3.238 \(\int \frac {1}{a-b \sinh ^4(c+d x)} \, dx\)

Optimal. Leaf size=115 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \sqrt {\sqrt {a}+\sqrt {b}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3209, 1166, 208} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \sqrt {\sqrt {a}+\sqrt {b}}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Sinh[c + d*x]^4)^(-1),x]

[Out]

ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] - Sqrt[b]]*d) + ArcTanh[(Sqrt
[Sqrt[a] + Sqrt[b]]*Tanh[c + d*x])/a^(1/4)]/(2*a^(3/4)*Sqrt[Sqrt[a] + Sqrt[b]]*d)

Rule 208

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

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 3209

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dis
t[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x
]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{a-2 a x^2+(a-b) x^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}-\frac {\left (1+\frac {\sqrt {b}}{\sqrt {a}}\right ) \operatorname {Subst}\left (\int \frac {1}{-a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tanh (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}} d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 128, normalized size = 1.11 \[ \frac {\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {\tan ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tanh (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\sqrt {\sqrt {a} \sqrt {b}-a}}}{2 \sqrt {a} d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*Sinh[c + d*x]^4)^(-1),x]

[Out]

(-(ArcTan[((Sqrt[a] - Sqrt[b])*Tanh[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ArcTan
h[((Sqrt[a] + Sqrt[b])*Tanh[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/Sqrt[a + Sqrt[a]*Sqrt[b]])/(2*Sqrt[a]*d)

________________________________________________________________________________________

fricas [B]  time = 3.68, size = 975, normalized size = 8.48 \[ -\frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} \log \left (-2 \, {\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - 1}{{\left (a^{2} - a b\right )} d^{2}}} - b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

-1/4*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(2*(a^3 - a^2*b)
*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*
x + c)^2 + 2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(((a^2 - a*b)*d^2*sqrt(b/
((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2)) - b) + 1/4*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4
*b + a^3*b^2)*d^4)) + 1)/((a^2 - a*b)*d^2))*log(2*(a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) +
b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - 2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 -
2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + 1)/((a^2 - a
*b)*d^2)) - b) + 1/4*sqrt(-((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2))*lo
g(-2*(a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*
x + c) + b*sinh(d*x + c)^2 + 2*((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + a*b*d)*sqrt(-((a^2
- a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2)) - b) - 1/4*sqrt(-((a^2 - a*b)*d^2*
sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - 1)/((a^2 - a*b)*d^2))*log(-2*(a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*
b + a^3*b^2)*d^4)) + b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - 2*((a^4 - a^3*b
)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + a*b*d)*sqrt(-((a^2 - a*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2
)*d^4)) - 1)/((a^2 - a*b)*d^2)) - b)

________________________________________________________________________________________

giac [A]  time = 0.16, size = 1, normalized size = 0.01 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

0

________________________________________________________________________________________

maple [C]  time = 0.08, size = 102, normalized size = 0.89 \[ \frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}-4 a \,\textit {\_Z}^{6}+\left (6 a -16 b \right ) \textit {\_Z}^{4}-4 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 \textit {\_R}^{4}-3 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -3 \textit {\_R}^{5} a +3 \textit {\_R}^{3} a -8 \textit {\_R}^{3} b -\textit {\_R} a}}{4 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*sinh(d*x+c)^4),x)

[Out]

1/4/d*sum((-_R^6+3*_R^4-3*_R^2+1)/(_R^7*a-3*_R^5*a+3*_R^3*a-8*_R^3*b-_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootO
f(a*_Z^8-4*a*_Z^6+(6*a-16*b)*_Z^4-4*a*_Z^2+a))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{b \sinh \left (d x + c\right )^{4} - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a-b*sinh(d*x+c)^4),x, algorithm="maxima")

[Out]

-integrate(1/(b*sinh(d*x + c)^4 - a), x)

________________________________________________________________________________________

mupad [B]  time = 10.10, size = 1787, normalized size = 15.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*sinh(c + d*x)^4),x)

[Out]

log((((((524288*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) +
21*a*b^2*exp(2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)) + (1048576*a*d^3*((a^2 - (a^3*b)^(1/2))
/(a^3*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c +
2*d*x) + 48*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*((a^2 - (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))/4 + (262
144*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)
))/(b^5*(a - b)))*((a^2 - (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))/4 + (32768*(128*a*b - 128*a^2 - 15*b^2 + 25
6*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c + 2*d*x)))/(a*b^5*(a - b)))*((a^2 - (a^3*b)
^(1/2))/(16*(a^4*d^2 - a^3*b*d^2)))^(1/2) - log((((((524288*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^
3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp(2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a -
b)) - (1048576*a*d^3*((a^2 - (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 +
4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*((a^2 - (a^3*
b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))/4 - (262144*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2
*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*((a^2 - (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))
/4 + (32768*(128*a*b - 128*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c
+ 2*d*x)))/(a*b^5*(a - b)))*((a^2 - (a^3*b)^(1/2))/(16*(a^4*d^2 - a^3*b*d^2)))^(1/2) - log((((((524288*d^2*(3
1*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b^3*exp(2*c + 2*d*x) + 21*a*b^2*exp(2*c + 2*d
*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)) - (1048576*a*d^3*((a^2 + (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1
/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp
(2*c + 2*d*x)))/(b^5*(a - b)))*((a^2 + (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))/4 - (262144*d*(72*a*b - 64*a^2
- 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 288*a*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*((a^
2 + (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))/4 + (32768*(128*a*b - 128*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x)
+ 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c + 2*d*x)))/(a*b^5*(a - b)))*((a^2 + (a^3*b)^(1/2))/(16*(a^4*d^2 -
a^3*b*d^2)))^(1/2) + log((((((524288*d^2*(31*a*b^2 - 128*a^2*b + 128*a^3 - b^3 + 256*a^3*exp(2*c + 2*d*x) + b
^3*exp(2*c + 2*d*x) + 21*a*b^2*exp(2*c + 2*d*x) - 240*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)) + (1048576*a*d^3*
((a^2 + (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2)*(45*a*b^2 - 104*a^2*b + 64*a^3 - 3*b^3 + 4*b^3*exp(2*c + 2*d*x
) - 50*a*b^2*exp(2*c + 2*d*x) + 48*a^2*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*((a^2 + (a^3*b)^(1/2))/(a^3*d^2*(a
- b)))^(1/2))/4 + (262144*d*(72*a*b - 64*a^2 - 9*b^2 + 256*a^2*exp(2*c + 2*d*x) + 31*b^2*exp(2*c + 2*d*x) - 28
8*a*b*exp(2*c + 2*d*x)))/(b^5*(a - b)))*((a^2 + (a^3*b)^(1/2))/(a^3*d^2*(a - b)))^(1/2))/4 + (32768*(128*a*b -
128*a^2 - 15*b^2 + 256*a^2*exp(2*c + 2*d*x) + 29*b^2*exp(2*c + 2*d*x) - 304*a*b*exp(2*c + 2*d*x)))/(a*b^5*(a
- b)))*((a^2 + (a^3*b)^(1/2))/(16*(a^4*d^2 - a^3*b*d^2)))^(1/2)

________________________________________________________________________________________

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(1/(a-b*sinh(d*x+c)**4),x)

[Out]

Timed out

________________________________________________________________________________________