3.8 \(\int \frac{1}{(b \coth (c+d x))^{7/2}} \, dx\)
Optimal. Leaf size=100 \[ -\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b \coth (c+d x)}}{\sqrt{b}}\right )}{b^{7/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \coth (c+d x)}}{\sqrt{b}}\right )}{b^{7/2} d}-\frac{2}{5 b d (b \coth (c+d x))^{5/2}} \]
[Out]
-(ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(7/2)*d)) + ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(7/2)*d) - 2/
(5*b*d*(b*Coth[c + d*x])^(5/2)) - 2/(b^3*d*Sqrt[b*Coth[c + d*x]])
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Rubi [A] time = 0.0705722, antiderivative size = 100, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used =
{3474, 3476, 329, 298, 203, 206} \[ -\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b \coth (c+d x)}}{\sqrt{b}}\right )}{b^{7/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \coth (c+d x)}}{\sqrt{b}}\right )}{b^{7/2} d}-\frac{2}{5 b d (b \coth (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(b*Coth[c + d*x])^(-7/2),x]
[Out]
-(ArcTan[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(7/2)*d)) + ArcTanh[Sqrt[b*Coth[c + d*x]]/Sqrt[b]]/(b^(7/2)*d) - 2/
(5*b*d*(b*Coth[c + d*x])^(5/2)) - 2/(b^3*d*Sqrt[b*Coth[c + d*x]])
Rule 3474
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Rule 3476
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
Rule 298
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
& !GtQ[a/b, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
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 \frac{1}{(b \coth (c+d x))^{7/2}} \, dx &=-\frac{2}{5 b d (b \coth (c+d x))^{5/2}}+\frac{\int \frac{1}{(b \coth (c+d x))^{3/2}} \, dx}{b^2}\\ &=-\frac{2}{5 b d (b \coth (c+d x))^{5/2}}-\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}+\frac{\int \sqrt{b \coth (c+d x)} \, dx}{b^4}\\ &=-\frac{2}{5 b d (b \coth (c+d x))^{5/2}}-\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{-b^2+x^2} \, dx,x,b \coth (c+d x)\right )}{b^3 d}\\ &=-\frac{2}{5 b d (b \coth (c+d x))^{5/2}}-\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{-b^2+x^4} \, dx,x,\sqrt{b \coth (c+d x)}\right )}{b^3 d}\\ &=-\frac{2}{5 b d (b \coth (c+d x))^{5/2}}-\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \coth (c+d x)}\right )}{b^3 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \coth (c+d x)}\right )}{b^3 d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b \coth (c+d x)}}{\sqrt{b}}\right )}{b^{7/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \coth (c+d x)}}{\sqrt{b}}\right )}{b^{7/2} d}-\frac{2}{5 b d (b \coth (c+d x))^{5/2}}-\frac{2}{b^3 d \sqrt{b \coth (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0964058, size = 38, normalized size = 0.38 \[ -\frac{2 \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};\coth ^2(c+d x)\right )}{5 b d (b \coth (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(b*Coth[c + d*x])^(-7/2),x]
[Out]
(-2*Hypergeometric2F1[-5/4, 1, -1/4, Coth[c + d*x]^2])/(5*b*d*(b*Coth[c + d*x])^(5/2))
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Maple [A] time = 0.013, size = 83, normalized size = 0.8 \begin{align*} -{\frac{1}{d}\arctan \left ({\sqrt{b{\rm coth} \left (dx+c\right )}{\frac{1}{\sqrt{b}}}} \right ){b}^{-{\frac{7}{2}}}}+{\frac{1}{d}{\it Artanh} \left ({\sqrt{b{\rm coth} \left (dx+c\right )}{\frac{1}{\sqrt{b}}}} \right ){b}^{-{\frac{7}{2}}}}-{\frac{2}{5\,bd} \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{-{\frac{5}{2}}}}-2\,{\frac{1}{{b}^{3}d\sqrt{b{\rm coth} \left (dx+c\right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(b*coth(d*x+c))^(7/2),x)
[Out]
-arctan((b*coth(d*x+c))^(1/2)/b^(1/2))/b^(7/2)/d+arctanh((b*coth(d*x+c))^(1/2)/b^(1/2))/b^(7/2)/d-2/5/b/d/(b*c
oth(d*x+c))^(5/2)-2/b^3/d/(b*coth(d*x+c))^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
integrate((b*coth(d*x + c))^(-7/2), x)
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Fricas [B] time = 2.59864, size = 5797, normalized size = 57.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*coth(d*x+c))^(7/2),x, algorithm="fricas")
[Out]
[-1/20*(10*(cosh(d*x + c)^6 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6 + 3*(5*cosh(d*x + c)^2 + 1)*si
nh(d*x + c)^4 + 3*cosh(d*x + c)^4 + 4*(5*cosh(d*x + c)^3 + 3*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*cosh(d*x +
c)^4 + 6*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 3*cosh(d*x + c)^2 + 6*(cosh(d*x + c)^5 + 2*cosh(d*x + c)^3 + c
