Optimal. Leaf size=175 \[ -\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}+\frac {40 d^3 \sinh (a+b x)}{9 b^4}-\frac {40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}+\frac {2 d^2 (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {(c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
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Rubi [A] time = 0.23, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3311, 3296, 2637, 3310} \[ -\frac {40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}+\frac {2 d^2 (c+d x) \sinh ^2(a+b x) \cosh (a+b x)}{9 b^3}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {2 d (c+d x)^2 \sinh (a+b x)}{b^2}-\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}+\frac {40 d^3 \sinh (a+b x)}{9 b^4}-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {(c+d x)^3 \sinh ^2(a+b x) \cosh (a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3310
Rule 3311
Rubi steps
\begin {align*} \int (c+d x)^3 \sinh ^3(a+b x) \, dx &=\frac {(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {2}{3} \int (c+d x)^3 \sinh (a+b x) \, dx+\frac {\left (2 d^2\right ) \int (c+d x) \sinh ^3(a+b x) \, dx}{3 b^2}\\ &=-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {(2 d) \int (c+d x)^2 \cosh (a+b x) \, dx}{b}-\frac {\left (4 d^2\right ) \int (c+d x) \sinh (a+b x) \, dx}{9 b^2}\\ &=-\frac {4 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac {2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}-\frac {\left (4 d^2\right ) \int (c+d x) \sinh (a+b x) \, dx}{b^2}+\frac {\left (4 d^3\right ) \int \cosh (a+b x) \, dx}{9 b^3}\\ &=-\frac {40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {4 d^3 \sinh (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac {2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}+\frac {\left (4 d^3\right ) \int \cosh (a+b x) \, dx}{b^3}\\ &=-\frac {40 d^2 (c+d x) \cosh (a+b x)}{9 b^3}-\frac {2 (c+d x)^3 \cosh (a+b x)}{3 b}+\frac {40 d^3 \sinh (a+b x)}{9 b^4}+\frac {2 d (c+d x)^2 \sinh (a+b x)}{b^2}+\frac {2 d^2 (c+d x) \cosh (a+b x) \sinh ^2(a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh (a+b x) \sinh ^2(a+b x)}{3 b}-\frac {2 d^3 \sinh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \sinh ^3(a+b x)}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 127, normalized size = 0.73 \[ \frac {-162 b (c+d x) \cosh (a+b x) \left (b^2 (c+d x)^2+6 d^2\right )+6 b (c+d x) \cosh (3 (a+b x)) \left (3 b^2 (c+d x)^2+2 d^2\right )-4 d \sinh (a+b x) \left (\cosh (2 (a+b x)) \left (9 b^2 (c+d x)^2+2 d^2\right )-117 b^2 (c+d x)^2-242 d^2\right )}{216 b^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 345, normalized size = 1.97 \[ \frac {3 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{3} + 2 \, b c d^{2} + {\left (9 \, b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \sinh \left (b x + a\right )^{3} - 81 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} + 6 \, b c d^{2} + 3 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \cosh \left (b x + a\right ) + 3 \, {\left (81 \, b^{2} d^{3} x^{2} + 162 \, b^{2} c d^{2} x + 81 \, b^{2} c^{2} d + 162 \, d^{3} - {\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 414, normalized size = 2.37 \[ \frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x - 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} - 18 \, b^{2} c d^{2} x - 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 2 \, d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )}}{216 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x - 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} - 6 \, b^{2} c d^{2} x - 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} - 6 \, d^{3}\right )} e^{\left (b x + a\right )}}{8 \, b^{4}} - \frac {3 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{8 \, b^{4}} + \frac {{\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 27 \, b^{3} c^{2} d x + 9 \, b^{2} d^{3} x^{2} + 9 \, b^{3} c^{3} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 2 \, d^{3}\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{216 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 634, normalized size = 3.62 \[ \frac {\frac {d^{3} \left (-\frac {2 \left (b x +a \right )^{3} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{3}+2 \left (b x +a \right )^{2} \sinh \left (b x +a \right )-\frac {40 \left (b x +a \right ) \cosh \left (b x +a \right )}{9}+\frac {40 \sinh \left (b x +a \right )}{9}-\frac {\left (b x +a \right )^{2} \left (\sinh ^{3}\left (b x +a \right )\right )}{3}+\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{9}-\frac {2 \left (\sinh ^{3}\left (b x +a \right )\right )}{27}\right )}{b^{3}}-\frac {3 d^{3} a \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{27}\right )}{b^{3}}+\frac {3 d^{3} a^{2} \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\left (\sinh ^{3}\left (b x +a \right )\right )}{9}\right )}{b^{3}}-\frac {d^{3} a^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right )}{3}\right ) \cosh \left (b x +a \right )}{b^{3}}+\frac {3 c \,d^{2} \left (-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{3}+\frac {4 \left (b x +a \right ) \sinh \left (b x +a \right )}{3}-\frac {40 \cosh \left (b x +a \right )}{27}-\frac {2 \left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )}{9}+\frac {2 \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{27}\right )}{b^{2}}-\frac {6 c \,d^{2} a \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\left (\sinh ^{3}\left (b x +a \right )\right )}{9}\right )}{b^{2}}+\frac {3 c \,d^{2} a^{2} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right )}{3}\right ) \cosh \left (b x +a \right )}{b^{2}}+\frac {3 c^{2} d \left (-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )}{3}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{3}+\frac {2 \sinh \left (b x +a \right )}{3}-\frac {\left (\sinh ^{3}\left (b x +a \right )\right )}{9}\right )}{b}-\frac {3 c^{2} d a \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right )}{3}\right ) \cosh \left (b x +a \right )}{b}+c^{3} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (b x +a \right )\right )}{3}\right ) \cosh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 435, normalized size = 2.49 \[ \frac {1}{24} \, c^{2} d {\left (\frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{2}} - \frac {27 \, {\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2}} - \frac {27 \, {\left (b x + 1\right )} e^{\left (-b x - a\right )}}{b^{2}} + \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{2}}\right )} + \frac {1}{24} \, c^{3} {\left (\frac {e^{\left (3 \, b x + 3 \, a\right )}}{b} - \frac {9 \, e^{\left (b x + a\right )}}{b} - \frac {9 \, e^{\left (-b x - a\right )}}{b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{b}\right )} + \frac {1}{72} \, c d^{2} {\left (\frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{b^{3}} - \frac {81 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{b^{3}} + \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{3}}\right )} + \frac {1}{216} \, d^{3} {\left (\frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{b^{4}} - \frac {81 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{b^{4}} + \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 364, normalized size = 2.08 \[ \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (3\,b^2\,c^3+14\,c\,d^2\right )}{3\,b^3}-\frac {{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (63\,b^2\,c^2\,d+122\,d^3\right )}{27\,b^4}-\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3+20\,c\,d^2\right )}{9\,b^3}+\frac {2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^4}-\frac {2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^3}-\frac {2\,d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b}-\frac {7\,d^3\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}-\frac {14\,c\,d^2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d+14\,d^3\right )}{3\,b^3}-\frac {2\,c\,d^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{b}+\frac {d^3\,x^3\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {2\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b}+\frac {4\,c\,d^2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.22, size = 495, normalized size = 2.83 \[ \begin {cases} \frac {c^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} + \frac {3 c^{2} d x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \cosh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \cosh ^{3}{\left (a + b x \right )}}{b} + \frac {d^{3} x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 d^{3} x^{3} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {7 c^{2} d \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 c^{2} d \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {14 c d^{2} x \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{2} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} - \frac {7 d^{3} x^{2} \sinh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 d^{3} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b^{2}} + \frac {14 c d^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {40 c d^{2} \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 d^{3} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{3 b^{3}} - \frac {40 d^{3} x \cosh ^{3}{\left (a + b x \right )}}{9 b^{3}} - \frac {122 d^{3} \sinh ^{3}{\left (a + b x \right )}}{27 b^{4}} + \frac {40 d^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{9 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sinh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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