Optimal. Leaf size=400 \[ -\frac {a^2 f \cosh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}-\frac {a f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}+\frac {a f \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^2}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {a f x}{4 b^2 d}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \]
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Rubi [A] time = 0.48, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5579, 3310, 3296, 2638, 5565, 5446, 2635, 8, 5561, 2190, 2279, 2391} \[ -\frac {a f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^4 d^2}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}+\frac {a f \sinh (c+d x) \cosh (c+d x)}{4 b^2 d^2}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {a f x}{4 b^2 d}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 2638
Rule 3296
Rule 3310
Rule 5446
Rule 5561
Rule 5565
Rule 5579
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \cosh ^3(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {f \cosh ^3(c+d x)}{9 b d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}+\frac {a^2 \int (e+f x) \cosh (c+d x) \, dx}{b^3}-\frac {a \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {2 \int (e+f x) \cosh (c+d x) \, dx}{3 b}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}\\ &=\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {f \cosh ^3(c+d x)}{9 b d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {\left (a^2 f\right ) \int \sinh (c+d x) \, dx}{b^3 d}+\frac {(a f) \int \sinh ^2(c+d x) \, dx}{2 b^2 d}-\frac {(2 f) \int \sinh (c+d x) \, dx}{3 b d}\\ &=\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}-\frac {(a f) \int 1 \, dx}{4 b^2 d}+\frac {\left (a \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (a \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}\\ &=-\frac {a f x}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}+\frac {\left (a \left (a^2+b^2\right ) f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}+\frac {\left (a \left (a^2+b^2\right ) f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^2}\\ &=-\frac {a f x}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^2}{2 b^4 f}-\frac {a^2 f \cosh (c+d x)}{b^3 d^2}-\frac {2 f \cosh (c+d x)}{3 b d^2}-\frac {f \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {a^2 (e+f x) \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 b d}+\frac {a f \cosh (c+d x) \sinh (c+d x)}{4 b^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x) \sinh ^2(c+d x)}{2 b^2 d}\\ \end {align*}
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Mathematica [A] time = 2.76, size = 551, normalized size = 1.38 \[ -\frac {36 b^2 f \left (a \left (\text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+\text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+(c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )-c \log (a+b \sinh (c+d x))-\frac {1}{2} (c+d x)^2\right )-b d x \sinh (c+d x)+b \cosh (c+d x)\right )+12 d e \left (-3 b \left (2 a^2+b^2\right ) \sinh (c+d x)+3 a \left (2 a^2+b^2\right ) \log (a+b \sinh (c+d x))+3 a b^2 \sinh ^2(c+d x)-2 b^3 \sinh ^3(c+d x)\right )+f \left (36 a \left (2 a^2+b^2\right ) \left (\text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )+\text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )+(c+d x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )-c \log (a+b \sinh (c+d x))-\frac {1}{2} (c+d x)^2\right )-18 b d x \left (4 a^2+b^2\right ) \sinh (c+d x)+18 b \left (4 a^2+b^2\right ) \cosh (c+d x)-9 a b^2 \sinh (2 (c+d x))+18 a b^2 d x \cosh (2 (c+d x))-6 b^3 d x \sinh (3 (c+d x))+2 b^3 \cosh (3 (c+d x))\right )-36 b^2 d e (b \sinh (c+d x)-a \log (a+b \sinh (c+d x)))}{72 b^4 d^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 2465, normalized size = 6.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 1102, normalized size = 2.76 \[ -\frac {a^{3} e x}{b^{4}}-\frac {a e x}{b^{2}}+\frac {a^{3} f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{4}}-\frac {2 a^{3} f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{4}}+\frac {2 a^{3} f c x}{d \,b^{4}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{4}}-\frac {a^{3} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{4}}-\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{4}}-\frac {a^{3} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{4}}+\frac {a f \,x^{2}}{2 b^{2}}+\frac {\left (3 d f x +3 d e -f \right ) {\mathrm e}^{3 d x +3 c}}{72 b \,d^{2}}-\frac {\left (3 d f x +3 d e +f \right ) {\mathrm e}^{-3 d x -3 c}}{72 b \,d^{2}}+\frac {\left (4 a^{2} d f x +3 b^{2} d f x +4 a^{2} d e +3 b^{2} d e -4 a^{2} f -3 f \,b^{2}\right ) {\mathrm e}^{d x +c}}{8 b^{3} d^{2}}+\frac {2 a f c x}{d \,b^{2}}+\frac {a f c \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b^{2}}-\frac {2 a f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b^{2}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2}}-\frac {a f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2}}-\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,b^{2}}-\frac {a f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} b^{2}}+\frac {a^{3} f \,x^{2}}{2 b^{4}}+\frac {a f \,c^{2}}{d^{2} b^{2}}-\frac {a e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{2}}+\frac {2 a e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{2}}-\frac {a f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2}}-\frac {a f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{2}}-\frac {a \left (2 d f x +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 b^{2} d^{2}}-\frac {\left (4 a^{2}+3 b^{2}\right ) \left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{8 b^{3} d^{2}}-\frac {a \left (2 d f x +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 b^{2} d^{2}}-\frac {a^{3} f \dilog \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{4}}-\frac {a^{3} f \dilog \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} b^{4}}+\frac {a^{3} f \,c^{2}}{d^{2} b^{4}}-\frac {a^{3} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,b^{4}}+\frac {2 a^{3} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{24} \, e {\left (\frac {{\left (3 \, a b e^{\left (-d x - c\right )} - b^{2} - 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{b^{3} d} + \frac {24 \, {\left (a^{3} + a b^{2}\right )} {\left (d x + c\right )}}{b^{4} d} + \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + b^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{b^{3} d} + \frac {24 \, {\left (a^{3} + a b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{4} d}\right )} - \frac {1}{144} \, f {\left (\frac {{\left (72 \, {\left (a^{3} d^{2} e^{\left (3 \, c\right )} + a b^{2} d^{2} e^{\left (3 \, c\right )}\right )} x^{2} - 2 \, {\left (3 \, b^{3} d x e^{\left (6 \, c\right )} - b^{3} e^{\left (6 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 9 \, {\left (2 \, a b^{2} d x e^{\left (5 \, c\right )} - a b^{2} e^{\left (5 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 18 \, {\left (4 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, c\right )} - {\left (4 \, a^{2} b d e^{\left (4 \, c\right )} + 3 \, b^{3} d e^{\left (4 \, c\right )}\right )} x\right )} e^{\left (d x\right )} + 18 \, {\left (4 \, a^{2} b e^{\left (2 \, c\right )} + 3 \, b^{3} e^{\left (2 \, c\right )} + {\left (4 \, a^{2} b d e^{\left (2 \, c\right )} + 3 \, b^{3} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (-d x\right )} + 9 \, {\left (2 \, a b^{2} d x e^{c} + a b^{2} e^{c}\right )} e^{\left (-2 \, d x\right )} + 2 \, {\left (3 \, b^{3} d x + b^{3}\right )} e^{\left (-3 \, d x\right )}\right )} e^{\left (-3 \, c\right )}}{b^{4} d^{2}} - 9 \, \int \frac {32 \, {\left ({\left (a^{4} e^{c} + a^{2} b^{2} e^{c}\right )} x e^{\left (d x\right )} - {\left (a^{3} b + a b^{3}\right )} x\right )}}{b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{4} e^{\left (d x + c\right )} - b^{5}}\,{d x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (e+f\,x\right )}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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