3.70 \(\int \text{csch}^5(e+f x) \sqrt{a+b \sinh ^2(e+f x)} \, dx\)

Optimal. Leaf size=144 \[ -\frac{(a-b) (3 a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{8 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 a f}+\frac{(3 a+b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 a f} \]

[Out]

-((a - b)*(3*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*a^(3/2)*f) + ((3*a +
b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*C
oth[e + f*x]*Csch[e + f*x]^3)/(4*a*f)

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Rubi [A]  time = 0.150017, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3186, 382, 378, 377, 206} \[ -\frac{(a-b) (3 a+b) \tanh ^{-1}\left (\frac{\sqrt{a} \cosh (e+f x)}{\sqrt{a+b \cosh ^2(e+f x)-b}}\right )}{8 a^{3/2} f}-\frac{\coth (e+f x) \text{csch}^3(e+f x) \left (a+b \cosh ^2(e+f x)-b\right )^{3/2}}{4 a f}+\frac{(3 a+b) \coth (e+f x) \text{csch}(e+f x) \sqrt{a+b \cosh ^2(e+f x)-b}}{8 a f} \]

Antiderivative was successfully verified.

[In]

Int[Csch[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]

[Out]

-((a - b)*(3*a + b)*ArcTanh[(Sqrt[a]*Cosh[e + f*x])/Sqrt[a - b + b*Cosh[e + f*x]^2]])/(8*a^(3/2)*f) + ((3*a +
b)*Sqrt[a - b + b*Cosh[e + f*x]^2]*Coth[e + f*x]*Csch[e + f*x])/(8*a*f) - ((a - b + b*Cosh[e + f*x]^2)^(3/2)*C
oth[e + f*x]*Csch[e + f*x]^3)/(4*a*f)

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 382

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[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

Rule 378

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

Rule 377

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

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])

Rubi steps

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

Mathematica [A]  time = 0.538088, size = 129, normalized size = 0.9 \[ \frac{\left (-6 a^2+4 a b+2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cosh (e+f x)}{\sqrt{2 a+b \cosh (2 (e+f x))-b}}\right )-\sqrt{2} \sqrt{a} \coth (e+f x) \text{csch}(e+f x) \sqrt{2 a+b \cosh (2 (e+f x))-b} \left (2 a \text{csch}^2(e+f x)-3 a+b\right )}{16 a^{3/2} f} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[e + f*x]^5*Sqrt[a + b*Sinh[e + f*x]^2],x]

[Out]

((-6*a^2 + 4*a*b + 2*b^2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Cosh[e + f*x])/Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]] - Sqrt[
2]*Sqrt[a]*Sqrt[2*a - b + b*Cosh[2*(e + f*x)]]*Coth[e + f*x]*Csch[e + f*x]*(-3*a + b + 2*a*Csch[e + f*x]^2))/(
16*a^(3/2)*f)

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Maple [B]  time = 0.099, size = 381, normalized size = 2.7 \begin{align*}{\frac{1}{16\, \left ( \sinh \left ( fx+e \right ) \right ) ^{4}\cosh \left ( fx+e \right ) f}\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( 6\,\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( \sinh \left ( fx+e \right ) \right ) ^{2}{a}^{5/2}-3\,{a}^{3}\ln \left ({\frac{ \left ( a+b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}+a-b}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{4}+2\,b\ln \left ({\frac{ \left ( a+b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}+a-b}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sinh \left ( fx+e \right ) \right ) ^{4}{a}^{2}+\ln \left ({\frac{1}{ \left ( \sinh \left ( fx+e \right ) \right ) ^{2}} \left ( \left ( a+b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{b \left ( \cosh \left ( fx+e \right ) \right ) ^{4}+ \left ( a-b \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}+a-b \right ) } \right ){b}^{2} \left ( \sinh \left ( fx+e \right ) \right ) ^{4}a-2\,b\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}} \left ( \sinh \left ( fx+e \right ) \right ) ^{2}{a}^{3/2}-4\,\sqrt{ \left ( a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2} \right ) \left ( \cosh \left ( fx+e \right ) \right ) ^{2}}{a}^{5/2} \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{a+b \left ( \sinh \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x)

[Out]

1/16*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*(6*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^2*a^(5
/2)-3*a^3*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)^2)*si
nh(f*x+e)^4+2*b*ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(f*x+e)
^2)*sinh(f*x+e)^4*a^2+ln(((a+b)*cosh(f*x+e)^2+2*a^(1/2)*(b*cosh(f*x+e)^4+(a-b)*cosh(f*x+e)^2)^(1/2)+a-b)/sinh(
f*x+e)^2)*b^2*sinh(f*x+e)^4*a-2*b*((a+b*sinh(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*sinh(f*x+e)^2*a^(3/2)-4*((a+b*sinh
(f*x+e)^2)*cosh(f*x+e)^2)^(1/2)*a^(5/2))/sinh(f*x+e)^4/a^(5/2)/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (f x + e\right )^{2} + a} \operatorname{csch}\left (f x + e\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sinh(f*x + e)^2 + a)*csch(f*x + e)^5, x)

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Fricas [B]  time = 4.09698, size = 8303, normalized size = 57.66 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^8 + 8*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (3*a^
2 - 2*a*b - b^2)*sinh(f*x + e)^8 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^6 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f
*x + e)^2 - 3*a^2 + 2*a*b + b^2)*sinh(f*x + e)^6 + 8*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - 3*(3*a^2 - 2*a
*b - b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 2*(35*(3*a^2 - 2*a*b - b^
2)*cosh(f*x + e)^4 - 30*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 9*a^2 - 6*a*b - 3*b^2)*sinh(f*x + e)^4 + 8*(7*
(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^5 - 10*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 + 3*(3*a^2 - 2*a*b - b^2)*cos
h(f*x + e))*sinh(f*x + e)^3 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x +
e)^6 - 15*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 9*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 - 3*a^2 + 2*a*b + b^
2)*sinh(f*x + e)^2 + 3*a^2 - 2*a*b - b^2 + 8*((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^7 - 3*(3*a^2 - 2*a*b - b^2)*
cosh(f*x + e)^5 + 3*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - (3*a^2 - 2*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e)
)*sqrt(a)*log(-((a + b)*cosh(f*x + e)^4 + 4*(a + b)*cosh(f*x + e)*sinh(f*x + e)^3 + (a + b)*sinh(f*x + e)^4 +
2*(3*a - b)*cosh(f*x + e)^2 + 2*(3*(a + b)*cosh(f*x + e)^2 + 3*a - b)*sinh(f*x + e)^2 + 2*sqrt(2)*(cosh(f*x +
e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(a)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^
2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)) + 4*((a + b)*cosh(f*x + e)^3
+ (3*a - b)*cosh(f*x + e))*sinh(f*x + e) + a + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f
*x + e)^4 + 2*(3*cosh(f*x + e)^2 - 1)*sinh(f*x + e)^2 - 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 - cosh(f*x + e)
)*sinh(f*x + e) + 1)) - 2*sqrt(2)*((3*a^2 - a*b)*cosh(f*x + e)^6 + 6*(3*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)
^5 + (3*a^2 - a*b)*sinh(f*x + e)^6 - (11*a^2 - a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - a*b)*cosh(f*x + e)^2 - 11*a
^2 + a*b)*sinh(f*x + e)^4 + 4*(5*(3*a^2 - a*b)*cosh(f*x + e)^3 - (11*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e)^3
- (11*a^2 - a*b)*cosh(f*x + e)^2 + (15*(3*a^2 - a*b)*cosh(f*x + e)^4 - 6*(11*a^2 - a*b)*cosh(f*x + e)^2 - 11*
a^2 + a*b)*sinh(f*x + e)^2 + 3*a^2 - a*b + 2*(3*(3*a^2 - a*b)*cosh(f*x + e)^5 - 2*(11*a^2 - a*b)*cosh(f*x + e)
^3 - (11*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh
(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(a^2*f*cosh(f*x + e)^8 + 8*a^2*f*cosh(f*x + e
)*sinh(f*x + e)^7 + a^2*f*sinh(f*x + e)^8 - 4*a^2*f*cosh(f*x + e)^6 + 6*a^2*f*cosh(f*x + e)^4 + 4*(7*a^2*f*cos
h(f*x + e)^2 - a^2*f)*sinh(f*x + e)^6 + 8*(7*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^5 -
4*a^2*f*cosh(f*x + e)^2 + 2*(35*a^2*f*cosh(f*x + e)^4 - 30*a^2*f*cosh(f*x + e)^2 + 3*a^2*f)*sinh(f*x + e)^4 +
8*(7*a^2*f*cosh(f*x + e)^5 - 10*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*f + 4*(7*
a^2*f*cosh(f*x + e)^6 - 15*a^2*f*cosh(f*x + e)^4 + 9*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^2 + 8*(a^2*f
*cosh(f*x + e)^7 - 3*a^2*f*cosh(f*x + e)^5 + 3*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*sinh(f*x + e)), 1/
8*(((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^8 + 8*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)*sinh(f*x + e)^7 + (3*a^2 - 2
*a*b - b^2)*sinh(f*x + e)^8 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^6 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x +
e)^2 - 3*a^2 + 2*a*b + b^2)*sinh(f*x + e)^6 + 8*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - 3*(3*a^2 - 2*a*b -
b^2)*cosh(f*x + e))*sinh(f*x + e)^5 + 6*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 2*(35*(3*a^2 - 2*a*b - b^2)*co
sh(f*x + e)^4 - 30*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 9*a^2 - 6*a*b - 3*b^2)*sinh(f*x + e)^4 + 8*(7*(3*a^
2 - 2*a*b - b^2)*cosh(f*x + e)^5 - 10*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 + 3*(3*a^2 - 2*a*b - b^2)*cosh(f*x
+ e))*sinh(f*x + e)^3 - 4*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 + 4*(7*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^6
- 15*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^4 + 9*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^2 - 3*a^2 + 2*a*b + b^2)*si
nh(f*x + e)^2 + 3*a^2 - 2*a*b - b^2 + 8*((3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^7 - 3*(3*a^2 - 2*a*b - b^2)*cosh(
f*x + e)^5 + 3*(3*a^2 - 2*a*b - b^2)*cosh(f*x + e)^3 - (3*a^2 - 2*a*b - b^2)*cosh(f*x + e))*sinh(f*x + e))*sqr
t(-a)*arctan(sqrt(2)*(cosh(f*x + e)^2 + 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 + 1)*sqrt(-a)*sqrt((b*
cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e
)^2))/(b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(2*a - b)*cosh(f*x + e)^2
+ 2*(3*b*cosh(f*x + e)^2 + 2*a - b)*sinh(f*x + e)^2 + 4*(b*cosh(f*x + e)^3 + (2*a - b)*cosh(f*x + e))*sinh(f*
x + e) + b)) + sqrt(2)*((3*a^2 - a*b)*cosh(f*x + e)^6 + 6*(3*a^2 - a*b)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2
- a*b)*sinh(f*x + e)^6 - (11*a^2 - a*b)*cosh(f*x + e)^4 + (15*(3*a^2 - a*b)*cosh(f*x + e)^2 - 11*a^2 + a*b)*s
inh(f*x + e)^4 + 4*(5*(3*a^2 - a*b)*cosh(f*x + e)^3 - (11*a^2 - a*b)*cosh(f*x + e))*sinh(f*x + e)^3 - (11*a^2
- a*b)*cosh(f*x + e)^2 + (15*(3*a^2 - a*b)*cosh(f*x + e)^4 - 6*(11*a^2 - a*b)*cosh(f*x + e)^2 - 11*a^2 + a*b)*
sinh(f*x + e)^2 + 3*a^2 - a*b + 2*(3*(3*a^2 - a*b)*cosh(f*x + e)^5 - 2*(11*a^2 - a*b)*cosh(f*x + e)^3 - (11*a^
2 - a*b)*cosh(f*x + e))*sinh(f*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2
- 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(a^2*f*cosh(f*x + e)^8 + 8*a^2*f*cosh(f*x + e)*sinh(f*x
+ e)^7 + a^2*f*sinh(f*x + e)^8 - 4*a^2*f*cosh(f*x + e)^6 + 6*a^2*f*cosh(f*x + e)^4 + 4*(7*a^2*f*cosh(f*x + e)^
2 - a^2*f)*sinh(f*x + e)^6 + 8*(7*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^5 - 4*a^2*f*cos
h(f*x + e)^2 + 2*(35*a^2*f*cosh(f*x + e)^4 - 30*a^2*f*cosh(f*x + e)^2 + 3*a^2*f)*sinh(f*x + e)^4 + 8*(7*a^2*f*
cosh(f*x + e)^5 - 10*a^2*f*cosh(f*x + e)^3 + 3*a^2*f*cosh(f*x + e))*sinh(f*x + e)^3 + a^2*f + 4*(7*a^2*f*cosh(
f*x + e)^6 - 15*a^2*f*cosh(f*x + e)^4 + 9*a^2*f*cosh(f*x + e)^2 - a^2*f)*sinh(f*x + e)^2 + 8*(a^2*f*cosh(f*x +
e)^7 - 3*a^2*f*cosh(f*x + e)^5 + 3*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*sinh(f*x + e))]

________________________________________________________________________________________

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(f*x+e)**5*(a+b*sinh(f*x+e)**2)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.57355, size = 3086, normalized size = 21.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(f*x+e)^5*(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

1/4*(3*a^2 - 2*a*b - b^2)*arctan(-1/2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e)
- 2*b*e^(2*f*x + 2*e) + b) - sqrt(b))/sqrt(-a))/(sqrt(-a)*a*f) - 1/2*(3*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4
*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a^2 - 2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4
*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*a*b - (sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f
*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^7*b^2 - 21*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*
f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*a^2*sqrt(b) - 18*(sqrt(b)*e^(2*f*x + 2*e) - sqr
t(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*a*b^(3/2) + 7*(sqrt(b)*e^(2*f*x + 2*e)
- sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^6*b^(5/2) - 44*(sqrt(b)*e^(2*f*x +
2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^3 - 121*(sqrt(b)*e^(2*f*x
+ 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a^2*b + 122*(sqrt(b)*e^(2
*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*a*b^2 - 21*(sqrt(b)*e
^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^5*b^3 - 292*(sqrt(b)
*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^3*sqrt(b) + 55
9*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^4*a^2*b^
(3/2) - 270*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b)
)^4*a*b^(5/2) + 35*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*
e) + b))^4*b^(7/2) - 176*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*
x + 2*e) + b))^3*a^4 + 872*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*
f*x + 2*e) + b))^3*a^3*b - 823*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e
^(2*f*x + 2*e) + b))^3*a^2*b^2 + 290*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) -
2*b*e^(2*f*x + 2*e) + b))^3*a*b^3 - 35*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e
) - 2*b*e^(2*f*x + 2*e) + b))^3*b^4 + 528*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2
*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^4*sqrt(b) - 936*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^
(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^3*b^(3/2) + 577*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4*f*x + 4*e
) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a^2*b^(5/2) - 158*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(4
*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*a*b^(7/2) + 21*(sqrt(b)*e^(2*f*x + 2*e) - sqrt
(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2*b^(9/2) + 192*(sqrt(b)*e^(2*f*x + 2*e)
- sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^5 - 656*(sqrt(b)*e^(2*f*x + 2*e)
- sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^4*b + 580*(sqrt(b)*e^(2*f*x + 2*e
) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^3*b^2 - 211*(sqrt(b)*e^(2*f*x +
2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a^2*b^3 + 38*(sqrt(b)*e^(2*f*
x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*a*b^4 - 7*(sqrt(b)*e^(2*f*
x + 2*e) - sqrt(b*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*b^5 - 192*a^5*sqrt(b) + 30
4*a^4*b^(3/2) - 180*a^3*b^(5/2) + 37*a^2*b^(7/2) - 2*a*b^(9/2) + b^(11/2))/(((sqrt(b)*e^(2*f*x + 2*e) - sqrt(b
*e^(4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))^2 - 2*(sqrt(b)*e^(2*f*x + 2*e) - sqrt(b*e^(
4*f*x + 4*e) + 4*a*e^(2*f*x + 2*e) - 2*b*e^(2*f*x + 2*e) + b))*sqrt(b) - 4*a + b)^4*a*f)