3.35 \(\int \frac{1}{a+b \sinh ^2(c+d x)} \, dx\)
Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a-b}} \]
[Out]
ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)
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Rubi [A] time = 0.0270021, antiderivative size = 40, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used =
{3181, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a-b}} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Sinh[c + d*x]^2)^(-1),x]
[Out]
ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)
Rule 3181
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[1/(a + (a + b)*ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
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]
Rubi steps
\begin{align*} \int \frac{1}{a+b \sinh ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{a-b} d}\\ \end{align*}
Mathematica [A] time = 0.0709516, size = 40, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh (c+d x)}{\sqrt{a}}\right )}{\sqrt{a} d \sqrt{a-b}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Sinh[c + d*x]^2)^(-1),x]
[Out]
ArcTanh[(Sqrt[a - b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[a - b]*d)
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Maple [B] time = 0.042, size = 267, normalized size = 6.7 \begin{align*}{\frac{1}{d}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}}-{\frac{b}{d}{\it Artanh} \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{-b \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }+a-2\,b \right ) a}}}}-{\frac{1}{d}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}}-{\frac{b}{d}\arctan \left ({a\tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ){\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \right ){\frac{1}{\sqrt{-b \left ( a-b \right ) }}}{\frac{1}{\sqrt{ \left ( 2\,\sqrt{-b \left ( a-b \right ) }-a+2\,b \right ) a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*sinh(d*x+c)^2),x)
[Out]
1/d/((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*(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))*b-1/d/((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*(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))*b
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sinh(d*x+c)^2),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.92759, size = 1068, normalized size = 26.7 \begin{align*} \left [\frac{\log \left (\frac{b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \,{\left (b^{2} \cosh \left (d x + c\right )^{3} +{\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \,{\left (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 \, a - b\right )} \sqrt{a^{2} - a b}}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \,{\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (b \cosh \left (d x + c\right )^{3} +{\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right )}{2 \, \sqrt{a^{2} - a b} d}, -\frac{\sqrt{-a^{2} + a b} \arctan \left (-\frac{{\left (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 \, a - b\right )} \sqrt{-a^{2} + a b}}{2 \,{\left (a^{2} - a b\right )}}\right )}{{\left (a^{2} - a b\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sinh(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/2*log((b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sinh(d*x + c)^4 + 2*(2*a*b - b^2)*co
sh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 + 2*a*b - b^2)*sinh(d*x + c)^2 + 8*a^2 - 8*a*b + b^2 + 4*(b^2*cosh(d*
x + c)^3 + (2*a*b - b^2)*cosh(d*x + c))*sinh(d*x + c) - 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 - b)*sqrt(a^2 - a*b))/(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*si
nh(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))/(sqrt(a^2 - a*b)*d), -sqrt(-a^2 + a*b)*arctan(-1/2*(b*
cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 + 2*a - b)*sqrt(-a^2 + a*b)/(a^2 - a*b))
/((a^2 - a*b)*d)]
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Sympy [A] time = 118.484, size = 3859, normalized size = 96.48 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sinh(d*x+c)**2),x)
[Out]
Piecewise((zoo*x/sinh(c)**2, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-tanh(c/2 + d*x/2)/(2*d) - 1/(2*d*tanh(c/2 + d
*x/2)))/b, Eq(a, 0)), (2*tanh(c/2 + d*x/2)/(b*d*tanh(c/2 + d*x/2)**2 + b*d), Eq(a, b)), (x/(a + b*sinh(c)**2),
Eq(d, 0)), (-a**3*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh
(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b +
b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + a**3*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(
1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) +
32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - a**3*sqrt(1 -
2*b/a + 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d -
18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a
*b**2*d*sqrt(-a*b + b**2)) + a**3*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a + 2*sqrt(-a*b + b
**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b
*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 13*a**2*b*sqrt(1 - 2*b/a - 2*sqrt(-a*b +
b**2)/a)*log(-sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d
*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b*
*2)) - 13*a**2*b*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/
2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b*
*2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 5*a**2*b*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(-sqr
t(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2)
+ 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 5*a**2*b*sq
rt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4
*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d +
16*a*b**2*d*sqrt(-a*b + b**2)) - 5*a**2*sqrt(-a*b + b**2)*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1
- 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) +
32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 5*a**2*sqrt(-a
*b + b**2)*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*
x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) -
16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 3*a**2*sqrt(-a*b + b**2)*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a
)*log(-sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a
*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 3
*a**2*sqrt(-a*b + b**2)*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) +
tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a
*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 28*a*b**2*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*
log(-sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b
+ b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 28*
a*b**2*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2)
)/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a
*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 4*a*b**2*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1 - 2*b/
a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2
*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 4*a*b**2*sqrt(1 - 2*b
/a + 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a*
*3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2
*d*sqrt(-a*b + b**2)) + 20*a*b*sqrt(-a*b + b**2)*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1 - 2*b/a -
2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b*
*2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 20*a*b*sqrt(-a*b + b**2)
*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a
**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*
d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 4*a*b*sqrt(-a*b + b**2)*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(-sqrt
(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2)
+ 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 4*a*b*sqrt(-
a*b + b**2)*sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a + 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d
*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) -
16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 16*b**3*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1 -
2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*
a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) - 16*b**3*sqrt(1 -
2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18
*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b
**2*d*sqrt(-a*b + b**2)) - 16*b**2*sqrt(-a*b + b**2)*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(-sqrt(1 - 2*b
/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**
2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)) + 16*b**2*sqrt(-a*b +
b**2)*sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a)*log(sqrt(1 - 2*b/a - 2*sqrt(-a*b + b**2)/a) + tanh(c/2 + d*x/2))
/(2*a**4*d - 18*a**3*b*d + 8*a**3*d*sqrt(-a*b + b**2) + 32*a**2*b**2*d - 24*a**2*b*d*sqrt(-a*b + b**2) - 16*a*
b**3*d + 16*a*b**2*d*sqrt(-a*b + b**2)), True))
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Giac [A] time = 1.25799, size = 63, normalized size = 1.58 \begin{align*} \frac{\arctan \left (\frac{b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt{-a^{2} + a b}}\right )}{\sqrt{-a^{2} + a b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*sinh(d*x+c)^2),x, algorithm="giac")
[Out]
arctan(1/2*(b*e^(2*d*x + 2*c) + 2*a - b)/sqrt(-a^2 + a*b))/(sqrt(-a^2 + a*b)*d)