3.334 \(\int \frac{\cosh (c+d x)}{(a+b \sinh ^2(c+d x))^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} d}+\frac{\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )} \]

[Out]

ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Sinh[c + d*x]/(2*a*d*(a + b*Sinh[c + d*x]^2))

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Rubi [A]  time = 0.0466733, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3190, 199, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} d}+\frac{\sinh (c+d x)}{2 a d \left (a+b \sinh ^2(c+d x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]*d) + Sinh[c + d*x]/(2*a*d*(a + b*Sinh[c + d*x]^2))

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.0456064, size = 64, normalized size = 0.97 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{b} \sinh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b}}+\frac{\sinh (c+d x)}{2 a \left (a+b \sinh ^2(c+d x)\right )}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]/(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

(ArcTan[(Sqrt[b]*Sinh[c + d*x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sinh[c + d*x]/(2*a*(a + b*Sinh[c + d*x]^2)))/d

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Maple [A]  time = 0.013, size = 57, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( dx+c \right ) }{2\,da \left ( a+b \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) }}+{\frac{1}{2\,da}\arctan \left ({b\sinh \left ( dx+c \right ){\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x)

[Out]

1/2*sinh(d*x+c)/a/d/(a+b*sinh(d*x+c)^2)+1/2/d/a/(a*b)^(1/2)*arctan(sinh(d*x+c)*b/(a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{e^{\left (3 \, d x + 3 \, c\right )} - e^{\left (d x + c\right )}}{a b d e^{\left (4 \, d x + 4 \, c\right )} + a b d + 2 \,{\left (2 \, a^{2} d e^{\left (2 \, c\right )} - a b d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \frac{1}{2} \, \int \frac{2 \,{\left (e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}\right )}}{a b e^{\left (4 \, d x + 4 \, c\right )} + a b + 2 \,{\left (2 \, a^{2} e^{\left (2 \, c\right )} - a b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

(e^(3*d*x + 3*c) - e^(d*x + c))/(a*b*d*e^(4*d*x + 4*c) + a*b*d + 2*(2*a^2*d*e^(2*c) - a*b*d*e^(2*c))*e^(2*d*x)
) + 1/2*integrate(2*(e^(3*d*x + 3*c) + e^(d*x + c))/(a*b*e^(4*d*x + 4*c) + a*b + 2*(2*a^2*e^(2*c) - a*b*e^(2*c
))*e^(2*d*x)), x)