osh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-b)*arctan((cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x +
c)^2)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c))/(b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*s
inh(d*x + c)^2 + b)) + 5*(cosh(d*x + c)^6 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6 + 3*(5*cosh(d*x
+ c)^2 + 1)*sinh(d*x + c)^4 + 3*cosh(d*x + c)^4 + 4*(5*cosh(d*x + c)^3 + 3*cosh(d*x + c))*sinh(d*x + c)^3 + 3*
(5*cosh(d*x + c)^4 + 6*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 3*cosh(d*x + c)^2 + 6*(cosh(d*x + c)^5 + 2*cosh(
d*x + c)^3 + cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(-b)*log(-(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)^3*sinh(d*x
+ c) + 6*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 - 2*(cosh(
d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 - 1)*sqrt(-b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c))
- 2*b)/(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x +
c)*sinh(d*x + c)^3 + sinh(d*x + c)^4)) + 16*(3*cosh(d*x + c)^6 + 18*cosh(d*x + c)*sinh(d*x + c)^5 + 3*sinh(d*x
+ c)^6 + (45*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^4 + cosh(d*x + c)^4 + 4*(15*cosh(d*x + c)^3 + cosh(d*x + c))*
sinh(d*x + c)^3 + (45*cosh(d*x + c)^4 + 6*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - cosh(d*x + c)^2 + 2*(9*cosh(d
*x + c)^5 + 2*cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) - 3)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)))/(b^4*d*
cosh(d*x + c)^6 + 6*b^4*d*cosh(d*x + c)*sinh(d*x + c)^5 + b^4*d*sinh(d*x + c)^6 + 3*b^4*d*cosh(d*x + c)^4 + 3*
b^4*d*cosh(d*x + c)^2 + b^4*d + 3*(5*b^4*d*cosh(d*x + c)^2 + b^4*d)*sinh(d*x + c)^4 + 4*(5*b^4*d*cosh(d*x + c)
^3 + 3*b^4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^4*d*cosh(d*x + c)^4 + 6*b^4*d*cosh(d*x + c)^2 + b^4*d)*si
nh(d*x + c)^2 + 6*(b^4*d*cosh(d*x + c)^5 + 2*b^4*d*cosh(d*x + c)^3 + b^4*d*cosh(d*x + c))*sinh(d*x + c)), -1/2
0*(10*(cosh(d*x + c)^6 + 6*cosh(d*x + c)*sinh(d*x + c)^5 + sinh(d*x + c)^6 + 3*(5*cosh(d*x + c)^2 + 1)*sinh(d*
x + c)^4 + 3*cosh(d*x + c)^4 + 4*(5*cosh(d*x + c)^3 + 3*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*cosh(d*x + c)^4
+ 6*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 3*cosh(d*x + c)^2 + 6*(cosh(d*x + c)^5 + 2*cosh(d*x + c)^3 + cosh(d
*x + c))*sinh(d*x + c) + 1)*sqrt(b)*arctan(sqrt(b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c))/(b*cosh(d*x + c)^2 + 2*
b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + b)) - 5*(cosh(d*x + c)^6 + 6*cosh(d*x + c)*sinh(d*x + c)^5
+ sinh(d*x + c)^6 + 3*(5*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^4 + 3*cosh(d*x + c)^4 + 4*(5*cosh(d*x + c)^3 + 3*
cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*cosh(d*x + c)^4 + 6*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^2 + 3*cosh(d*x +
c)^2 + 6*(cosh(d*x + c)^5 + 2*cosh(d*x + c)^3 + cosh(d*x + c))*sinh(d*x + c) + 1)*sqrt(b)*log(2*b*cosh(d*x + c
)^4 + 8*b*cosh(d*x + c)^3*sinh(d*x + c) + 12*b*cosh(d*x + c)^2*sinh(d*x + c)^2 + 8*b*cosh(d*x + c)*sinh(d*x +
c)^3 + 2*b*sinh(d*x + c)^4 + 2*(cosh(d*x + c)^4 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 + (6*cosh(
d*x + c)^2 - 1)*sinh(d*x + c)^2 - cosh(d*x + c)^2 + 2*(2*cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c))*sqrt(
b)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)) - b) + 16*(3*cosh(d*x + c)^6 + 18*cosh(d*x + c)*sinh(d*x + c)^5 + 3*sin
h(d*x + c)^6 + (45*cosh(d*x + c)^2 + 1)*sinh(d*x + c)^4 + cosh(d*x + c)^4 + 4*(15*cosh(d*x + c)^3 + cosh(d*x +
c))*sinh(d*x + c)^3 + (45*cosh(d*x + c)^4 + 6*cosh(d*x + c)^2 - 1)*sinh(d*x + c)^2 - cosh(d*x + c)^2 + 2*(9*c
osh(d*x + c)^5 + 2*cosh(d*x + c)^3 - cosh(d*x + c))*sinh(d*x + c) - 3)*sqrt(b*cosh(d*x + c)/sinh(d*x + c)))/(b
^4*d*cosh(d*x + c)^6 + 6*b^4*d*cosh(d*x + c)*sinh(d*x + c)^5 + b^4*d*sinh(d*x + c)^6 + 3*b^4*d*cosh(d*x + c)^4
+ 3*b^4*d*cosh(d*x + c)^2 + b^4*d + 3*(5*b^4*d*cosh(d*x + c)^2 + b^4*d)*sinh(d*x + c)^4 + 4*(5*b^4*d*cosh(d*x
+ c)^3 + 3*b^4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^4*d*cosh(d*x + c)^4 + 6*b^4*d*cosh(d*x + c)^2 + b^4*
d)*sinh(d*x + c)^2 + 6*(b^4*d*cosh(d*x + c)^5 + 2*b^4*d*cosh(d*x + c)^3 + b^4*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(1/(b*coth(d*x+c))**(7/2),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(1/(b*coth(d*x+c))^(7/2),x, algorithm="giac")
[Out]
Exception raised: TypeError