Optimal. Leaf size=74 \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3269, 380}
\begin {gather*} \frac {a^2 \sinh (c+d x)}{d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 380
Rule 3269
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right )^2 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^2+a (a+2 b) x^2+b (2 a+b) x^4+b^2 x^6\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 83, normalized size = 1.12 \begin {gather*} \frac {\left (2800 a^2-616 a b+102 b^2+\left (560 a^2+448 a b-111 b^2\right ) \cosh (2 (c+d x))+6 (28 a-b) b \cosh (4 (c+d x))+15 b^2 \cosh (6 (c+d x))\right ) \sinh (c+d x)}{3360 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.84, size = 97, normalized size = 1.31
method | result | size |
default | \(\frac {\left (-\frac {1}{64} b^{2}+\frac {1}{8} a b \right ) \sinh \left (5 d x +5 c \right )}{5 d}+\frac {\left (-\frac {3}{64} b^{2}+\frac {1}{8} a b +\frac {1}{4} a^{2}\right ) \sinh \left (3 d x +3 c \right )}{3 d}+\frac {\left (\frac {3}{64} b^{2}-\frac {1}{4} a b +\frac {3}{4} a^{2}\right ) \sinh \left (d x +c \right )}{d}+\frac {b^{2} \sinh \left (7 d x +7 c \right )}{448 d}\) | \(97\) |
risch | \(\frac {b^{2} {\mathrm e}^{7 d x +7 c}}{896 d}+\frac {b \,{\mathrm e}^{5 d x +5 c} a}{80 d}-\frac {b^{2} {\mathrm e}^{5 d x +5 c}}{640 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a b}{48 d}-\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{128 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}-\frac {a b \,{\mathrm e}^{d x +c}}{8 d}+\frac {3 \,{\mathrm e}^{d x +c} b^{2}}{128 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}+\frac {{\mathrm e}^{-d x -c} a b}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} b^{2}}{128 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} a b}{48 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{128 d}-\frac {b \,{\mathrm e}^{-5 d x -5 c} a}{80 d}+\frac {b^{2} {\mathrm e}^{-5 d x -5 c}}{640 d}-\frac {b^{2} {\mathrm e}^{-7 d x -7 c}}{896 d}\) | \(293\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs.
\(2 (68) = 136\).
time = 0.27, size = 242, normalized size = 3.27 \begin {gather*} -\frac {1}{4480} \, b^{2} {\left (\frac {{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} - 105 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {105 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 7 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{240} \, a 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^{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs.
\(2 (68) = 136\).
time = 0.37, size = 188, normalized size = 2.54 \begin {gather*} \frac {15 \, b^{2} \sinh \left (d x + c\right )^{7} + 21 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 105 \, {\left (b^{2} \cosh \left (d x + c\right )^{6} + {\left (8 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (63) = 126\).
time = 0.67, size = 136, normalized size = 1.84 \begin {gather*} \begin {cases} - \frac {2 a^{2} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {4 a b \sinh ^{5}{\left (c + d x \right )}}{15 d} + \frac {2 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} - \frac {2 b^{2} \sinh ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{2} \cosh ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs.
\(2 (68) = 136\).
time = 0.42, size = 196, normalized size = 2.65 \begin {gather*} \frac {b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (8 \, a b - b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} + \frac {{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} - \frac {{\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} - \frac {{\left (8 \, a b - b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 80, normalized size = 1.08 \begin {gather*} \frac {35\,a^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3+105\,a^2\,\mathrm {sinh}\left (c+d\,x\right )+42\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5+70\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+15\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^7+21\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^5}{105\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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