3.74 \(\int \frac {\cosh ^3(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\)

Optimal. Leaf size=76 \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} d \sqrt {a+b}}+\frac {(a-b) \sinh (c+d x)}{a^2 d}+\frac {\sinh ^3(c+d x)}{3 a d} \]

[Out]

(a-b)*sinh(d*x+c)/a^2/d+1/3*sinh(d*x+c)^3/a/d+b^2*arctan(sinh(d*x+c)*a^(1/2)/(a+b)^(1/2))/a^(5/2)/d/(a+b)^(1/2
)

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Rubi [A]  time = 0.09, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4147, 390, 205} \[ \frac {b^2 \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{a^{5/2} d \sqrt {a+b}}+\frac {(a-b) \sinh (c+d x)}{a^2 d}+\frac {\sinh ^3(c+d x)}{3 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]*d) + ((a - b)*Sinh[c + d*x])/(a^2*d) +
Sinh[c + d*x]^3/(3*a*d)

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]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 4147

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^
((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n
/2] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 0.27, size = 79, normalized size = 1.04 \[ \frac {a^{3/2} \sinh (3 (c+d x))-\frac {12 b^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \text {csch}(c+d x)}{\sqrt {a}}\right )}{\sqrt {a+b}}+3 \sqrt {a} (3 a-4 b) \sinh (c+d x)}{12 a^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*b^2*ArcTan[(Sqrt[a + b]*Csch[c + d*x])/Sqrt[a]])/Sqrt[a + b] + 3*Sqrt[a]*(3*a - 4*b)*Sinh[c + d*x] + a^(
3/2)*Sinh[3*(c + d*x)])/(12*a^(5/2)*d)

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fricas [B]  time = 0.47, size = 1616, normalized size = 21.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

[1/24*((a^3 + a^2*b)*cosh(d*x + c)^6 + 6*(a^3 + a^2*b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3 + a^2*b)*sinh(d*x
+ c)^6 + 3*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3 - a^2*b - 4*a*b^2 + 5*(a^3 + a^2*b)*cosh(d*x +
c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + a^2*b)*cosh(d*x + c)^3 + 3*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c))*sinh(
d*x + c)^3 - a^3 - a^2*b - 3*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^2 + 3*(5*(a^3 + a^2*b)*cosh(d*x + c)^4 -
3*a^3 + a^2*b + 4*a*b^2 + 6*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 12*(b^2*cosh(d*x + c)
^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x + c)*sinh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(-a^
2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d
*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*b)*cosh(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^2 - a*b) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x
+ c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 +
4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 6*((a^3 + a^2*b)*cosh(d*x + c)^5 + 2*(3
*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^3 - (3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + a^3*
b)*d*cosh(d*x + c)^3 + 3*(a^4 + a^3*b)*d*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^4 + a^3*b)*d*cosh(d*x + c)*sinh(
d*x + c)^2 + (a^4 + a^3*b)*d*sinh(d*x + c)^3), 1/24*((a^3 + a^2*b)*cosh(d*x + c)^6 + 6*(a^3 + a^2*b)*cosh(d*x
+ c)*sinh(d*x + c)^5 + (a^3 + a^2*b)*sinh(d*x + c)^6 + 3*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^4 + 3*(3*a^3
- a^2*b - 4*a*b^2 + 5*(a^3 + a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 + a^2*b)*cosh(d*x + c)^3 + 3*
(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - a^2*b - 3*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x
+ c)^2 + 3*(5*(a^3 + a^2*b)*cosh(d*x + c)^4 - 3*a^3 + a^2*b + 4*a*b^2 + 6*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x +
c)^2)*sinh(d*x + c)^2 + 24*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x + c)*s
inh(d*x + c)^2 + b^2*sinh(d*x + c)^3)*sqrt(a^2 + a*b)*arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d
*x + c)^2 + a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/s
qrt(a^2 + a*b)) + 24*(b^2*cosh(d*x + c)^3 + 3*b^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*b^2*cosh(d*x + c)*sinh(d*x
+ c)^2 + b^2*sinh(d*x + c)^3)*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*b)*(cosh(d*x + c) + sinh(d*x + c))/(a +
b)) + 6*((a^3 + a^2*b)*cosh(d*x + c)^5 + 2*(3*a^3 - a^2*b - 4*a*b^2)*cosh(d*x + c)^3 - (3*a^3 - a^2*b - 4*a*b
^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + a^3*b)*d*cosh(d*x + c)^3 + 3*(a^4 + a^3*b)*d*cosh(d*x + c)^2*sinh(d*
x + c) + 3*(a^4 + a^3*b)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^4 + a^3*b)*d*sinh(d*x + c)^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[31,78]Warning, need to choose a branch for the root of a polynomial with parameters. This
might be wrong.The choice was done assuming [a,b]=[-13,-93]Warning, need to choose a branch for the root of a
polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-65,-82]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming
[a,b]=[97,-56]Warning, need to choose a branch for the root of a polynomial with parameters. This might be wr
ong.The choice was done assuming [a,b]=[80,44]Warning, need to choose a branch for the root of a polynomial wi
th parameters. This might be wrong.The choice was done assuming [a,b]=[22,73]Warning, need to choose a branch
for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[36,86]Wa
rning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice wa
s done assuming [a,b]=[-59,-45]Warning, need to choose a branch for the root of a polynomial with parameters.
This might be wrong.The choice was done assuming [a,b]=[15,66]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[55,80]Undef/Unsigned In
f encountered in limitEvaluation time: 1.25Limit: Max order reached or unable to make series expansion Error:
Bad Argument Value

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maple [B]  time = 0.49, size = 256, normalized size = 3.37 \[ -\frac {1}{3 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {1}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {b}{d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {1}{3 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {1}{d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {b}{d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {b^{2} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{d \,a^{\frac {5}{2}} \sqrt {a +b}}+\frac {b^{2} \arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{d \,a^{\frac {5}{2}} \sqrt {a +b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/d/a/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/d/a/(tanh(1/2*d*x+1/2*c)-1)^2-1/d/a/(tanh(1/2*d*x+1/2*c)-1)+1/d/a^2/(ta
nh(1/2*d*x+1/2*c)-1)*b-1/3/d/a/(tanh(1/2*d*x+1/2*c)+1)^3+1/2/d/a/(tanh(1/2*d*x+1/2*c)+1)^2-1/d/a/(tanh(1/2*d*x
+1/2*c)+1)+1/d/a^2/(tanh(1/2*d*x+1/2*c)+1)*b+1/d*b^2/a^(5/2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*
x+1/2*c)-2*b^(1/2))/a^(1/2))+1/d*b^2/a^(5/2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)+2*b^(1/
2))/a^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (3 \, {\left (3 \, a e^{\left (4 \, c\right )} - 4 \, b e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 3 \, {\left (3 \, a e^{\left (2 \, c\right )} - 4 \, b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + a e^{\left (6 \, d x + 6 \, c\right )} - a\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, a^{2} d} + \frac {1}{8} \, \int \frac {16 \, {\left (b^{2} e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (d x + c\right )}\right )}}{a^{3} e^{\left (4 \, d x + 4 \, c\right )} + a^{3} + 2 \, {\left (a^{3} e^{\left (2 \, c\right )} + 2 \, a^{2} b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^3/(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

1/24*(3*(3*a*e^(4*c) - 4*b*e^(4*c))*e^(4*d*x) - 3*(3*a*e^(2*c) - 4*b*e^(2*c))*e^(2*d*x) + a*e^(6*d*x + 6*c) -
a)*e^(-3*d*x - 3*c)/(a^2*d) + 1/8*integrate(16*(b^2*e^(3*d*x + 3*c) + b^2*e^(d*x + c))/(a^3*e^(4*d*x + 4*c) +
a^3 + 2*(a^3*e^(2*c) + 2*a^2*b*e^(2*c))*e^(2*d*x)), x)

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mupad [B]  time = 2.21, size = 332, normalized size = 4.37 \[ \frac {{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,a\,d}-\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,a\,d}-\frac {\sqrt {b^4}\,\left (2\,\mathrm {atan}\left (\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {2\,b^2}{a^8\,d\,{\left (a+b\right )}^2\,\sqrt {b^4}}-\frac {4\,\left (2\,a^2\,b^4\,d\,\sqrt {b^4}+2\,a^3\,b^3\,d\,\sqrt {b^4}\right )}{a^6\,b^5\,\left (a+b\right )\,\sqrt {a^6\,d^2+b\,a^5\,d^2}\,\sqrt {a^5\,d^2\,\left (a+b\right )}}\right )-\frac {2\,b^2\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}}{a^8\,d\,{\left (a+b\right )}^2\,\sqrt {b^4}}\right )\,\left (\frac {a^7\,\sqrt {a^6\,d^2+b\,a^5\,d^2}}{4}+\frac {a^6\,b\,\sqrt {a^6\,d^2+b\,a^5\,d^2}}{4}\right )\right )-2\,\mathrm {atan}\left (\frac {b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^5\,d^2\,\left (a+b\right )}}{2\,a^2\,d\,\left (a+b\right )\,\sqrt {b^4}}\right )\right )}{2\,\sqrt {a^6\,d^2+b\,a^5\,d^2}}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a-4\,b\right )}{8\,a^2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a-4\,b\right )}{8\,a^2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(3*c + 3*d*x)/(24*a*d) - exp(- 3*c - 3*d*x)/(24*a*d) - ((b^4)^(1/2)*(2*atan((exp(d*x)*exp(c)*((2*b^2)/(a^8*
d*(a + b)^2*(b^4)^(1/2)) - (4*(2*a^2*b^4*d*(b^4)^(1/2) + 2*a^3*b^3*d*(b^4)^(1/2)))/(a^6*b^5*(a + b)*(a^6*d^2 +
a^5*b*d^2)^(1/2)*(a^5*d^2*(a + b))^(1/2))) - (2*b^2*exp(3*c)*exp(3*d*x))/(a^8*d*(a + b)^2*(b^4)^(1/2)))*((a^7
*(a^6*d^2 + a^5*b*d^2)^(1/2))/4 + (a^6*b*(a^6*d^2 + a^5*b*d^2)^(1/2))/4)) - 2*atan((b^2*exp(d*x)*exp(c)*(a^5*d
^2*(a + b))^(1/2))/(2*a^2*d*(a + b)*(b^4)^(1/2)))))/(2*(a^6*d^2 + a^5*b*d^2)^(1/2)) + (exp(c + d*x)*(3*a - 4*b
))/(8*a^2*d) - (exp(- c - d*x)*(3*a - 4*b))/(8*a^2*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**3/(a+b*sech(d*x+c)**2),x)

[Out]

Integral(cosh(c + d*x)**3/(a + b*sech(c + d*x)**2), x)

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