3.416 \(\int \frac{\cosh ^3(c+d x)}{(a+b \sqrt{\sinh (c+d x)})^2} \, dx\)

Optimal. Leaf size=142 \[ -\frac{8 a^3 \sqrt{\sinh (c+d x)}}{b^5 d}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}+\frac{2 a \left (a^4+b^4\right )}{b^6 d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \left (5 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^6 d}-\frac{4 a \sinh ^{\frac{3}{2}}(c+d x)}{3 b^3 d}+\frac{\sinh ^2(c+d x)}{2 b^2 d} \]

[Out]

(2*(5*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^6*d) + (2*a*(a^4 + b^4))/(b^6*d*(a + b*Sqrt[Sinh[c + d*x]]
)) - (8*a^3*Sqrt[Sinh[c + d*x]])/(b^5*d) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (4*a*Sinh[c + d*x]^(3/2))/(3*b^3*d)
+ Sinh[c + d*x]^2/(2*b^2*d)

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Rubi [A]  time = 0.163606, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {3223, 1890, 1620} \[ -\frac{8 a^3 \sqrt{\sinh (c+d x)}}{b^5 d}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}+\frac{2 a \left (a^4+b^4\right )}{b^6 d \left (a+b \sqrt{\sinh (c+d x)}\right )}+\frac{2 \left (5 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^6 d}-\frac{4 a \sinh ^{\frac{3}{2}}(c+d x)}{3 b^3 d}+\frac{\sinh ^2(c+d x)}{2 b^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(5*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]])/(b^6*d) + (2*a*(a^4 + b^4))/(b^6*d*(a + b*Sqrt[Sinh[c + d*x]]
)) - (8*a^3*Sqrt[Sinh[c + d*x]])/(b^5*d) + (3*a^2*Sinh[c + d*x])/(b^4*d) - (4*a*Sinh[c + d*x]^(3/2))/(3*b^3*d)
+ Sinh[c + d*x]^2/(2*b^2*d)

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 1890

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

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int \frac{\cosh ^3(c+d x)}{\left (a+b \sqrt{\sinh (c+d x)}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{\left (a+b \sqrt{x}\right )^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{x \left (1+x^4\right )}{(a+b x)^2} \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (-\frac{4 a^3}{b^5}+\frac{3 a^2 x}{b^4}-\frac{2 a x^2}{b^3}+\frac{x^3}{b^2}-\frac{a \left (a^4+b^4\right )}{b^5 (a+b x)^2}+\frac{5 a^4+b^4}{b^5 (a+b x)}\right ) \, dx,x,\sqrt{\sinh (c+d x)}\right )}{d}\\ &=\frac{2 \left (5 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )}{b^6 d}+\frac{2 a \left (a^4+b^4\right )}{b^6 d \left (a+b \sqrt{\sinh (c+d x)}\right )}-\frac{8 a^3 \sqrt{\sinh (c+d x)}}{b^5 d}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{4 a \sinh ^{\frac{3}{2}}(c+d x)}{3 b^3 d}+\frac{\sinh ^2(c+d x)}{2 b^2 d}\\ \end{align*}

Mathematica [A]  time = 0.464117, size = 123, normalized size = 0.87 \[ \frac{18 a^2 b^2 \sinh (c+d x)+12 \left (\frac{a \left (a^4+b^4\right )}{a+b \sqrt{\sinh (c+d x)}}+\left (5 a^4+b^4\right ) \log \left (a+b \sqrt{\sinh (c+d x)}\right )\right )-48 a^3 b \sqrt{\sinh (c+d x)}-8 a b^3 \sinh ^{\frac{3}{2}}(c+d x)+3 b^4 \sinh ^2(c+d x)}{6 b^6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*((5*a^4 + b^4)*Log[a + b*Sqrt[Sinh[c + d*x]]] + (a*(a^4 + b^4))/(a + b*Sqrt[Sinh[c + d*x]])) - 48*a^3*b*Sq
rt[Sinh[c + d*x]] + 18*a^2*b^2*Sinh[c + d*x] - 8*a*b^3*Sinh[c + d*x]^(3/2) + 3*b^4*Sinh[c + d*x]^2)/(6*b^6*d)

