Optimal. Leaf size=46 \[ \frac {a \sinh (c+d x)}{d}+\frac {(a+b) \sinh ^3(c+d x)}{3 d}+\frac {b \sinh ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 46, 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, 380}
\begin {gather*} \frac {(a+b) \sinh ^3(c+d x)}{3 d}+\frac {a \sinh (c+d x)}{d}+\frac {b \sinh ^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 380
Rule 3269
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a+(a+b) x^2+b x^4\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a \sinh (c+d x)}{d}+\frac {(a+b) \sinh ^3(c+d x)}{3 d}+\frac {b \sinh ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 48, normalized size = 1.04 \begin {gather*} \frac {(100 a-11 b+4 (5 a+2 b) \cosh (2 (c+d x))+3 b \cosh (4 (c+d x))) \sinh (c+d x)}{120 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.64, size = 55, normalized size = 1.20
method | result | size |
default | \(\frac {\left (-\frac {b}{8}+\frac {3 a}{4}\right ) \sinh \left (d x +c \right )}{d}+\frac {\left (\frac {b}{16}+\frac {a}{4}\right ) \sinh \left (3 d x +3 c \right )}{3 d}+\frac {b \sinh \left (5 d x +5 c \right )}{80 d}\) | \(55\) |
risch | \(\frac {b \,{\mathrm e}^{5 d x +5 c}}{160 d}+\frac {{\mathrm e}^{3 d x +3 c} a}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} b}{96 d}+\frac {3 a \,{\mathrm e}^{d x +c}}{8 d}-\frac {b \,{\mathrm e}^{d x +c}}{16 d}-\frac {3 \,{\mathrm e}^{-d x -c} a}{8 d}+\frac {{\mathrm e}^{-d x -c} b}{16 d}-\frac {{\mathrm e}^{-3 d x -3 c} a}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} b}{96 d}-\frac {b \,{\mathrm e}^{-5 d x -5 c}}{160 d}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (42) = 84\).
time = 0.26, size = 136, normalized size = 2.96 \begin {gather*} \frac {1}{480} \, b {\left (\frac {{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 30 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {30 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a {\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 82, normalized size = 1.78 \begin {gather*} \frac {3 \, b \sinh \left (d x + c\right )^{5} + 5 \, {\left (6 \, b \cosh \left (d x + c\right )^{2} + 4 \, a + b\right )} \sinh \left (d x + c\right )^{3} + 15 \, {\left (b \cosh \left (d x + c\right )^{4} + {\left (4 \, a + b\right )} \cosh \left (d x + c\right )^{2} + 12 \, a - 2 \, b\right )} \sinh \left (d x + c\right )}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (39) = 78\).
time = 0.28, size = 85, normalized size = 1.85 \begin {gather*} \begin {cases} - \frac {2 a \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {2 b \sinh ^{5}{\left (c + d x \right )}}{15 d} + \frac {b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \cosh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (42) = 84\).
time = 0.42, size = 108, normalized size = 2.35 \begin {gather*} \frac {b e^{\left (5 \, d x + 5 \, c\right )}}{160 \, d} + \frac {{\left (4 \, a + b\right )} e^{\left (3 \, d x + 3 \, c\right )}}{96 \, d} + \frac {{\left (6 \, a - b\right )} e^{\left (d x + c\right )}}{16 \, d} - \frac {{\left (6 \, a - b\right )} e^{\left (-d x - c\right )}}{16 \, d} - \frac {{\left (4 \, a + b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{96 \, d} - \frac {b e^{\left (-5 \, d x - 5 \, c\right )}}{160 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.34, size = 48, normalized size = 1.04 \begin {gather*} \frac {15\,a\,\mathrm {sinh}\left (c+d\,x\right )+5\,a\,{\mathrm {sinh}\left (c+d\,x\right )}^3+5\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+3\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5}{15\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________