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

Optimal. Leaf size=125 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b}}} \]

[Out]

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

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Rubi [A]  time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3215, 1093, 205, 208} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [4]{b} d \sqrt {\sqrt {a}+\sqrt {b}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] - Sqrt[b]]*b^(1/4)*d) + ArcTan
h[(b^(1/4)*Cosh[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]]/(2*Sqrt[a]*Sqrt[Sqrt[a] + Sqrt[b]]*b^(1/4)*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 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 1093

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

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(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 - 2*b*ff^2*x^2 + b*ff^4*x^4
)^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 (c+d x)}{a-b \sinh ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 \sqrt {a} d}+\frac {\sqrt {b} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cosh (c+d x)\right )}{2 \sqrt {a} d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt [4]{b} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} \cosh (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt [4]{b} d}\\ \end {align*}

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Mathematica [C]  time = 0.17, size = 221, normalized size = 1.77 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8 b-4 \text {$\#$1}^6 b-16 \text {$\#$1}^4 a+6 \text {$\#$1}^4 b-4 \text {$\#$1}^2 b+b\& ,\frac {2 \text {$\#$1}^3 \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {$\#$1}^3 c+\text {$\#$1}^3 d x-2 \text {$\#$1} \log \left (-\text {$\#$1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {$\#$1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\text {$\#$1} c-\text {$\#$1} d x}{\text {$\#$1}^6 b-3 \text {$\#$1}^4 b-8 \text {$\#$1}^2 a+3 \text {$\#$1}^2 b-b}\& \right ]}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-(c*#1) - d*x*#1 - 2*Log[-Cosh[(c +
d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1 + c*#1^3 + d*x*#1^3 + 2*Log[-Cos
h[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^3)/(-b - 8*a*#1^2 + 3*b*#
1^2 - 3*b*#1^4 + b*#1^6) & ]/d

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fricas [B]  time = 0.94, size = 979, normalized size = 7.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.31, size = 329, normalized size = 2.63 \[ \frac {\frac {{\left (\sqrt {-b^{2} - \sqrt {a b} b} a^{2} b + 8 \, \sqrt {-b^{2} - \sqrt {a b} b} a b^{2} - \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} a b - 8 \, \sqrt {a b} \sqrt {-b^{2} - \sqrt {a b} b} b^{2}\right )} {\left | b \right |} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b + \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{a^{3} b^{3} + 7 \, a^{2} b^{4} - 8 \, a b^{5}} + \frac {{\left (\sqrt {-b^{2} + \sqrt {a b} b} a^{2} b + 8 \, \sqrt {-b^{2} + \sqrt {a b} b} a b^{2} + \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} a b + 8 \, \sqrt {a b} \sqrt {-b^{2} + \sqrt {a b} b} b^{2}\right )} {\left | b \right |} \arctan \left (\frac {e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}}{2 \, \sqrt {-\frac {b - \sqrt {{\left (a - b\right )} b + b^{2}}}{b}}}\right )}{a^{3} b^{3} + 7 \, a^{2} b^{4} - 8 \, a b^{5}}}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((sqrt(-b^2 - sqrt(a*b)*b)*a^2*b + 8*sqrt(-b^2 - sqrt(a*b)*b)*a*b^2 - sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*a
*b - 8*sqrt(a*b)*sqrt(-b^2 - sqrt(a*b)*b)*b^2)*abs(b)*arctan(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(b + sqrt(
(a - b)*b + b^2))/b))/(a^3*b^3 + 7*a^2*b^4 - 8*a*b^5) + (sqrt(-b^2 + sqrt(a*b)*b)*a^2*b + 8*sqrt(-b^2 + sqrt(a
*b)*b)*a*b^2 + sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*a*b + 8*sqrt(a*b)*sqrt(-b^2 + sqrt(a*b)*b)*b^2)*abs(b)*arcta
n(1/2*(e^(d*x + c) + e^(-d*x - c))/sqrt(-(b - sqrt((a - b)*b + b^2))/b))/(a^3*b^3 + 7*a^2*b^4 - 8*a*b^5))/d

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maple [A]  time = 0.10, size = 126, normalized size = 1.01 \[ \frac {\arctan \left (\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \sqrt {a b}-2 a}{4 \sqrt {-a b +\sqrt {a b}\, a}}\right )}{2 d \sqrt {-a b +\sqrt {a b}\, a}}-\frac {\arctan \left (\frac {-2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \sqrt {a b}+2 a}{4 \sqrt {-a b -\sqrt {a b}\, a}}\right )}{2 d \sqrt {-a b -\sqrt {a b}\, a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 8.15, size = 1007, normalized size = 8.06 \[ \ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}+\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}-\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b-\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}}-\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}-\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}+\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}}+\ln \left (\frac {\left (\frac {\left (\frac {4194304\,a^2\,d^2\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (3\,a+b\right )}{b^5\,{\left (a-b\right )}^2}+\frac {16777216\,a^3\,d^3\,{\mathrm {e}}^{c+d\,x}\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{b^5\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {2097152\,a^2\,d\,{\mathrm {e}}^{c+d\,x}}{b^6\,\left (a-b\right )}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{a^2\,b\,d^2\,\left (a-b\right )}}}{4}-\frac {262144\,a\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\,\left (a+b\right )}{b^6\,{\left (a-b\right )}^2}\right )\,\sqrt {-\frac {a\,b+\sqrt {a^3\,b}}{16\,\left (a^3\,b\,d^2-a^2\,b^2\,d^2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^5*(a - b)^2) + (16777216*a^3*d^3*exp(c + d*x)*(-
(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/(b^5*(a - b)))*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^
(1/2))/4 - (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/4 -
(262144*a*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^6*(a - b)^2))*(-(a*b - (a^3*b)^(1/2))/(16*(a^3*b*d^2 - a^2*b^2*d
^2)))^(1/2) - log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^5*(a - b)^2) - (16777216*a^3*d^3*e
xp(c + d*x)*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/(b^5*(a - b)))*(-(a*b - (a^3*b)^(1/2))/(a^2*b*
d^2*(a - b)))^(1/2))/4 + (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*(-(a*b - (a^3*b)^(1/2))/(a^2*b*d^2*(a - b
)))^(1/2))/4 - (262144*a*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^6*(a - b)^2))*(-(a*b - (a^3*b)^(1/2))/(16*(a^3*b*d
^2 - a^2*b^2*d^2)))^(1/2) - log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^5*(a - b)^2) - (1677
7216*a^3*d^3*exp(c + d*x)*(-(a*b + (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/(b^5*(a - b)))*(-(a*b + (a^3*b)^
(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/4 + (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*(-(a*b + (a^3*b)^(1/2))/(a^
2*b*d^2*(a - b)))^(1/2))/4 - (262144*a*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^6*(a - b)^2))*(-(a*b + (a^3*b)^(1/2)
)/(16*(a^3*b*d^2 - a^2*b^2*d^2)))^(1/2) + log((((((4194304*a^2*d^2*(exp(2*c + 2*d*x) + 1)*(3*a + b))/(b^5*(a -
b)^2) + (16777216*a^3*d^3*exp(c + d*x)*(-(a*b + (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/(b^5*(a - b)))*(-(
a*b + (a^3*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/4 - (2097152*a^2*d*exp(c + d*x))/(b^6*(a - b)))*(-(a*b + (a^3
*b)^(1/2))/(a^2*b*d^2*(a - b)))^(1/2))/4 - (262144*a*(exp(2*c + 2*d*x) + 1)*(a + b))/(b^6*(a - b)^2))*(-(a*b +
(a^3*b)^(1/2))/(16*(a^3*b*d^2 - a^2*b^2*d^2)))^(1/2)

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

[Out]

Timed out

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