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Fricas [B]  time = 1.6321, size = 3372, normalized size = 51.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[1/4*(4*a*b*cosh(d*x + c)^3 + 12*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 4*a*b*sinh(d*x + c)^3 - 4*a*b*cosh(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*(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*b)*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*cosh(d*x + c)^2 - 2*a - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - (2*a + b)*co
sh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(
d*x + c)^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a*b) + 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)) + 4*(3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d
*x + c))/(a^2*b^2*d*cosh(d*x + c)^4 + 4*a^2*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*b^2*d*sinh(d*x + c)^4 +
a^2*b^2*d + 2*(2*a^3*b - a^2*b^2)*d*cosh(d*x + c)^2 + 2*(3*a^2*b^2*d*cosh(d*x + c)^2 + (2*a^3*b - a^2*b^2)*d)*
sinh(d*x + c)^2 + 4*(a^2*b^2*d*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c)), 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 - 2*a*b*cosh(d*x + c) + (b*co
sh(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*c
osh(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*b)*arctan(1/2*sqrt(a*b)*(cosh(d*x + c) + sinh(d*x + c))/a) + (b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*si
nh(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)*sqrt(a*b)*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(a*b)/(a*b)) + 2*(3*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c))/(a^2*b^2*d*cosh(d*
x + c)^4 + 4*a^2*b^2*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^2*b^2*d*sinh(d*x + c)^4 + a^2*b^2*d + 2*(2*a^3*b - a^
2*b^2)*d*cosh(d*x + c)^2 + 2*(3*a^2*b^2*d*cosh(d*x + c)^2 + (2*a^3*b - a^2*b^2)*d)*sinh(d*x + c)^2 + 4*(a^2*b^
2*d*cosh(d*x + c)^3 + (2*a^3*b - a^2*b^2)*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [A]  time = 31.6704, size = 428, normalized size = 6.48 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \cosh{\left (c \right )}}{\sinh ^{4}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{\sinh{\left (c + d x \right )}}{a^{2} d} & \text{for}\: b = 0 \\- \frac{1}{3 b^{2} d \sinh ^{3}{\left (c + d x \right )}} & \text{for}\: a = 0 \\\frac{x \cosh{\left (c \right )}}{\left (a + b \sinh ^{2}{\left (c \right )}\right )^{2}} & \text{for}\: d = 0 \\\frac{2 i \sqrt{a} b \sqrt{\frac{1}{b}} \sinh{\left (c + d x \right )}}{4 i a^{\frac{5}{2}} b d \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} d \sqrt{\frac{1}{b}} \sinh ^{2}{\left (c + d x \right )}} + \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sinh{\left (c + d x \right )} \right )}}{4 i a^{\frac{5}{2}} b d \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} d \sqrt{\frac{1}{b}} \sinh ^{2}{\left (c + d x \right )}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sinh{\left (c + d x \right )} \right )}}{4 i a^{\frac{5}{2}} b d \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} d \sqrt{\frac{1}{b}} \sinh ^{2}{\left (c + d x \right )}} + \frac{b \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sinh{\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{4 i a^{\frac{5}{2}} b d \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} d \sqrt{\frac{1}{b}} \sinh ^{2}{\left (c + d x \right )}} - \frac{b \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sinh{\left (c + d x \right )} \right )} \sinh ^{2}{\left (c + d x \right )}}{4 i a^{\frac{5}{2}} b d \sqrt{\frac{1}{b}} + 4 i a^{\frac{3}{2}} b^{2} d \sqrt{\frac{1}{b}} \sinh ^{2}{\left (c + d x \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Piecewise((zoo*x*cosh(c)/sinh(c)**4, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (sinh(c + d*x)/(a**2*d), Eq(b, 0)), (-1/
(3*b**2*d*sinh(c + d*x)**3), Eq(a, 0)), (x*cosh(c)/(a + b*sinh(c)**2)**2, Eq(d, 0)), (2*I*sqrt(a)*b*sqrt(1/b)*
sinh(c + d*x)/(4*I*a**(5/2)*b*d*sqrt(1/b) + 4*I*a**(3/2)*b**2*d*sqrt(1/b)*sinh(c + d*x)**2) + a*log(-I*sqrt(a)
*sqrt(1/b) + sinh(c + d*x))/(4*I*a**(5/2)*b*d*sqrt(1/b) + 4*I*a**(3/2)*b**2*d*sqrt(1/b)*sinh(c + d*x)**2) - a*
log(I*sqrt(a)*sqrt(1/b) + sinh(c + d*x))/(4*I*a**(5/2)*b*d*sqrt(1/b) + 4*I*a**(3/2)*b**2*d*sqrt(1/b)*sinh(c +
d*x)**2) + b*log(-I*sqrt(a)*sqrt(1/b) + sinh(c + d*x))*sinh(c + d*x)**2/(4*I*a**(5/2)*b*d*sqrt(1/b) + 4*I*a**(
3/2)*b**2*d*sqrt(1/b)*sinh(c + d*x)**2) - b*log(I*sqrt(a)*sqrt(1/b) + sinh(c + d*x))*sinh(c + d*x)**2/(4*I*a**
(5/2)*b*d*sqrt(1/b) + 4*I*a**(3/2)*b**2*d*sqrt(1/b)*sinh(c + d*x)**2), True))

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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(cosh(d*x+c)/(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError