2*c)^2-4*a-4*b)^2/a^3-1/32*(-4*a-4*b)/a^2/tanh(1/2*d*x+1/2*c)^2-1/64/a/tanh(1/2*d*
x+1/2*c)^4+1/16/a^3*(6*a^2+8*a*b+16*b^2)*ln(tanh(1/2*d*x+1/2*c))-b^3/a^3/(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(csch(d*x+c)^5/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/4*((3*a*e^(7*c) + 4*b*e^(7*c))*e^(7*d*x) - (11*a*e^(5*c) + 4*b*e^(5*c))*e^(5*d*x) - (11*a*e^(3*c) + 4*b*e^(3
*c))*e^(3*d*x) + (3*a*e^c + 4*b*e^c)*e^(d*x))/(a^2*d*e^(8*d*x + 8*c) - 4*a^2*d*e^(6*d*x + 6*c) + 6*a^2*d*e^(4*
d*x + 4*c) - 4*a^2*d*e^(2*d*x + 2*c) + a^2*d) - 1/8*(3*a^2 + 4*a*b + 8*b^2)*log((e^(d*x + c) + 1)*e^(-c))/(a^3
*d) + 1/8*(3*a^2 + 4*a*b + 8*b^2)*log((e^(d*x + c) - 1)*e^(-c))/(a^3*d) - 32*integrate(1/16*(b^3*e^(3*d*x + 3*
c) - b^3*e^(d*x + c))/(a^3*b*e^(4*d*x + 4*c) + a^3*b + 2*(2*a^4*e^(2*c) - a^3*b*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. 2976 vs. \(2 (116) = 232\).
time = 0.55, size = 5809, normalized size = 44.68 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*(3*a^2 + 4*a*b)*cosh(d*x + c)^7 + 14*(3*a^2 + 4*a*b)*cosh(d*x + c)*sinh(d*x + c)^6 + 2*(3*a^2 + 4*a*b)
*sinh(d*x + c)^7 - 2*(11*a^2 + 4*a*b)*cosh(d*x + c)^5 + 2*(21*(3*a^2 + 4*a*b)*cosh(d*x + c)^2 - 11*a^2 - 4*a*b
)*sinh(d*x + c)^5 + 10*(7*(3*a^2 + 4*a*b)*cosh(d*x + c)^3 - (11*a^2 + 4*a*b)*cosh(d*x + c))*sinh(d*x + c)^4 -
2*(11*a^2 + 4*a*b)*cosh(d*x + c)^3 + 2*(35*(3*a^2 + 4*a*b)*cosh(d*x + c)^4 - 10*(11*a^2 + 4*a*b)*cosh(d*x + c)
^2 - 11*a^2 - 4*a*b)*sinh(d*x + c)^3 + 2*(21*(3*a^2 + 4*a*b)*cosh(d*x + c)^5 - 10*(11*a^2 + 4*a*b)*cosh(d*x +
c)^3 - 3*(11*a^2 + 4*a*b)*cosh(d*x + c))*sinh(d*x + c)^2 + 4*(b^2*cosh(d*x + c)^8 + 8*b^2*cosh(d*x + c)*sinh(d
*x + c)^7 + b^2*sinh(d*x + c)^8 - 4*b^2*cosh(d*x + c)^6 + 4*(7*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^6 + 6*
b^2*cosh(d*x + c)^4 + 8*(7*b^2*cosh(d*x + c)^3 - 3*b^2*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*b^2*cosh(d*x + c
)^4 - 30*b^2*cosh(d*x + c)^2 + 3*b^2)*sinh(d*x + c)^4 - 4*b^2*cosh(d*x + c)^2 + 8*(7*b^2*cosh(d*x + c)^5 - 10*
b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^2*cosh(d*x + c)^6 - 15*b^2*cosh(d*x + c)^4
+ 9*b^2*cosh(d*x + c)^2 - b^2)*sinh(d*x + c)^2 + b^2 + 8*(b^2*cosh(d*x + c)^7 - 3*b^2*cosh(d*x + c)^5 + 3*b^2
*cosh(d*x + c)^3 - b^2*cosh(d*x + c))*sinh(d*x + c))*sqrt(-b/(a - b))*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)*s
inh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a - 3*b)*cosh(d*x + c))*sinh(d*x + c) - 4*((a - b)*cosh(d*x + c)^3
+ 3*(a - b)*cosh(d*x + c)*sinh(d*x + c)^2 + (a - b)*sinh(d*x + c)^3 + (a - b)*cosh(d*x + c) + (3*(a - b)*cosh(
d*x + c)^2 + a - b)*sinh(d*x + c))*sqrt(-b/(a - b)) + 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)) + 2*(3*a^2 + 4*a*b)*cosh(d*x + c) - ((3*a^2 +
4*a*b + 8*b^2)*cosh(d*x + c)^8 + 8*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 + 4*a*b + 8*
b^2)*sinh(d*x + c)^8 - 4*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^
2 - 3*a^2 - 4*a*b - 8*b^2)*sinh(d*x + c)^6 + 8*(7*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^3 - 3*(3*a^2 + 4*a*b +
