3.1.15 \(\int \text {csch}(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\) [15]

Optimal. Leaf size=52 \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {(2 a-b) b \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d} \]

[Out]

-a^2*arctanh(cosh(d*x+c))/d+(2*a-b)*b*cosh(d*x+c)/d+1/3*b^2*cosh(d*x+c)^3/d

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Rubi [A]
time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3265, 398, 212} \begin {gather*} -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {b (2 a-b) \cosh (c+d x)}{d}+\frac {b^2 \cosh ^3(c+d x)}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*ArcTanh[Cosh[c + d*x]])/d) + ((2*a - b)*b*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 398

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 3265

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 104, normalized size = 2.00 \begin {gather*} \frac {2 a b \cosh (c) \cosh (d x)}{d}-\frac {3 b^2 \cosh (c+d x)}{4 d}+\frac {b^2 \cosh (3 (c+d x))}{12 d}-\frac {a^2 \log \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {a^2 \log \left (\sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {2 a b \sinh (c) \sinh (d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b*Cosh[c]*Cosh[d*x])/d - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Log[Cosh[c/2
+ (d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d + (2*a*b*Sinh[c]*Sinh[d*x])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(50)=100\).
time = 1.16, size = 112, normalized size = 2.15

method	result	size
default	\(\frac {b^{2} \left (\frac {\left (\cosh ^{3}\left (d x +c \right )\right )}{3}+\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-2 b^{2} \left (\cosh \left (d x +c \right )-2 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )-2 a^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )+4 a b \arctanh \left ({\mathrm e}^{d x +c}\right )-2 b^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )}{d}\)	\(112\)
risch	\(\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}+\frac {a b \,{\mathrm e}^{d x +c}}{d}-\frac {3 \,{\mathrm e}^{d x +c} b^{2}}{8 d}+\frac {{\mathrm e}^{-d x -c} a b}{d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(b^2*(1/3*cosh(d*x+c)^3+cosh(d*x+c)-2*arctanh(exp(d*x+c)))+2*a*b*(cosh(d*x+c)-2*arctanh(exp(d*x+c)))-2*b^2
*(cosh(d*x+c)-2*arctanh(exp(d*x+c)))-2*a^2*arctanh(exp(d*x+c))+4*a*b*arctanh(exp(d*x+c))-2*b^2*arctanh(exp(d*x
+c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (50) = 100\).
time = 0.27, size = 102, normalized size = 1.96 \begin {gather*} \frac {1}{24} \, b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b {\left (\frac {e^{\left (d x + c\right )}}{d} + \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + a*b*(e^(d*x + c)/d +
e^(-d*x - c)/d) + a^2*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs. \(2 (50) = 100\).
time = 0.46, size = 492, normalized size = 9.46 \begin {gather*} \frac {b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} - 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \, {\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \, {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*(8*a*b - 3*b^2)*cosh
(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*(8*a*b
- 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(8*a*b - 3*b^2)*cosh(d*x + c)^2 + 3*(5*b^2*cosh(d*x + c)^4 + 6*(8*
a*b - 3*b^2)*cosh(d*x + c)^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^2 + b^2 - 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x
+ c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*
x + c) + 1) + 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c
)^2 + a^2*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(b^2*cosh(d*x + c)^5 + 2*(8*a*b - 3*b^2)
*cosh(d*x + c)^3 + (8*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)^2*sinh
(d*x + c) + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + d*sinh(d*x + c)^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*sinh(c + d*x)**2)**2*csch(c + d*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (50) = 100\).
time = 0.43, size = 110, normalized size = 2.12 \begin {gather*} \frac {b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b e^{\left (d x + c\right )} - 9 \, b^{2} e^{\left (d x + c\right )} - 24 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 24 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + {\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(b^2*e^(3*d*x + 3*c) + 24*a*b*e^(d*x + c) - 9*b^2*e^(d*x + c) - 24*a^2*log(e^(d*x + c) + 1) + 24*a^2*log(
abs(e^(d*x + c) - 1)) + (24*a*b*e^(2*d*x + 2*c) - 9*b^2*e^(2*d*x + 2*c) + b^2)*e^(-3*d*x - 3*c))/d

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Mupad [B]
time = 0.16, size = 116, normalized size = 2.23 \begin {gather*} \frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}-\frac {2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}+\frac {b\,{\mathrm {e}}^{-c-d\,x}\,\left (8\,a-3\,b\right )}{8\,d}+\frac {b\,{\mathrm {e}}^{c+d\,x}\,\left (8\,a-3\,b\right )}{8\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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