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Maple [C]  time = 0.207, size = 481, normalized size = 3.4 \begin{align*} -4\,{\frac{\tanh \left ( 1/2\,dx+c/2 \right ){a}^{4}}{d{b}^{4} \left ({a}^{2} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+2\,{b}^{2}\tanh \left ( 1/2\,dx+c/2 \right ) -{a}^{2} \right ) }}-4\,{\frac{\tanh \left ( 1/2\,dx+c/2 \right ) }{d \left ({a}^{2} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+2\,{b}^{2}\tanh \left ( 1/2\,dx+c/2 \right ) -{a}^{2} \right ) }}+5\,{\frac{\ln \left ({a}^{2} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+2\,{b}^{2}\tanh \left ( 1/2\,dx+c/2 \right ) -{a}^{2} \right ){a}^{4}}{{b}^{6}d}}+{\frac{1}{d{b}^{2}}\ln \left ({a}^{2} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}+2\,{b}^{2}\tanh \left ( 1/2\,dx+c/2 \right ) -{a}^{2} \right ) }+{\frac{1}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-3\,{\frac{{a}^{2}}{d{b}^{4} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }}-{\frac{1}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-5\,{\frac{\ln \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ){a}^{4}}{{b}^{6}d}}-{\frac{1}{d{b}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}-3\,{\frac{{a}^{2}}{d{b}^{4} \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+{\frac{1}{2\,d{b}^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-5\,{\frac{\ln \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ){a}^{4}}{{b}^{6}d}}-{\frac{1}{d{b}^{2}}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{1}{d}\mbox{{\tt ` int/indef0`}} \left ( -2\,{\frac{ \left ( \cosh \left ( dx+c \right ) \right ) ^{2}ab\sqrt{\sinh \left ( dx+c \right ) }}{{b}^{4} \left ( \sinh \left ( dx+c \right ) \right ) ^{2}-2\,{a}^{2}{b}^{2}\sinh \left ( dx+c \right ) +{a}^{4}}},\sinh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^(1/2))^2,x)

[Out]

-4/d/b^4*tanh(1/2*d*x+1/2*c)/(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh(1/2*d*x+1/2*c)-a^2)*a^4-4/d*tanh(1/2*d*x+1/
2*c)/(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh(1/2*d*x+1/2*c)-a^2)+5/d/b^6*ln(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh
(1/2*d*x+1/2*c)-a^2)*a^4+1/d/b^2*ln(a^2*tanh(1/2*d*x+1/2*c)^2+2*b^2*tanh(1/2*d*x+1/2*c)-a^2)+1/2/d/b^2/(tanh(1
/2*d*x+1/2*c)+1)^2-3/d/b^4/(tanh(1/2*d*x+1/2*c)+1)*a^2-1/2/d/b^2/(tanh(1/2*d*x+1/2*c)+1)-5/d/b^6*ln(tanh(1/2*d
*x+1/2*c)+1)*a^4-1/d/b^2*ln(tanh(1/2*d*x+1/2*c)+1)+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)^2-3/d/b^4/(tanh(1/2*d*x+1
/2*c)-1)*a^2+1/2/d/b^2/(tanh(1/2*d*x+1/2*c)-1)-5/d/b^6*ln(tanh(1/2*d*x+1/2*c)-1)*a^4-1/d/b^2*ln(tanh(1/2*d*x+1
/2*c)-1)+`int/indef0`(-2*cosh(d*x+c)^2*a*b*sinh(d*x+c)^(1/2)/(b^4*sinh(d*x+c)^2-2*a^2*b^2*sinh(d*x+c)+a^4),sin
h(d*x+c))/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )^{3}}{{\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)^3/(a+b*sinh(d*x+c)^(1/2))^2,x, algorithm="maxima")

