3.325 \(\int \frac{\text{sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx\)
Optimal. Leaf size=92 \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^2}+\frac{(a-3 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^2}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b)} \]
[Out]
((a - 3*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^2*d) + (b^(3/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]
*(a - b)^2*d) + (Sech[c + d*x]*Tanh[c + d*x])/(2*(a - b)*d)
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Rubi [A] time = 0.10543, antiderivative size = 92, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used =
{3190, 414, 522, 203, 205} \[ \frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d (a-b)^2}+\frac{(a-3 b) \tan ^{-1}(\sinh (c+d x))}{2 d (a-b)^2}+\frac{\tanh (c+d x) \text{sech}(c+d x)}{2 d (a-b)} \]
Antiderivative was successfully verified.
[In]
Int[Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]
[Out]
((a - 3*b)*ArcTan[Sinh[c + d*x]])/(2*(a - b)^2*d) + (b^(3/2)*ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]])/(Sqrt[a]
*(a - b)^2*d) + (Sech[c + d*x]*Tanh[c + d*x])/(2*(a - b)*d)
Rule 3190
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[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 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 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 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{sech}^3(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d}-\frac{\operatorname{Subst}\left (\int \frac{-a+2 b-b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\sinh (c+d x)\right )}{2 (a-b) d}\\ &=\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d}+\frac{(a-3 b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{2 (a-b)^2 d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\sinh (c+d x)\right )}{(a-b)^2 d}\\ &=\frac{(a-3 b) \tan ^{-1}(\sinh (c+d x))}{2 (a-b)^2 d}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} (a-b)^2 d}+\frac{\text{sech}(c+d x) \tanh (c+d x)}{2 (a-b) d}\\ \end{align*}
Mathematica [A] time = 0.222276, size = 91, normalized size = 0.99 \[ \frac{-2 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} \text{csch}(c+d x)}{\sqrt{b}}\right )+2 \sqrt{a} (a-3 b) \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+\sqrt{a} (a-b) \tanh (c+d x) \text{sech}(c+d x)}{2 \sqrt{a} d (a-b)^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Sech[c + d*x]^3/(a + b*Sinh[c + d*x]^2),x]
[Out]
(-2*b^(3/2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[b]] + 2*Sqrt[a]*(a - 3*b)*ArcTan[Tanh[(c + d*x)/2]] + Sqrt[a]*
(a - b)*Sech[c + d*x]*Tanh[c + d*x])/(2*Sqrt[a]*(a - b)^2*d)
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Maple [B] time = 0.068, size = 662, normalized size = 7.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^3/(a+b*sinh(d*x+c)^2),x)
[Out]
-1/d*b^2/(a-b)^2*a/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b
*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/d*b^2/(a-b)^2/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2
*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))+1/d*b^3/(a-b)^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2
)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)+a-2*b)*a)^(1/2))-1/d*b^2/(a-b)^2*a/(-b*(a-b))^(1/2)/((2*(
-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1/d*b^2/(a-
b)^2/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2))+1
/d*b^3/(a-b)^2/(-b*(a-b))^(1/2)/((2*(-b*(a-b))^(1/2)-a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(-b*(a-b
))^(1/2)-a+2*b)*a)^(1/2))-1/d/(a-b)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3*a+1/d/(a-b)^2/(tanh(1/
2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)^3*b+1/d/(a-b)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)*a-1/d/
(a-b)^2/(tanh(1/2*d*x+1/2*c)^2+1)^2*tanh(1/2*d*x+1/2*c)*b+1/d/(a-b)^2*arctan(tanh(1/2*d*x+1/2*c))*a-3/d/(a-b)^
2*arctan(tanh(1/2*d*x+1/2*c))*b
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (a e^{c} - 3 \, b e^{c}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) e^{\left (-c\right )}}{a^{2} d - 2 \, a b d + b^{2} d} + \frac{e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}{a d - b d +{\left (a d e^{\left (4 \, c\right )} - b d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (a d e^{\left (2 \, c\right )} - b d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + 8 \, \int \frac{b^{2} e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (d x + c\right )}}{4 \,{\left (a^{2} b - 2 \, a b^{2} + b^{3} +{\left (a^{2} b e^{\left (4 \, c\right )} - 2 \, a b^{2} e^{\left (4 \, c\right )} + b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \,{\left (2 \, a^{3} e^{\left (2 \, c\right )} - 5 \, a^{2} b e^{\left (2 \, c\right )} + 4 \, a b^{2} e^{\left (2 \, c\right )} - b^{3} 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(sech(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
[Out]
(a*e^c - 3*b*e^c)*arctan(e^(d*x + c))*e^(-c)/(a^2*d - 2*a*b*d + b^2*d) + (e^(3*d*x + 3*c) - e^(d*x + c))/(a*d
- b*d + (a*d*e^(4*c) - b*d*e^(4*c))*e^(4*d*x) + 2*(a*d*e^(2*c) - b*d*e^(2*c))*e^(2*d*x)) + 8*integrate(1/4*(b^
2*e^(3*d*x + 3*c) + b^2*e^(d*x + c))/(a^2*b - 2*a*b^2 + b^3 + (a^2*b*e^(4*c) - 2*a*b^2*e^(4*c) + b^3*e^(4*c))*
e^(4*d*x) + 2*(2*a^3*e^(2*c) - 5*a^2*b*e^(2*c) + 4*a*b^2*e^(2*c) - b^3*e^(2*c))*e^(2*d*x)), x)
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Fricas [B] time = 1.92306, size = 4370, normalized size = 47.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/2*(2*(a - b)*cosh(d*x + c)^3 + 6*(a - b)*cosh(d*x + c)*sinh(d*x + c)^2 + 2*(a - b)*sinh(d*x + c)^3 + (b*cos
h(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x +
c)^2 + b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) + b)*sqrt(-b/a)*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 + b)*cosh(d*x + c)^2 + 2*(3*b*cos
h(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) + 4*(a
*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 - a*cosh(d*x + c) + (3*a*cosh(d*x + c
)^2 - a)*sinh(d*x + c))*sqrt(-b/a) + 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*((a - 3*b)*cosh(d*x + c)^4 + 4*(a - 3*b)*cosh(d*x + c)*sinh
(d*x + c)^3 + (a - 3*b)*sinh(d*x + c)^4 + 2*(a - 3*b)*cosh(d*x + c)^2 + 2*(3*(a - 3*b)*cosh(d*x + c)^2 + a - 3
*b)*sinh(d*x + c)^2 + 4*((a - 3*b)*cosh(d*x + c)^3 + (a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + a - 3*b)*arctan(
cosh(d*x + c) + sinh(d*x + c)) - 2*(a - b)*cosh(d*x + c) + 2*(3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c)
)/((a^2 - 2*a*b + b^2)*d*cosh(d*x + c)^4 + 4*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - 2*a*
b + b^2)*d*sinh(d*x + c)^4 + 2*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c
)^2 + (a^2 - 2*a*b + b^2)*d)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*d + 4*((a^2 - 2*a*b + b^2)*d*cosh(d*x + c)^
3 + (a^2 - 2*a*b + b^2)*d*cosh(d*x + c))*sinh(d*x + c)), ((a - b)*cosh(d*x + c)^3 + 3*(a - b)*cosh(d*x + c)*si
nh(d*x + c)^2 + (a - b)*sinh(d*x + c)^3 + (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x
+ c)^4 + 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + b*cosh(d*x
+ c))*sinh(d*x + c) + b)*sqrt(b/a)*arctan(1/2*sqrt(b/a)*(cosh(d*x + c) + sinh(d*x + c))) + (b*cosh(d*x + c)^4
+ 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x + c)^4 + 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 + b)*s
inh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 + b*cosh(d*x + c))*sinh(d*x + c) + b)*sqrt(b/a)*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 - b)*cosh(d*x + c) + (3*b*cosh(d*x + c)^2
+ 4*a - b)*sinh(d*x + c))*sqrt(b/a)/b) + ((a - 3*b)*cosh(d*x + c)^4 + 4*(a - 3*b)*cosh(d*x + c)*sinh(d*x + c)
^3 + (a - 3*b)*sinh(d*x + c)^4 + 2*(a - 3*b)*cosh(d*x + c)^2 + 2*(3*(a - 3*b)*cosh(d*x + c)^2 + a - 3*b)*sinh(
d*x + c)^2 + 4*((a - 3*b)*cosh(d*x + c)^3 + (a - 3*b)*cosh(d*x + c))*sinh(d*x + c) + a - 3*b)*arctan(cosh(d*x
+ c) + sinh(d*x + c)) - (a - b)*cosh(d*x + c) + (3*(a - b)*cosh(d*x + c)^2 - a + b)*sinh(d*x + c))/((a^2 - 2*a
*b + b^2)*d*cosh(d*x + c)^4 + 4*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - 2*a*b + b^2)*d*si
nh(d*x + c)^4 + 2*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c)^2 + 2*(3*(a^2 - 2*a*b + b^2)*d*cosh(d*x + c)^2 + (a^2 -
2*a*b + b^2)*d)*sinh(d*x + c)^2 + (a^2 - 2*a*b + b^2)*d + 4*((a^2 - 2*a*b + b^2)*d*cosh(d*x + c)^3 + (a^2 - 2*
a*b + b^2)*d*cosh(d*x + c))*sinh(d*x + c))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{3}{\left (c + d x \right )}}{a + b \sinh ^{2}{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)**3/(a+b*sinh(d*x+c)**2),x)
[Out]
Integral(sech(c + d*x)**3/(a + b*sinh(c + d*x)**2), x)
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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(sech(d*x+c)^3/(a+b*sinh(d*x+c)^2),x, algorithm="giac")
[Out]
Exception raised: TypeError