8*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 6*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 + 4*a*b + 8*
b^2)*cosh(d*x + c)^4 - 30*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2 + 9*a^2 + 12*a*b + 24*b^2)*sinh(d*x + c)^4 +
8*(7*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^5 - 10*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 4*a*b
+ 8*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 + 4*a*b + 8*
b^2)*cosh(d*x + c)^6 - 15*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^4 + 9*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2
- 3*a^2 - 4*a*b - 8*b^2)*sinh(d*x + c)^2 + 3*a^2 + 4*a*b + 8*b^2 + 8*((3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^7
- 3*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^5 + 3*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^3 - (3*a^2 + 4*a*b + 8*b
^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((3*a^2 + 4*a*b + 8*b^2)*cosh(d*x +
c)^8 + 8*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^2 + 4*a*b + 8*b^2)*sinh(d*x + c)^8 - 4*
(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^6 + 4*(7*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2 - 3*a^2 - 4*a*b - 8*b^2
)*sinh(d*x + c)^6 + 8*(7*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^3 - 3*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c))*si
nh(d*x + c)^5 + 6*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^4 + 2*(35*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^4 - 30
*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2 + 9*a^2 + 12*a*b + 24*b^2)*sinh(d*x + c)^4 + 8*(7*(3*a^2 + 4*a*b + 8*
b^2)*cosh(d*x + c)^5 - 10*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c))*s
inh(d*x + c)^3 - 4*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2 + 4*(7*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^6 - 15
*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^4 + 9*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^2 - 3*a^2 - 4*a*b - 8*b^2)*
sinh(d*x + c)^2 + 3*a^2 + 4*a*b + 8*b^2 + 8*((3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^7 - 3*(3*a^2 + 4*a*b + 8*b^
2)*cosh(d*x + c)^5 + 3*(3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c)^3 - (3*a^2 + 4*a*b + 8*b^2)*cosh(d*x + c))*sinh(d
*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(7*(3*a^2 + 4*a*b)*cosh(d*x + c)^6 - 5*(11*a^2 + 4*a*b)*co
sh(d*x + c)^4 - 3*(11*a^2 + 4*a*b)*cosh(d*x + c)^2 + 3*a^2 + 4*a*b)*sinh(d*x + c))/(a^3*d*cosh(d*x + c)^8 + 8*
a^3*d*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*d*sinh(d*x + c)^8 - 4*a^3*d*cosh(d*x + c)^6 + 6*a^3*d*cosh(d*x + c)^
4 + 4*(7*a^3*d*cosh(d*x + c)^2 - a^3*d)*sinh(d*x + c)^6 - 4*a^3*d*cosh(d*x + c)^2 + 8*(7*a^3*d*cosh(d*x + c)^3
- 3*a^3*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*a^3*d*cosh(d*x + c)^4 - 30*a^3*d*cosh(d*x + c)^2 + 3*a^3*d)*
sinh(d*x + c)^4 + a^3*d + 8*(7*a^3*d*cosh(d*x + c)^5 - 10*a^3*d*cosh(d*x + c)^3 + 3*a^3*d*cosh(d*x + c))*sinh(
d*x + c)^3 + 4*(7*a^3*d*cosh(d*x + c)^6 - 15*a^...

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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(csch(d*x+c)**5/(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(csch(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,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 = 5.66, size = 1639, normalized size = 12.61 \begin {gather*} \frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+4\,b\,a\right )}{4\,a^3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (243\,a^{12}\,\sqrt {-a^6\,d^2}+18432\,b^{12}\,\sqrt {-a^6\,d^2}+6912\,a^2\,b^{10}\,\sqrt {-a^6\,d^2}-30720\,a^3\,b^9\,\sqrt {-a^6\,d^2}-26880\,a^4\,b^8\,\sqrt {-a^6\,d^2}-24192\,a^5\,b^7\,\sqrt {-a^6\,d^2}+5024\,a^6\,b^6\,\sqrt {-a^6\,d^2}+13408\,a^7\,b^5\,\sqrt {-a^6\,d^2}+17160\,a^8\,b^4\,\sqrt {-a^6\,d^2}+9540\,a^9\,b^3\,\sqrt {-a^6\,d^2}+4563\,a^{10}\,b^2\,\sqrt {-a^6\,d^2}+9216\,a\,b^{11}\,\sqrt {-a^6\,d^2}+1134\,a^{11}\,b\,\sqrt {-a^6\,d^2}\right )}{81\,a^{13}\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}+2304\,a^3\,b^{10}\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}-3840\,a^6\,b^7\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}-1440\,a^7\,b^6\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}-864\,a^8\,b^5\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}+1600\,a^9\,b^4\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}+1200\,a^{10}\,b^3\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}+945\,a^{11}\,b^2\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}+270\,a^{12}\,b\,d\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}}\right )\,\sqrt {9\,a^4+24\,a^3\,b+64\,a^2\,b^2+64\,a\,b^3+64\,b^4}}{4\,\sqrt {-a^6\,d^2}}-\frac {6\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}}{a\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {\left (2\,\mathrm {atan}\left (\frac {b^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^6\,d^2\,\left (a-b\right )}\,\left (9\,a^5+15\,a^4\,b+40\,a^3\,b^2-48\,b^5\right )}{2\,a^3\,d\,\sqrt {b^5}\,\left (9\,a^6+6\,a^5\,b+25\,a^4\,b^2-40\,a^3\,b^3-48\,a\,b^5+48\,b^6\right )}\right )+2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {4\,\left (18\,a^9\,d\,\sqrt {b^5}-96\,a^4\,d\,{\left (b^5\right )}^{3/2}+96\,a^3\,b\,d\,{\left (b^5\right )}^{3/2}+12\,a^8\,b\,d\,\sqrt {b^5}-80\,a^6\,b^3\,d\,\sqrt {b^5}+50\,a^7\,b^2\,d\,\sqrt {b^5}\right )}{a^8\,b^4\,\left (a-b\right )\,\sqrt {a^7\,d^2-a^6\,b\,d^2}\,\left (a\,b-a^2\right )\,\sqrt {a^6\,d^2\,\left (a-b\right )}\,\left (9\,a^5+15\,a^4\,b+40\,a^3\,b^2-48\,b^5\right )}+\frac {2\,\left (40\,a^3\,b^5\,\sqrt {a^7\,d^2-a^6\,b\,d^2}-48\,b^8\,\sqrt {a^7\,d^2-a^6\,b\,d^2}+15\,a^4\,b^4\,\sqrt {a^7\,d^2-a^6\,b\,d^2}+9\,a^5\,b^3\,\sqrt {a^7\,d^2-a^6\,b\,d^2}\right )}{a^{11}\,b\,d\,\left (a-b\right )\,\sqrt {a^7\,d^2-a^6\,b\,d^2}\,\left (a\,b-a^2\right )\,\sqrt {b^5}\,\left (9\,a^6+6\,a^5\,b+25\,a^4\,b^2-40\,a^3\,b^3-48\,a\,b^5+48\,b^6\right )}\right )+\frac {2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (40\,a^3\,b^5\,\sqrt {a^7\,d^2-a^6\,b\,d^2}-48\,b^8\,\sqrt {a^7\,d^2-a^6\,b\,d^2}+15\,a^4\,b^4\,\sqrt {a^7\,d^2-a^6\,b\,d^2}+9\,a^5\,b^3\,\sqrt {a^7\,d^2-a^6\,b\,d^2}\right )}{a^{11}\,b\,d\,\left (a-b\right )\,\sqrt {a^7\,d^2-a^6\,b\,d^2}\,\left (a\,b-a^2\right )\,\sqrt {b^5}\,\left (9\,a^6+6\,a^5\,b+25\,a^4\,b^2-40\,a^3\,b^3-48\,a\,b^5+48\,b^6\right )}\right )\,\left (\frac {a^{11}\,b\,\sqrt {a^7\,d^2-a^6\,b\,d^2}}{4}+\frac {a^9\,b^3\,\sqrt {a^7\,d^2-a^6\,b\,d^2}}{4}-\frac {a^{10}\,b^2\,\sqrt {a^7\,d^2-a^6\,b\,d^2}}{2}\right )\right )\right )\,\sqrt {b^5}}{2\,\sqrt {a^7\,d^2-a^6\,b\,d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a-4\,b\right )}{2\,a^2\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^5*(a + b*sinh(c + d*x)^2)),x)

[Out]

(exp(c + d*x)*(4*a*b + 3*a^2))/(4*a^3*d*(exp(2*c + 2*d*x) - 1)) - (atan((exp(d*x)*exp(c)*(243*a^12*(-a^6*d^2)^
(1/2) + 18432*b^12*(-a^6*d^2)^(1/2) + 6912*a^2*b^10*(-a^6*d^2)^(1/2) - 30720*a^3*b^9*(-a^6*d^2)^(1/2) - 26880*
a^4*b^8*(-a^6*d^2)^(1/2) - 24192*a^5*b^7*(-a^6*d^2)^(1/2) + 5024*a^6*b^6*(-a^6*d^2)^(1/2) + 13408*a^7*b^5*(-a^
6*d^2)^(1/2) + 17160*a^8*b^4*(-a^6*d^2)^(1/2) + 9540*a^9*b^3*(-a^6*d^2)^(1/2) + 4563*a^10*b^2*(-a^6*d^2)^(1/2)
+ 9216*a*b^11*(-a^6*d^2)^(1/2) + 1134*a^11*b*(-a^6*d^2)^(1/2)))/(81*a^13*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*
b^4 + 64*a^2*b^2)^(1/2) + 2304*a^3*b^10*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) - 3840*a^6
*b^7*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) - 1440*a^7*b^6*d*(64*a*b^3 + 24*a^3*b + 9*a^4
+ 64*b^4 + 64*a^2*b^2)^(1/2) - 864*a^8*b^5*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 1600
*a^9*b^4*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 1200*a^10*b^3*d*(64*a*b^3 + 24*a^3*b +
9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2) + 945*a^11*b^2*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2)
+ 270*a^12*b*d*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b^4 + 64*a^2*b^2)^(1/2)))*(64*a*b^3 + 24*a^3*b + 9*a^4 + 64*b
^4 + 64*a^2*b^2)^(1/2))/(4*(-a^6*d^2)^(1/2)) - (6*exp(c + d*x))/(a*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x)
+ exp(6*c + 6*d*x) - 1)) - (4*exp(c + d*x))/(a*d*(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x)
+ exp(8*c + 8*d*x) + 1)) - ((2*atan((b^3*exp(d*x)*exp(c)*(a^6*d^2*(a - b))^(1/2)*(15*a^4*b + 9*a^5 - 48*b^5 +
40*a^3*b^2))/(2*a^3*d*(b^5)^(1/2)*(6*a^5*b - 48*a*b^5 + 9*a^6 + 48*b^6 - 40*a^3*b^3 + 25*a^4*b^2))) + 2*atan(
(exp(d*x)*exp(c)*((4*(18*a^9*d*(b^5)^(1/2) - 96*a^4*d*(b^5)^(3/2) + 96*a^3*b*d*(b^5)^(3/2) + 12*a^8*b*d*(b^5)^
(1/2) - 80*a^6*b^3*d*(b^5)^(1/2) + 50*a^7*b^2*d*(b^5)^(1/2)))/(a^8*b^4*(a - b)*(a^7*d^2 - a^6*b*d^2)^(1/2)*(a*
b - a^2)*(a^6*d^2*(a - b))^(1/2)*(15*a^4*b + 9*a^5 - 48*b^5 + 40*a^3*b^2)) + (2*(40*a^3*b^5*(a^7*d^2 - a^6*b*d
^2)^(1/2) - 48*b^8*(a^7*d^2 - a^6*b*d^2)^(1/2) + 15*a^4*b^4*(a^7*d^2 - a^6*b*d^2)^(1/2) + 9*a^5*b^3*(a^7*d^2 -
a^6*b*d^2)^(1/2)))/(a^11*b*d*(a - b)*(a^7*d^2 - a^6*b*d^2)^(1/2)*(a*b - a^2)*(b^5)^(1/2)*(6*a^5*b - 48*a*b^5
+ 9*a^6 + 48*b^6 - 40*a^3*b^3 + 25*a^4*b^2))) + (2*exp(3*c)*exp(3*d*x)*(40*a^3*b^5*(a^7*d^2 - a^6*b*d^2)^(1/2)
- 48*b^8*(a^7*d^2 - a^6*b*d^2)^(1/2) + 15*a^4*b^4*(a^7*d^2 - a^6*b*d^2)^(1/2) + 9*a^5*b^3*(a^7*d^2 - a^6*b*d^
2)^(1/2)))/(a^11*b*d*(a - b)*(a^7*d^2 - a^6*b*d^2)^(1/2)*(a*b - a^2)*(b^5)^(1/2)*(6*a^5*b - 48*a*b^5 + 9*a^6 +
48*b^6 - 40*a^3*b^3 + 25*a^4*b^2)))*((a^11*b*(a^7*d^2 - a^6*b*d^2)^(1/2))/4 + (a^9*b^3*(a^7*d^2 - a^6*b*d^2)^
(1/2))/4 - (a^10*b^2*(a^7*d^2 - a^6*b*d^2)^(1/2))/2)))*(b^5)^(1/2))/(2*(a^7*d^2 - a^6*b*d^2)^(1/2)) - (exp(c +
d*x)*(a - 4*b))/(2*a^2*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))

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