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

Optimal. Leaf size=49 \[ \frac{2 a}{b^2 d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d} \]

[Out]

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

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Rubi [A]  time = 0.0545739, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3223, 190, 43} \[ \frac{2 a}{b^2 d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rule 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0609125, size = 42, normalized size = 0.86 \[ \frac{2 \left (\frac{a}{a+b \sqrt{\sinh (c+d x)}}+\log \left (a+b \sqrt{\sinh (c+d x)}\right )\right )}{b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.021, size = 144, normalized size = 2.9 \begin{align*} -2\,{\frac{{a}^{2}}{d \left ({b}^{2}\sinh \left ( dx+c \right ) -{a}^{2} \right ){b}^{2}}}+{\frac{\ln \left ({b}^{2}\sinh \left ( dx+c \right ) -{a}^{2} \right ) }{d{b}^{2}}}+{\frac{a}{d{b}^{2}} \left ( b\sqrt{\sinh \left ( dx+c \right ) }-a \right ) ^{-1}}-{\frac{1}{d{b}^{2}}\ln \left ( b\sqrt{\sinh \left ( dx+c \right ) }-a \right ) }+{\frac{a}{d{b}^{2}} \left ( a+b\sqrt{\sinh \left ( dx+c \right ) } \right ) ^{-1}}+{\frac{1}{d{b}^{2}}\ln \left ( a+b\sqrt{\sinh \left ( dx+c \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)/(a+b*sinh(d*x+c)^(1/2))^2,x)

[Out]

-2/d*a^2/(b^2*sinh(d*x+c)-a^2)/b^2+1/d*ln(b^2*sinh(d*x+c)-a^2)/b^2+1/d*a/b^2/(b*sinh(d*x+c)^(1/2)-a)-1/d/b^2*l
n(b*sinh(d*x+c)^(1/2)-a)+a/b^2/d/(a+b*sinh(d*x+c)^(1/2))+ln(a+b*sinh(d*x+c)^(1/2))/b^2/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b \sqrt{\sinh \left (d x + c\right )} + a\right )}^{2}}\,{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)^(1/2))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.77643, size = 1370, normalized size = 27.96 \begin{align*} \frac{b^{2} d x + b^{2} c -{\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )^{2} -{\left (b^{2} d x + b^{2} c\right )} \sinh \left (d x + c\right )^{2} + 2 \,{\left (a^{2} d x + a^{2} c - 2 \, a^{2}\right )} \cosh \left (d x + c\right ) +{\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) + a^{2}\right )} \sinh \left (d x + c\right ) + 4 \,{\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt{\sinh \left (d x + c\right )}}{b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )}\right ) +{\left (b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - b^{2} + 2 \,{\left (b^{2} \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac{2 \,{\left (b^{2} \sinh \left (d x + c\right ) - a^{2}\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \,{\left (a^{2} d x + a^{2} c - 2 \, a^{2} -{\left (b^{2} d x + b^{2} c\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \,{\left (a b \cosh \left (d x + c\right ) + a b \sinh \left (d x + c\right )\right )} \sqrt{\sinh \left (d x + c\right )}}{b^{4} d \cosh \left (d x + c\right )^{2} + b^{4} d \sinh \left (d x + c\right )^{2} - 2 \, a^{2} b^{2} d \cosh \left (d x + c\right ) - b^{4} d + 2 \,{\left (b^{4} d \cosh \left (d x + c\right ) - a^{2} b^{2} d\right )} \sinh \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)/(a+b*sinh(d*x+c)^(1/2))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.92605, size = 151, normalized size = 3.08 \begin{align*} \begin{cases} \frac{x \cosh{\left (c \right )}}{a^{2}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{\sinh{\left (c + d x \right )}}{a^{2} d} & \text{for}\: b = 0 \\\frac{x \cosh{\left (c \right )}}{\left (a + b \sqrt{\sinh{\left (c \right )}}\right )^{2}} & \text{for}\: d = 0 \\\frac{2 a \log{\left (\frac{a}{b} + \sqrt{\sinh{\left (c + d x \right )}} \right )}}{a b^{2} d + b^{3} d \sqrt{\sinh{\left (c + d x \right )}}} + \frac{2 a}{a b^{2} d + b^{3} d \sqrt{\sinh{\left (c + d x \right )}}} + \frac{2 b \log{\left (\frac{a}{b} + \sqrt{\sinh{\left (c + d x \right )}} \right )} \sqrt{\sinh{\left (c + d x \right )}}}{a b^{2} d + b^{3} d \sqrt{\sinh{\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)**(1/2))**2,x)

[Out]

Piecewise((x*cosh(c)/a**2, Eq(b, 0) & Eq(d, 0)), (sinh(c + d*x)/(a**2*d), Eq(b, 0)), (x*cosh(c)/(a + b*sqrt(si
nh(c)))**2, Eq(d, 0)), (2*a*log(a/b + sqrt(sinh(c + d*x)))/(a*b**2*d + b**3*d*sqrt(sinh(c + d*x))) + 2*a/(a*b*
*2*d + b**3*d*sqrt(sinh(c + d*x))) + 2*b*log(a/b + sqrt(sinh(c + d*x)))*sqrt(sinh(c + d*x))/(a*b**2*d + b**3*d
*sqrt(sinh(c + d*x))), 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)^(1/2))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError