Optimal. Leaf size=64 \[ \frac {a^3 \cosh (c+d x)}{d}-\frac {3 a^2 b \text {sech}(c+d x)}{d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, 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 = {4218, 276}
\begin {gather*} \frac {a^3 \cosh (c+d x)}{d}-\frac {3 a^2 b \text {sech}(c+d x)}{d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 4218
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^3 \sinh (c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {\left (b+a x^2\right )^3}{x^6} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^3+\frac {b^3}{x^6}+\frac {3 a b^2}{x^4}+\frac {3 a^2 b}{x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {a^3 \cosh (c+d x)}{d}-\frac {3 a^2 b \text {sech}(c+d x)}{d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}-\frac {b^3 \text {sech}^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 93, normalized size = 1.45 \begin {gather*} \frac {8 \left (b+a \cosh ^2(c+d x)\right )^3 \left (-b^3-5 a b^2 \cosh ^2(c+d x)-15 a^2 b \cosh ^4(c+d x)+5 a^3 \cosh ^6(c+d x)\right ) \text {sech}^5(c+d x)}{5 d (a+2 b+a \cosh (2 (c+d x)))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 58, normalized size = 0.91
method | result | size |
derivativedivides | \(-\frac {\frac {b^{3} \mathrm {sech}\left (d x +c \right )^{5}}{5}+a \,b^{2} \mathrm {sech}\left (d x +c \right )^{3}+3 a^{2} b \,\mathrm {sech}\left (d x +c \right )-\frac {a^{3}}{\mathrm {sech}\left (d x +c \right )}}{d}\) | \(58\) |
default | \(-\frac {\frac {b^{3} \mathrm {sech}\left (d x +c \right )^{5}}{5}+a \,b^{2} \mathrm {sech}\left (d x +c \right )^{3}+3 a^{2} b \,\mathrm {sech}\left (d x +c \right )-\frac {a^{3}}{\mathrm {sech}\left (d x +c \right )}}{d}\) | \(58\) |
risch | \(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}-\frac {2 \,{\mathrm e}^{d x +c} b \left (15 a^{2} {\mathrm e}^{8 d x +8 c}+60 a^{2} {\mathrm e}^{6 d x +6 c}+20 a b \,{\mathrm e}^{6 d x +6 c}+90 a^{2} {\mathrm e}^{4 d x +4 c}+40 a b \,{\mathrm e}^{4 d x +4 c}+16 b^{2} {\mathrm e}^{4 d x +4 c}+60 a^{2} {\mathrm e}^{2 d x +2 c}+20 a b \,{\mathrm e}^{2 d x +2 c}+15 a^{2}\right )}{5 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{5}}\) | \(173\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 94, normalized size = 1.47 \begin {gather*} \frac {a^{3} \cosh \left (d x + c\right )}{d} - \frac {6 \, a^{2} b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} - \frac {8 \, a b^{2}}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} - \frac {32 \, b^{3}}{5 \, d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}} \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. 276 vs.
\(2 (62) = 124\).
time = 0.35, size = 276, normalized size = 4.31 \begin {gather*} \frac {5 \, a^{3} \cosh \left (d x + c\right )^{6} + 5 \, a^{3} \sinh \left (d x + c\right )^{6} + 30 \, {\left (a^{3} - 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{4} + 15 \, {\left (5 \, a^{3} \cosh \left (d x + c\right )^{2} + 2 \, a^{3} - 4 \, a^{2} b\right )} \sinh \left (d x + c\right )^{4} + 50 \, a^{3} - 180 \, a^{2} b - 80 \, a b^{2} - 32 \, b^{3} + 5 \, {\left (15 \, a^{3} - 48 \, a^{2} b - 16 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 5 \, {\left (15 \, a^{3} \cosh \left (d x + c\right )^{4} + 15 \, a^{3} - 48 \, a^{2} b - 16 \, a b^{2} + 36 \, {\left (a^{3} - 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2}}{10 \, {\left (d \cosh \left (d x + c\right )^{5} + 5 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \, d \cosh \left (d x + c\right )^{3} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \sinh {\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 101, normalized size = 1.58 \begin {gather*} \frac {5 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - \frac {4 \, {\left (15 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{4} + 20 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 16 \, b^{3}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5}}}{10 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.50, size = 288, normalized size = 4.50 \begin {gather*} \frac {a^3\,{\mathrm {e}}^{c+d\,x}}{2\,d}+\frac {a^3\,{\mathrm {e}}^{-c-d\,x}}{2\,d}+\frac {64\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {8\,{\mathrm {e}}^{c+d\,x}\,\left (5\,a\,b^2-4\,b^3\right )}{5\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {6\,a^2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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