Optimal. Leaf size=250 \[ -\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac {12 a b d^3 \sinh (e+f x)}{f^4}-\frac {6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3398, 3377,
2717, 3392, 32, 3391} \begin {gather*} \frac {a^2 (c+d x)^4}{4 d}+\frac {12 a b d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac {6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac {12 a b d^3 \sinh (e+f x)}{f^4}+\frac {3 b^2 d^2 (c+d x) \sinh (e+f x) \cosh (e+f x)}{4 f^3}-\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}+\frac {b^2 (c+d x)^3 \sinh (e+f x) \cosh (e+f x)}{2 f}-\frac {b^2 (c+d x)^4}{8 d}-\frac {3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d^3 x^2}{8 f^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2717
Rule 3377
Rule 3391
Rule 3392
Rule 3398
Rubi steps
\begin {align*} \int (c+d x)^3 (a+b \sinh (e+f x))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \sinh (e+f x)+b^2 (c+d x)^3 \sinh ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \sinh (e+f x) \, dx+b^2 \int (c+d x)^3 \sinh ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {2 a b (c+d x)^3 \cosh (e+f x)}{f}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}-\frac {1}{2} b^2 \int (c+d x)^3 \, dx+\frac {\left (3 b^2 d^2\right ) \int (c+d x) \sinh ^2(e+f x) \, dx}{2 f^2}-\frac {(6 a b d) \int (c+d x)^2 \cosh (e+f x) \, dx}{f}\\ &=\frac {a^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^4}{8 d}+\frac {2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}+\frac {\left (12 a b d^2\right ) \int (c+d x) \sinh (e+f x) \, dx}{f^2}-\frac {\left (3 b^2 d^2\right ) \int (c+d x) \, dx}{4 f^2}\\ &=-\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac {6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}-\frac {\left (12 a b d^3\right ) \int \cosh (e+f x) \, dx}{f^3}\\ &=-\frac {3 b^2 c d^2 x}{4 f^2}-\frac {3 b^2 d^3 x^2}{8 f^2}+\frac {a^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^4}{8 d}+\frac {12 a b d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac {2 a b (c+d x)^3 \cosh (e+f x)}{f}-\frac {12 a b d^3 \sinh (e+f x)}{f^4}-\frac {6 a b d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac {3 b^2 d^2 (c+d x) \cosh (e+f x) \sinh (e+f x)}{4 f^3}+\frac {b^2 (c+d x)^3 \cosh (e+f x) \sinh (e+f x)}{2 f}-\frac {3 b^2 d^3 \sinh ^2(e+f x)}{8 f^4}-\frac {3 b^2 d (c+d x)^2 \sinh ^2(e+f x)}{4 f^2}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 235, normalized size = 0.94 \begin {gather*} \frac {32 a b f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (6+f^2 x^2\right )\right ) \cosh (e+f x)-3 b^2 d \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (1+2 f^2 x^2\right )\right ) \cosh (2 (e+f x))+2 \left (\left (2 a^2-b^2\right ) f^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-48 a b d \left (c^2 f^2+2 c d f^2 x+d^2 \left (2+f^2 x^2\right )\right ) \sinh (e+f x)+b^2 f (c+d x) \left (2 c^2 f^2+4 c d f^2 x+d^2 \left (3+2 f^2 x^2\right )\right ) \sinh (2 (e+f x))\right )}{16 f^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. \(1060\) vs.
\(2(234)=468\).