[Out]

integrate(cosh(d*x + c)^3/(b*sqrt(sinh(d*x + c)) + a)^2, x)

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Fricas [B]  time = 7.24589, size = 5027, normalized size = 35.4 \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)^3/(a+b*sinh(d*x+c)^(1/2))^2,x, algorithm="fricas")

[Out]

1/24*(3*b^6*cosh(d*x + c)^6 + 3*b^6*sinh(d*x + c)^6 + 30*a^2*b^4*cosh(d*x + c)^5 + 30*a^2*b^4*cosh(d*x + c) -
3*b^6 + 6*(3*b^6*cosh(d*x + c) + 5*a^2*b^4)*sinh(d*x + c)^5 - 3*(24*a^4*b^2 + b^6 + 8*(5*a^4*b^2 + b^6)*d*x +
8*(5*a^4*b^2 + b^6)*c)*cosh(d*x + c)^4 + 3*(15*b^6*cosh(d*x + c)^2 + 50*a^2*b^4*cosh(d*x + c) - 24*a^4*b^2 - b
^6 - 8*(5*a^4*b^2 + b^6)*d*x - 8*(5*a^4*b^2 + b^6)*c)*sinh(d*x + c)^4 - 24*(4*a^6 + 7*a^2*b^4 - 2*(5*a^6 + a^2
*b^4)*d*x - 2*(5*a^6 + a^2*b^4)*c)*cosh(d*x + c)^3 + 12*(5*b^6*cosh(d*x + c)^3 + 25*a^2*b^4*cosh(d*x + c)^2 -
8*a^6 - 14*a^2*b^4 + 4*(5*a^6 + a^2*b^4)*d*x + 4*(5*a^6 + a^2*b^4)*c - (24*a^4*b^2 + b^6 + 8*(5*a^4*b^2 + b^6)
*d*x + 8*(5*a^4*b^2 + b^6)*c)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(24*a^4*b^2 + b^6 + 8*(5*a^4*b^2 + b^6)*d*x +
8*(5*a^4*b^2 + b^6)*c)*cosh(d*x + c)^2 + 3*(15*b^6*cosh(d*x + c)^4 + 100*a^2*b^4*cosh(d*x + c)^3 + 24*a^4*b^2
+ b^6 + 8*(5*a^4*b^2 + b^6)*d*x - 6*(24*a^4*b^2 + b^6 + 8*(5*a^4*b^2 + b^6)*d*x + 8*(5*a^4*b^2 + b^6)*c)*cosh
(d*x + c)^2 + 8*(5*a^4*b^2 + b^6)*c - 24*(4*a^6 + 7*a^2*b^4 - 2*(5*a^6 + a^2*b^4)*d*x - 2*(5*a^6 + a^2*b^4)*c)
*cosh(d*x + c))*sinh(d*x + c)^2 + 24*((5*a^4*b^2 + b^6)*cosh(d*x + c)^4 + (5*a^4*b^2 + b^6)*sinh(d*x + c)^4 -
2*(5*a^6 + a^2*b^4)*cosh(d*x + c)^3 - 2*(5*a^6 + a^2*b^4 - 2*(5*a^4*b^2 + b^6)*cosh(d*x + c))*sinh(d*x + c)^3
- (5*a^4*b^2 + b^6)*cosh(d*x + c)^2 - (5*a^4*b^2 + b^6 - 6*(5*a^4*b^2 + b^6)*cosh(d*x + c)^2 + 6*(5*a^6 + a^2*
b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + 2*(2*(5*a^4*b^2 + b^6)*cosh(d*x + c)^3 - 3*(5*a^6 + a^2*b^4)*cosh(d*x +
c)^2 - (5*a^4*b^2 + b^6)*cosh(d*x + c))*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))*s
qrt(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))) + 24*((5*a^4*b^2 + b^6)*cosh(d*x + c)^4 + (5*a^4*b^2 + b^6)*sinh(d*x + c)^4 - 2*(5*
a^6 + a^2*b^4)*cosh(d*x + c)^3 - 2*(5*a^6 + a^2*b^4 - 2*(5*a^4*b^2 + b^6)*cosh(d*x + c))*sinh(d*x + c)^3 - (5*
a^4*b^2 + b^6)*cosh(d*x + c)^2 - (5*a^4*b^2 + b^6 - 6*(5*a^4*b^2 + b^6)*cosh(d*x + c)^2 + 6*(5*a^6 + a^2*b^4)*
cosh(d*x + c))*sinh(d*x + c)^2 + 2*(2*(5*a^4*b^2 + b^6)*cosh(d*x + c)^3 - 3*(5*a^6 + a^2*b^4)*cosh(d*x + c)^2
- (5*a^4*b^2 + b^6)*cosh(d*x + c))*sinh(d*x + c))*log(2*(b^2*sinh(d*x + c) - a^2)/(cosh(d*x + c) - sinh(d*x +
c))) + 6*(3*b^6*cosh(d*x + c)^5 + 25*a^2*b^4*cosh(d*x + c)^4 + 5*a^2*b^4 - 2*(24*a^4*b^2 + b^6 + 8*(5*a^4*b^2
+ b^6)*d*x + 8*(5*a^4*b^2 + b^6)*c)*cosh(d*x + c)^3 - 12*(4*a^6 + 7*a^2*b^4 - 2*(5*a^6 + a^2*b^4)*d*x - 2*(5*a
^6 + a^2*b^4)*c)*cosh(d*x + c)^2 + (24*a^4*b^2 + b^6 + 8*(5*a^4*b^2 + b^6)*d*x + 8*(5*a^4*b^2 + b^6)*c)*cosh(d
*x + c))*sinh(d*x + c) - 16*(a*b^5*cosh(d*x + c)^5 + a*b^5*sinh(d*x + c)^5 + 10*a^3*b^3*cosh(d*x + c)^4 - 10*a
^3*b^3*cosh(d*x + c)^2 + a*b^5*cosh(d*x + c) + 5*(a*b^5*cosh(d*x + c) + 2*a^3*b^3)*sinh(d*x + c)^4 - 2*(15*a^5
*b + 4*a*b^5)*cosh(d*x + c)^3 + 2*(5*a*b^5*cosh(d*x + c)^2 + 20*a^3*b^3*cosh(d*x + c) - 15*a^5*b - 4*a*b^5)*si
nh(d*x + c)^3 + 2*(5*a*b^5*cosh(d*x + c)^3 + 30*a^3*b^3*cosh(d*x + c)^2 - 5*a^3*b^3 - 3*(15*a^5*b + 4*a*b^5)*c
osh(d*x + c))*sinh(d*x + c)^2 + (5*a*b^5*cosh(d*x + c)^4 + 40*a^3*b^3*cosh(d*x + c)^3 - 20*a^3*b^3*cosh(d*x +
c) + a*b^5 - 6*(15*a^5*b + 4*a*b^5)*cosh(d*x + c)^2)*sinh(d*x + c))*sqrt(sinh(d*x + c)))/(b^8*d*cosh(d*x + c)^
4 + b^8*d*sinh(d*x + c)^4 - 2*a^2*b^6*d*cosh(d*x + c)^3 - b^8*d*cosh(d*x + c)^2 + 2*(2*b^8*d*cosh(d*x + c) - a
^2*b^6*d)*sinh(d*x + c)^3 + (6*b^8*d*cosh(d*x + c)^2 - 6*a^2*b^6*d*cosh(d*x + c) - b^8*d)*sinh(d*x + c)^2 + 2*
(2*b^8*d*cosh(d*x + c)^3 - 3*a^2*b^6*d*cosh(d*x + c)^2 - b^8*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(cosh(d*x+c)**3/(a+b*sinh(d*x+c)**(1/2))**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(cosh(d*x+c)^3/(a+b*sinh(d*x+c)^(1/2))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError