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

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

[Out]

-((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)

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Rubi [A]  time = 0.203545, antiderivative size = 130, normalized size of antiderivative = 1., 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 = {3186, 414, 527, 522, 206, 205} \[ -\frac{b^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \cosh (c+d x)}{\sqrt{a-b}}\right )}{a^3 d \sqrt{a-b}}-\frac{\left (3 a^2+4 a b+8 b^2\right ) \tanh ^{-1}(\cosh (c+d x))}{8 a^3 d}+\frac{(3 a+4 b) \coth (c+d x) \text{csch}(c+d x)}{8 a^2 d}-\frac{\coth (c+d x) \text{csch}^3(c+d x)}{4 a d} \]

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 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]

Rule 414

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]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 527

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 522

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 206

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

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]

Rubi steps

\begin{align*} \int \frac{\text{csch}^5(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=-\frac{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 6.28295, size = 574, normalized size = 4.42 \[ \frac{\left (3 a^2+4 a b+8 b^2\right ) \text{csch}^2(c+d x) \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right ) (2 a+b \cosh (2 (c+d x))-b)}{16 a^3 d \left (a \text{csch}^2(c+d x)+b\right )}-\frac{b^{5/2} \text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left (\frac{\text{sech}\left (\frac{1}{2} (c+d x)\right ) \left (\sqrt{b} \cosh \left (\frac{1}{2} (c+d x)\right )-i \sqrt{a} \sinh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a-b}}\right )}{2 a^3 d \sqrt{a-b} \left (a \text{csch}^2(c+d x)+b\right )}-\frac{b^{5/2} \text{csch}^2(c+d x) (2 a+b \cosh (2 (c+d x))-b) \tan ^{-1}\left (\frac{\text{sech}\left (\frac{1}{2} (c+d x)\right ) \left (\sqrt{b} \cosh \left (\frac{1}{2} (c+d x)\right )+i \sqrt{a} \sinh \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a-b}}\right )}{2 a^3 d \sqrt{a-b} \left (a \text{csch}^2(c+d x)+b\right )}+\frac{(3 a+4 b) \text{csch}^2(c+d x) \text{csch}^2\left (\frac{1}{2} (c+d x)\right ) (2 a+b \cosh (2 (c+d x))-b)}{64 a^2 d \left (a \text{csch}^2(c+d x)+b\right )}+\frac{(3 a+4 b) \text{csch}^2(c+d x) \text{sech}^2\left (\frac{1}{2} (c+d x)\right ) (2 a+b \cosh (2 (c+d x))-b)}{64 a^2 d \left (a \text{csch}^2(c+d x)+b\right )}-\frac{\text{csch}^2(c+d x) \text{csch}^4\left (\frac{1}{2} (c+d x)\right ) (2 a+b \cosh (2 (c+d x))-b)}{128 a d \left (a \text{csch}^2(c+d x)+b\right )}+\frac{\text{csch}^2(c+d x) \text{sech}^4\left (\frac{1}{2} (c+d x)\right ) (2 a+b \cosh (2 (c+d x))-b)}{128 a d \left (a \text{csch}^2(c+d x)+b\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^(5/2)*ArcTan[(Sech[(c + d*x)/2]*(Sqrt[b]*Cosh[(c + d*x)/2] - I*Sqrt[a]*Sinh[(c + d*x)/2]))/Sqrt[a - b]]*(2
*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2)/(2*a^3*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2)) - (b^(5/2)*ArcTan
[(Sech[(c + d*x)/2]*(Sqrt[b]*Cosh[(c + d*x)/2] + I*Sqrt[a]*Sinh[(c + d*x)/2]))/Sqrt[a - b]]*(2*a - b + b*Cosh[
2*(c + d*x)])*Csch[c + d*x]^2)/(2*a^3*Sqrt[a - b]*d*(b + a*Csch[c + d*x]^2)) + ((3*a + 4*b)*(2*a - b + b*Cosh[
2*(c + d*x)])*Csch[(c + d*x)/2]^2*Csch[c + d*x]^2)/(64*a^2*d*(b + a*Csch[c + d*x]^2)) - ((2*a - b + b*Cosh[2*(
c + d*x)])*Csch[(c + d*x)/2]^4*Csch[c + d*x]^2)/(128*a*d*(b + a*Csch[c + d*x]^2)) + ((3*a^2 + 4*a*b + 8*b^2)*(
2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Log[Tanh[(c + d*x)/2]])/(16*a^3*d*(b + a*Csch[c + d*x]^2)) + ((
3*a + 4*b)*(2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Sech[(c + d*x)/2]^2)/(64*a^2*d*(b + a*Csch[c + d*x]
^2)) + ((2*a - b + b*Cosh[2*(c + d*x)])*Csch[c + d*x]^2*Sech[(c + d*x)/2]^4)/(128*a*d*(b + a*Csch[c + d*x]^2))