time = 0.61, size = 1061, normalized size = 4.24
method | result | size |
risch | \(\frac {a^{2} d^{3} x^{4}}{4}-\frac {d^{3} b^{2} x^{4}}{8}+a^{2} c \,d^{2} x^{3}-\frac {d^{2} b^{2} c \,x^{3}}{2}+\frac {3 a^{2} c^{2} d \,x^{2}}{2}-\frac {3 d \,b^{2} c^{2} x^{2}}{4}+c^{3} a^{2} x -\frac {b^{2} c^{3} x}{2}+\frac {a^{2} c^{4}}{4 d}-\frac {b^{2} c^{4}}{8 d}+\frac {b^{2} \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x -6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}-12 c \,d^{2} f^{2} x -6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -3 d^{3}\right ) {\mathrm e}^{2 f x +2 e}}{32 f^{4}}+\frac {a b \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x -3 d^{3} f^{2} x^{2}+c^{3} f^{3}-6 c \,d^{2} f^{2} x -3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f -6 d^{3}\right ) {\mathrm e}^{f x +e}}{f^{4}}+\frac {a b \left (d^{3} x^{3} f^{3}+3 c \,d^{2} f^{3} x^{2}+3 c^{2} d \,f^{3} x +3 d^{3} f^{2} x^{2}+c^{3} f^{3}+6 c \,d^{2} f^{2} x +3 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +6 d^{3}\right ) {\mathrm e}^{-f x -e}}{f^{4}}-\frac {b^{2} \left (4 d^{3} x^{3} f^{3}+12 c \,d^{2} f^{3} x^{2}+12 c^{2} d \,f^{3} x +6 d^{3} f^{2} x^{2}+4 c^{3} f^{3}+12 c \,d^{2} f^{2} x +6 c^{2} d \,f^{2}+6 d^{3} f x +6 c \,d^{2} f +3 d^{3}\right ) {\mathrm e}^{-2 f x -2 e}}{32 f^{4}}\) | \(531\) |
derivativedivides | \(\text {Expression too large to display}\) | \(1061\) |
default | \(\text {Expression too large to display}\) | \(1061\) |
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. 547 vs.
\(2 (244) = 488\).
time = 0.29, size = 547, normalized size = 2.19 \begin {gather*} \frac {1}{4} \, a^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} - \frac {3}{16} \, {\left (4 \, x^{2} - \frac {{\left (2 \, f x e^{\left (2 \, e\right )} - e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{2}} + \frac {{\left (2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{2}}\right )} b^{2} c^{2} d - \frac {1}{16} \, {\left (8 \, x^{3} - \frac {3 \, {\left (2 \, f^{2} x^{2} e^{\left (2 \, e\right )} - 2 \, f x e^{\left (2 \, e\right )} + e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{3}} + \frac {3 \, {\left (2 \, f^{2} x^{2} + 2 \, f x + 1\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{3}}\right )} b^{2} c d^{2} - \frac {1}{32} \, {\left (4 \, x^{4} - \frac {{\left (4 \, f^{3} x^{3} e^{\left (2 \, e\right )} - 6 \, f^{2} x^{2} e^{\left (2 \, e\right )} + 6 \, f x e^{\left (2 \, e\right )} - 3 \, e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )}}{f^{4}} + \frac {{\left (4 \, f^{3} x^{3} + 6 \, f^{2} x^{2} + 6 \, f x + 3\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{f^{4}}\right )} b^{2} d^{3} - \frac {1}{8} \, b^{2} c^{3} {\left (4 \, x - \frac {e^{\left (2 \, f x + 2 \, e\right )}}{f} + \frac {e^{\left (-2 \, f x - 2 \, e\right )}}{f}\right )} + a^{2} c^{3} x + 3 \, a b c^{2} d {\left (\frac {{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac {{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + 3 \, a b c d^{2} {\left (\frac {{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac {{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + a b d^{3} {\left (\frac {{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} + \frac {{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac {2 \, a b c^{3} \cosh \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 433, normalized size = 1.73 \begin {gather*} \frac {2 \, {\left (2 \, a^{2} - b^{2}\right )} d^{3} f^{4} x^{4} + 8 \, {\left (2 \, a^{2} - b^{2}\right )} c d^{2} f^{4} x^{3} + 12 \, {\left (2 \, a^{2} - b^{2}\right )} c^{2} d f^{4} x^{2} + 8 \, {\left (2 \, a^{2} - b^{2}\right )} c^{3} f^{4} x - 3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} + b^{2} d^{3}\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} - 3 \, {\left (2 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c d^{2} f^{2} x + 2 \, b^{2} c^{2} d f^{2} + b^{2} d^{3}\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )^{2} + 32 \, {\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + a b c^{3} f^{3} + 6 \, a b c d^{2} f + 3 \, {\left (a b c^{2} d f^{3} + 2 \, a b d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right ) - 4 \, {\left (24 \, a b d^{3} f^{2} x^{2} + 48 \, a b c d^{2} f^{2} x + 24 \, a b c^{2} d f^{2} + 48 \, a b d^{3} - {\left (2 \, b^{2} d^{3} f^{3} x^{3} + 6 \, b^{2} c d^{2} f^{3} x^{2} + 2 \, b^{2} c^{3} f^{3} + 3 \, b^{2} c d^{2} f + 3 \, {\left (2 \, b^{2} c^{2} d f^{3} + b^{2} d^{3} f\right )} x\right )} \cosh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )\right )} \sinh \left (f x + \cosh \left (1\right ) + \sinh \left (1\right )\right )}{16 \, f^{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. 779 vs.
