3.3.96 \(\int \cosh (c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\) [296]

Optimal. Leaf size=49 \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3269, 200} \begin {gather*} \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 3269

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

Rubi steps

\begin {align*} \int \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b x^2\right )^2 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^2+2 a b x^2+b^2 x^4\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.00 \begin {gather*} \frac {a^2 \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^5(c+d x)}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.72, size = 41, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {b^{2} \left (\sinh ^{5}\left (d x +c \right )\right )}{5}+\frac {2 a b \left (\sinh ^{3}\left (d x +c \right )\right )}{3}+a^{2} \sinh \left (d x +c \right )}{d}\)	\(41\)
default	\(\frac {\frac {b^{2} \left (\sinh ^{5}\left (d x +c \right )\right )}{5}+\frac {2 a b \left (\sinh ^{3}\left (d x +c \right )\right )}{3}+a^{2} \sinh \left (d x +c \right )}{d}\)	\(41\)
risch	\(\frac {b^{2} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {{\mathrm e}^{3 d x +3 c} a b}{12 d}-\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{32 d}+\frac {{\mathrm e}^{d x +c} a^{2}}{2 d}-\frac {a b \,{\mathrm e}^{d x +c}}{4 d}+\frac {{\mathrm e}^{d x +c} b^{2}}{16 d}-\frac {{\mathrm e}^{-d x -c} a^{2}}{2 d}+\frac {{\mathrm e}^{-d x -c} a b}{4 d}-\frac {{\mathrm e}^{-d x -c} b^{2}}{16 d}-\frac {{\mathrm e}^{-3 d x -3 c} a b}{12 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{32 d}-\frac {b^{2} {\mathrm e}^{-5 d x -5 c}}{160 d}\)	\(193\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 45, normalized size = 0.92 \begin {gather*} \frac {b^{2} \sinh \left (d x + c\right )^{5}}{5 \, d} + \frac {2 \, a b \sinh \left (d x + c\right )^{3}}{3 \, d} + \frac {a^{2} \sinh \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (45) = 90\).
time = 0.38, size = 106, normalized size = 2.16 \begin {gather*} \frac {3 \, b^{2} \sinh \left (d x + c\right )^{5} + 5 \, {\left (6 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 15 \, {\left (b^{2} \cosh \left (d x + c\right )^{4} + {\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 16 \, a^{2} - 8 \, a b + 2 \, b^{2}\right )} \sinh \left (d x + c\right )}{240 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(3*b^2*sinh(d*x + c)^5 + 5*(6*b^2*cosh(d*x + c)^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^3 + 15*(b^2*cosh(d*x +
c)^4 + (8*a*b - 3*b^2)*cosh(d*x + c)^2 + 16*a^2 - 8*a*b + 2*b^2)*sinh(d*x + c))/d

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Sympy [A]
time = 0.28, size = 58, normalized size = 1.18 \begin {gather*} \begin {cases} \frac {a^{2} \sinh {\left (c + d x \right )}}{d} + \frac {2 a b \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \sinh ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \cosh {\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*sinh(c + d*x)/d + 2*a*b*sinh(c + d*x)**3/(3*d) + b**2*sinh(c + d*x)**5/(5*d), Ne(d, 0)), (x*(a
+ b*sinh(c)**2)**2*cosh(c), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (45) = 90\).
time = 0.41, size = 134, normalized size = 2.73 \begin {gather*} \frac {b^{2} e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} - \frac {b^{2} e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} + \frac {{\left (8 \, a b - 3 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {{\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {{\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {{\left (8 \, a b - 3 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/160*b^2*e^(5*d*x + 5*c)/d - 1/160*b^2*e^(-5*d*x - 5*c)/d + 1/96*(8*a*b - 3*b^2)*e^(3*d*x + 3*c)/d + 1/16*(8*
a^2 - 4*a*b + b^2)*e^(d*x + c)/d - 1/16*(8*a^2 - 4*a*b + b^2)*e^(-d*x - c)/d - 1/96*(8*a*b - 3*b^2)*e^(-3*d*x
- 3*c)/d

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Mupad [B]
time = 0.83, size = 42, normalized size = 0.86 \begin {gather*} \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (15\,a^2+10\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+3\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^4\right )}{15\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sinh(c + d*x)*(15*a^2 + 3*b^2*sinh(c + d*x)^4 + 10*a*b*sinh(c + d*x)^2))/(15*d)

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