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Maple [A]  time = 0.066, size = 232, normalized size = 1.8 \begin{align*}{\frac{1}{64\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{b}{8\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{64\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}+{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{b}{8\,d{a}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{3}{8\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{b}{2\,d{a}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{{b}^{2}}{d{a}^{3}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{{b}^{3}}{d{a}^{3}}\arctan \left ({\frac{1}{4} \left ( 2\, \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a-2\,a+4\,b \right ){\frac{1}{\sqrt{ab-{b}^{2}}}}} \right ){\frac{1}{\sqrt{ab-{b}^{2}}}}} \end{align*}

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)

[Out]

1/64/d/a*tanh(1/2*d*x+1/2*c)^4-1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/8/d/a^2*tanh(1/2*d*x+1/2*c)^2*b-1/64/d/a/tanh(1
/2*d*x+1/2*c)^4+1/8/d/a/tanh(1/2*d*x+1/2*c)^2+1/8/d/a^2/tanh(1/2*d*x+1/2*c)^2*b+3/8/d/a*ln(tanh(1/2*d*x+1/2*c)
)+1/2/d/a^2*b*ln(tanh(1/2*d*x+1/2*c))+1/d/a^3*ln(tanh(1/2*d*x+1/2*c))*b^2-1/d/a^3*b^3/(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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (3 \, a e^{\left (7 \, c\right )} + 4 \, b e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} -{\left (11 \, a e^{\left (5 \, c\right )} + 4 \, b e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} -{\left (11 \, a e^{\left (3 \, c\right )} + 4 \, b e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (3 \, a e^{c} + 4 \, b e^{c}\right )} e^{\left (d x\right )}}{4 \,{\left (a^{2} d e^{\left (8 \, d x + 8 \, c\right )} - 4 \, a^{2} d e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a^{2} d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d\right )}} - \frac{{\left (3 \, a^{2} + 4 \, a b + 8 \, b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{8 \, a^{3} d} + \frac{{\left (3 \, a^{2} + 4 \, a b + 8 \, b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{8 \, a^{3} d} - 32 \, \int \frac{b^{3} e^{\left (3 \, d x + 3 \, c\right )} - b^{3} e^{\left (d x + c\right )}}{16 \,{\left (a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + a^{3} b + 2 \,{\left (2 \, a^{4} e^{\left (2 \, c\right )} - a^{3} b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}

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]  time = 3.03656, size = 14357, normalized size = 110.44 \begin{align*} \text{result too large to display} \end{align*}

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^3*d*cosh(d*x + c)^4 + 9*a^3*d*cosh(d*x + c)^2 - a^3*d)*sinh(d*x
+ c)^2 + 8*(a^3*d*cosh(d*x + c)^7 - 3*a^3*d*cosh(d*x + c)^5 + 3*a^3*d*cosh(d*x + c)^3 - a^3*d*cosh(d*x + c))*
sinh(d*x + c)), 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))*sin
h(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 - 8*(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))*arctan(1/2*sqrt(b/(a - b))
*(cosh(d*x + c) + sinh(d*x + c))) + 8*(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))*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(b/(a
- b))/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))*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) + 2*(7
*(3*a^2 + 4*a*b)*cosh(d*x + c)^6 - 5*(11*a^2 + 4*a*b)*cosh(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^3*d*cos
h(d*x + c)^4 + 9*a^3*d*cosh(d*x + c)^2 - a^3*d)*sinh(d*x + c)^2 + 8*(a^3*d*cosh(d*x + c)^7 - 3*a^3*d*cosh(d*x
+ c)^5 + 3*a^3*d*cosh(d*x + c)^3 - a^3*d*cosh(d*x + c))*sinh(d*x + c))]

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

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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