\(2 (255) = 510\).
time = 0.46, size = 779, normalized size = 3.12 \begin {gather*} \begin {cases} a^{2} c^{3} x + \frac {3 a^{2} c^{2} d x^{2}}{2} + a^{2} c d^{2} x^{3} + \frac {a^{2} d^{3} x^{4}}{4} + \frac {2 a b c^{3} \cosh {\left (e + f x \right )}}{f} + \frac {6 a b c^{2} d x \cosh {\left (e + f x \right )}}{f} - \frac {6 a b c^{2} d \sinh {\left (e + f x \right )}}{f^{2}} + \frac {6 a b c d^{2} x^{2} \cosh {\left (e + f x \right )}}{f} - \frac {12 a b c d^{2} x \sinh {\left (e + f x \right )}}{f^{2}} + \frac {12 a b c d^{2} \cosh {\left (e + f x \right )}}{f^{3}} + \frac {2 a b d^{3} x^{3} \cosh {\left (e + f x \right )}}{f} - \frac {6 a b d^{3} x^{2} \sinh {\left (e + f x \right )}}{f^{2}} + \frac {12 a b d^{3} x \cosh {\left (e + f x \right )}}{f^{3}} - \frac {12 a b d^{3} \sinh {\left (e + f x \right )}}{f^{4}} + \frac {b^{2} c^{3} x \sinh ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c^{3} x \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {b^{2} c^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} + \frac {3 b^{2} c^{2} d x^{2} \sinh ^{2}{\left (e + f x \right )}}{4} - \frac {3 b^{2} c^{2} d x^{2} \cosh ^{2}{\left (e + f x \right )}}{4} + \frac {3 b^{2} c^{2} d x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} c^{2} d \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {b^{2} c d^{2} x^{3} \sinh ^{2}{\left (e + f x \right )}}{2} - \frac {b^{2} c d^{2} x^{3} \cosh ^{2}{\left (e + f x \right )}}{2} + \frac {3 b^{2} c d^{2} x^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} c d^{2} x \sinh ^{2}{\left (e + f x \right )}}{4 f^{2}} - \frac {3 b^{2} c d^{2} x \cosh ^{2}{\left (e + f x \right )}}{4 f^{2}} + \frac {3 b^{2} c d^{2} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} + \frac {b^{2} d^{3} x^{4} \sinh ^{2}{\left (e + f x \right )}}{8} - \frac {b^{2} d^{3} x^{4} \cosh ^{2}{\left (e + f x \right )}}{8} + \frac {b^{2} d^{3} x^{3} \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{2 f} - \frac {3 b^{2} d^{3} x^{2} \sinh ^{2}{\left (e + f x \right )}}{8 f^{2}} - \frac {3 b^{2} d^{3} x^{2} \cosh ^{2}{\left (e + f x \right )}}{8 f^{2}} + \frac {3 b^{2} d^{3} x \sinh {\left (e + f x \right )} \cosh {\left (e + f x \right )}}{4 f^{3}} - \frac {3 b^{2} d^{3} \cosh ^{2}{\left (e + f x \right )}}{8 f^{4}} & \text {for}\: f \neq 0 \\\left (a + b \sinh {\left (e \right )}\right )^{2} \left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\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. 598 vs.
\(2 (234) = 468\).
time = 0.45, size = 598, normalized size = 2.39 \begin {gather*} \frac {1}{4} \, a^{2} d^{3} x^{4} - \frac {1}{8} \, b^{2} d^{3} x^{4} + a^{2} c d^{2} x^{3} - \frac {1}{2} \, b^{2} c d^{2} x^{3} + \frac {3}{2} \, a^{2} c^{2} d x^{2} - \frac {3}{4} \, b^{2} c^{2} d x^{2} + a^{2} c^{3} x - \frac {1}{2} \, b^{2} c^{3} x + \frac {{\left (4 \, b^{2} d^{3} f^{3} x^{3} + 12 \, b^{2} c d^{2} f^{3} x^{2} + 12 \, b^{2} c^{2} d f^{3} x - 6 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c^{3} f^{3} - 12 \, b^{2} c d^{2} f^{2} x - 6 \, b^{2} c^{2} d f^{2} + 6 \, b^{2} d^{3} f x + 6 \, b^{2} c d^{2} f - 3 \, b^{2} d^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )}}{32 \, f^{4}} + \frac {{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x - 3 \, a b d^{3} f^{2} x^{2} + a b c^{3} f^{3} - 6 \, a b c d^{2} f^{2} x - 3 \, a b c^{2} d f^{2} + 6 \, a b d^{3} f x + 6 \, a b c d^{2} f - 6 \, a b d^{3}\right )} e^{\left (f x + e\right )}}{f^{4}} + \frac {{\left (a b d^{3} f^{3} x^{3} + 3 \, a b c d^{2} f^{3} x^{2} + 3 \, a b c^{2} d f^{3} x + 3 \, a b d^{3} f^{2} x^{2} + a b c^{3} f^{3} + 6 \, a b c d^{2} f^{2} x + 3 \, a b c^{2} d f^{2} + 6 \, a b d^{3} f x + 6 \, a b c d^{2} f + 6 \, a b d^{3}\right )} e^{\left (-f x - e\right )}}{f^{4}} - \frac {{\left (4 \, b^{2} d^{3} f^{3} x^{3} + 12 \, b^{2} c d^{2} f^{3} x^{2} + 12 \, b^{2} c^{2} d f^{3} x + 6 \, b^{2} d^{3} f^{2} x^{2} + 4 \, b^{2} c^{3} f^{3} + 12 \, b^{2} c d^{2} f^{2} x + 6 \, b^{2} c^{2} d f^{2} + 6 \, b^{2} d^{3} f x + 6 \, b^{2} c d^{2} f + 3 \, b^{2} d^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{32 \, f^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.86, size = 481, normalized size = 1.92 \begin {gather*} a^2\,c^3\,x-\frac {b^2\,c^3\,x}{2}+\frac {a^2\,d^3\,x^4}{4}-\frac {b^2\,d^3\,x^4}{8}+\frac {3\,a^2\,c^2\,d\,x^2}{2}+a^2\,c\,d^2\,x^3-\frac {3\,b^2\,c^2\,d\,x^2}{4}-\frac {b^2\,c\,d^2\,x^3}{2}-\frac {3\,b^2\,d^3\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{16\,f^4}+\frac {b^2\,c^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}+\frac {2\,a\,b\,c^3\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {12\,a\,b\,d^3\,\mathrm {sinh}\left (e+f\,x\right )}{f^4}-\frac {3\,b^2\,d^3\,x^2\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{8\,f^2}+\frac {b^2\,d^3\,x^3\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}-\frac {3\,b^2\,c^2\,d\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{8\,f^2}+\frac {3\,b^2\,c\,d^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{8\,f^3}+\frac {3\,b^2\,d^3\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{8\,f^3}-\frac {3\,b^2\,c\,d^2\,x\,\mathrm {cosh}\left (2\,e+2\,f\,x\right )}{4\,f^2}+\frac {3\,b^2\,c^2\,d\,x\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}+\frac {12\,a\,b\,c\,d^2\,\mathrm {cosh}\left (e+f\,x\right )}{f^3}-\frac {6\,a\,b\,c^2\,d\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {12\,a\,b\,d^3\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f^3}+\frac {3\,b^2\,c\,d^2\,x^2\,\mathrm {sinh}\left (2\,e+2\,f\,x\right )}{4\,f}+\frac {2\,a\,b\,d^3\,x^3\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {6\,a\,b\,d^3\,x^2\,\mathrm {sinh}\left (e+f\,x\right )}{f^2}+\frac {6\,a\,b\,c\,d^2\,x^2\,\mathrm {cosh}\left (e+f\,x\right )}{f}+\frac {6\,a\,b\,c^2\,d\,x\,\mathrm {cosh}\left (e+f\,x\right )}{f}-\frac {12\,a\,b\,c\,d^2\,x\,\mathrm {sinh}\left (e+f\,x\right )}{f^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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