3.1.7 \(\int \text {csch}^2(c+d x) (a+b \sinh ^2(c+d x)) \, dx\) [7]

Optimal. Leaf size=16 \[ b x-\frac {a \coth (c+d x)}{d} \]

[Out]

b*x-a*coth(d*x+c)/d

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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3091, 8} \begin {gather*} b x-\frac {a \coth (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (a*Coth[c + d*x])/d

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3091

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A*Cos[e +
f*x]*((b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, x] && LtQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 16, normalized size = 1.00 \begin {gather*} b x-\frac {a \coth (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b*x - (a*Coth[c + d*x])/d

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Maple [A]
time = 1.02, size = 24, normalized size = 1.50

method	result	size
risch	\(b x -\frac {2 a}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x-2*a/d/(exp(2*d*x+2*c)-1)

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Maxima [A]
time = 0.26, size = 23, normalized size = 1.44 \begin {gather*} b x + \frac {2 \, a}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + 2*a/(d*(e^(-2*d*x - 2*c) - 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
time = 0.38, size = 36, normalized size = 2.25 \begin {gather*} -\frac {a \cosh \left (d x + c\right ) - {\left (b d x + a\right )} \sinh \left (d x + c\right )}{d \sinh \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*cosh(d*x + c) - (b*d*x + a)*sinh(d*x + c))/(d*sinh(d*x + c))

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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 ) \operatorname {csch}^{2}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.43, size = 28, normalized size = 1.75 \begin {gather*} \frac {{\left (d x + c\right )} b - \frac {2 \, a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d*x + c)*b - 2*a/(e^(2*d*x + 2*c) - 1))/d

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Mupad [B]
time = 0.57, size = 23, normalized size = 1.44 \begin {gather*} b\,x-\frac {2\,a}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x - (2*a)/(d*(exp(2*c + 2*d*x) - 1))

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