Optimal. Leaf size=175 \[ -\frac {40 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b} \]
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Rubi [A]
time = 0.12, 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 = {3392, 3377,
2718, 3391} \begin {gather*} -\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {40 d^3 \cosh (a+b x)}{9 b^4}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 d^2 (c+d x) \sinh (a+b x) \cosh ^2(a+b x)}{9 b^3}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {(c+d x)^3 \sinh (a+b x) \cosh ^2(a+b x)}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 3391
Rule 3392
Rubi steps
\begin {align*} \int (c+d x)^3 \cosh ^3(a+b x) \, dx &=-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {2}{3} \int (c+d x)^3 \cosh (a+b x) \, dx+\frac {\left (2 d^2\right ) \int (c+d x) \cosh ^3(a+b x) \, dx}{3 b^2}\\ &=-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {(2 d) \int (c+d x)^2 \sinh (a+b x) \, dx}{b}+\frac {\left (4 d^2\right ) \int (c+d x) \cosh (a+b x) \, dx}{9 b^2}\\ &=-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {4 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}+\frac {\left (4 d^2\right ) \int (c+d x) \cosh (a+b x) \, dx}{b^2}-\frac {\left (4 d^3\right ) \int \sinh (a+b x) \, dx}{9 b^3}\\ &=-\frac {4 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}-\frac {\left (4 d^3\right ) \int \sinh (a+b x) \, dx}{b^3}\\ &=-\frac {40 d^3 \cosh (a+b x)}{9 b^4}-\frac {2 d (c+d x)^2 \cosh (a+b x)}{b^2}-\frac {2 d^3 \cosh ^3(a+b x)}{27 b^4}-\frac {d (c+d x)^2 \cosh ^3(a+b x)}{3 b^2}+\frac {40 d^2 (c+d x) \sinh (a+b x)}{9 b^3}+\frac {2 (c+d x)^3 \sinh (a+b x)}{3 b}+\frac {2 d^2 (c+d x) \cosh ^2(a+b x) \sinh (a+b x)}{9 b^3}+\frac {(c+d x)^3 \cosh ^2(a+b x) \sinh (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.64, size = 122, normalized size = 0.70 \begin {gather*} \frac {-486 d \left (2 d^2+b^2 (c+d x)^2\right ) \cosh (a+b x)-2 d \left (2 d^2+9 b^2 (c+d x)^2\right ) \cosh (3 (a+b x))+12 b (c+d x) \left (82 d^2+15 b^2 (c+d x)^2+\left (2 d^2+3 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))\right ) \sinh (a+b x)}{216 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(708\) vs.
\(2(161)=322\).
time = 1.28, size = 709, normalized size = 4.05
method | result | size |
risch | \(\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x -9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}-18 b^{2} c \,d^{2} x -9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-2 d^{3}\right ) {\mathrm e}^{3 b x +3 a}}{216 b^{4}}+\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}+b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}-6 d^{3}\right ) {\mathrm e}^{b x +a}}{8 b^{4}}-\frac {3 \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+6 d^{3}\right ) {\mathrm e}^{-b x -a}}{8 b^{4}}-\frac {\left (9 d^{3} x^{3} b^{3}+27 b^{3} c \,d^{2} x^{2}+27 b^{3} c^{2} d x +9 b^{2} d^{3} x^{2}+9 b^{3} c^{3}+18 b^{2} c \,d^{2} x +9 b^{2} c^{2} d +6 b \,d^{3} x +6 b c \,d^{2}+2 d^{3}\right ) {\mathrm e}^{-3 b x -3 a}}{216 b^{4}}\) | \(415\) |
default | \(\frac {\frac {d^{3} \left (\left (3 b x +3 a \right )^{3} \sinh \left (3 b x +3 a \right )-3 \left (3 b x +3 a \right )^{2} \cosh \left (3 b x +3 a \right )+6 \left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-6 \cosh \left (3 b x +3 a \right )\right )}{b^{3}}-\frac {9 d^{3} a \left (\left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )+2 \sinh \left (3 b x +3 a \right )\right )}{b^{3}}+\frac {27 d^{3} a^{2} \left (\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )\right )}{b^{3}}-\frac {27 d^{3} a^{3} \sinh \left (3 b x +3 a \right )}{b^{3}}+\frac {9 c \,d^{2} \left (\left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )+2 \sinh \left (3 b x +3 a \right )\right )}{b^{2}}-\frac {54 c \,d^{2} a \left (\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )\right )}{b^{2}}+\frac {81 c \,d^{2} a^{2} \sinh \left (3 b x +3 a \right )}{b^{2}}+\frac {27 c^{2} d \left (\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )\right )}{b}-\frac {81 c^{2} d a \sinh \left (3 b x +3 a \right )}{b}+27 c^{3} \sinh \left (3 b x +3 a \right )}{324 b}+\frac {\frac {3 d^{3} \left (\left (b x +a \right )^{3} \sinh \left (b x +a \right )-3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )+6 \left (b x +a \right ) \sinh \left (b x +a \right )-6 \cosh \left (b x +a \right )\right )}{4 b^{3}}-\frac {9 d^{3} a \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{4 b^{3}}+\frac {9 d^{3} a^{2} \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{4 b^{3}}-\frac {3 d^{3} a^{3} \sinh \left (b x +a \right )}{4 b^{3}}+\frac {9 d^{2} c \left (\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )}{4 b^{2}}-\frac {9 d^{2} a c \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{2 b^{2}}+\frac {9 c \,d^{2} a^{2} \sinh \left (b x +a \right )}{4 b^{2}}+\frac {9 c^{2} d \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )}{4 b}-\frac {9 c^{2} d a \sinh \left (b x +a \right )}{4 b}+\frac {3 c^{3} \sinh \left (b x +a \right )}{4}}{b}\) | \(709\) |
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. 439 vs.
\(2 (161) = 322\).
time = 0.29, size = 439, normalized size = 2.51 \begin {gather*} \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 )} \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. 343 vs.
\(2 (161) = 322\).
time = 0.39, size = 343, normalized size = 1.96 \begin {gather*} -\frac {{\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 )^{3} + 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 ) \sinh \left (b x + a\right )^{2} - 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 )} \sinh \left (b x + a\right )^{3} + 243 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d + 2 \, d^{3}\right )} \cosh \left (b x + a\right ) - 9 \, {\left (9 \, b^{3} d^{3} x^{3} + 27 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{3} + 54 \, b c d^{2} + {\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 )^{2} + 27 \, {\left (b^{3} c^{2} d + 2 \, b d^{3}\right )} x\right )} \sinh \left (b x + a\right )}{108 \, b^{4}} \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. 495 vs.
\(2 (173) = 346\).
time = 0.54, size = 495, normalized size = 2.83 \begin {gather*} \begin {cases} - \frac {2 c^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {c^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c^{2} d x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \sinh ^{3}{\left (a + b x \right )}}{b} + \frac {3 c d^{2} x^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} - \frac {2 d^{3} x^{3} \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {d^{3} x^{3} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} + \frac {2 c^{2} d \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {7 c^{2} d \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {4 c d^{2} x \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {14 c d^{2} x \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 d^{3} x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b^{2}} - \frac {7 d^{3} x^{2} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} - \frac {40 c d^{2} \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 c d^{2} \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} - \frac {40 d^{3} x \sinh ^{3}{\left (a + b x \right )}}{9 b^{3}} + \frac {14 d^{3} x \sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {40 d^{3} \sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{9 b^{4}} - \frac {122 d^{3} \cosh ^{3}{\left (a + b x \right )}}{27 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 ) \cosh ^{3}{\left (a \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. 414 vs.
\(2 (161) = 322\).
time = 0.42, size = 414, normalized size = 2.37 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 364, normalized size = 2.08 \begin {gather*} \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (3\,b^2\,c^3+14\,c\,d^2\right )}{3\,b^3}-\frac {2\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (3\,b^2\,c^3+20\,c\,d^2\right )}{9\,b^3}-\frac {{\mathrm {cosh}\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 )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^4}-\frac {2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\left (9\,b^2\,c^2\,d+20\,d^3\right )}{9\,b^3}-\frac {7\,d^3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}-\frac {2\,d^3\,x^3\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3\,b}-\frac {14\,c\,d^2\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b^2}+\frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\left (9\,b^2\,c^2\,d+14\,d^3\right )}{3\,b^3}+\frac {d^3\,x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {2\,d^3\,x^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2}-\frac {2\,c\,d^2\,x^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{b}+\frac {3\,c\,d^2\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{b}+\frac {4\,c\,d^2\,x\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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