Optimal. Leaf size=400 \[ -\frac {b^2 f \cosh (c+d x)}{a^3 d^2}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \sinh (c+d x) \cosh (c+d x)}{4 a^2 d^2}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac {b f x}{4 a^2 d}-\frac {b f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b f \left (a^2+b^2\right ) \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^4 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {2 f \cosh (c+d x)}{3 a d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 a d} \]
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Rubi [A] time = 0.55, antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5594, 5579, 3310, 3296, 2638, 5565, 5446, 2635, 8, 5561, 2190, 2279, 2391} \[ -\frac {b f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b f \left (a^2+b^2\right ) \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}\right )}{a^4 d^2}-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^4 d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}+\frac {b f \sinh (c+d x) \cosh (c+d x)}{4 a^2 d^2}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac {b f x}{4 a^2 d}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {2 f \cosh (c+d x)}{3 a d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {(e+f x) \sinh (c+d x) \cosh ^2(c+d x)}{3 a d} \]
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
Rule 5594
Rubi steps
\begin {align*} \int \frac {(e+f x) \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx &=\int \frac {(e+f x) \cosh ^3(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac {\int (e+f x) \cosh ^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cosh ^3(c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=-\frac {f \cosh ^3(c+d x)}{9 a d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}+\frac {2 \int (e+f x) \cosh (c+d x) \, dx}{3 a}-\frac {b \int (e+f x) \cosh (c+d x) \sinh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x) \cosh (c+d x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx}{a^3}\\ &=\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {f \cosh ^3(c+d x)}{9 a d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {(2 f) \int \sinh (c+d x) \, dx}{3 a d}+\frac {(b f) \int \sinh ^2(c+d x) \, dx}{2 a^2 d}-\frac {\left (b^2 f\right ) \int \sinh (c+d x) \, dx}{a^3 d}\\ &=\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {2 f \cosh (c+d x)}{3 a d^2}-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}-\frac {(b f) \int 1 \, dx}{4 a^2 d}+\frac {\left (b \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (b \left (a^2+b^2\right ) f\right ) \int \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}\\ &=-\frac {b f x}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {2 f \cosh (c+d x)}{3 a d^2}-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}+\frac {\left (b \left (a^2+b^2\right ) f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^2}+\frac {\left (b \left (a^2+b^2\right ) f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^2}\\ &=-\frac {b f x}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^2}{2 a^4 f}-\frac {2 f \cosh (c+d x)}{3 a d^2}-\frac {b^2 f \cosh (c+d x)}{a^3 d^2}-\frac {f \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 (e+f x) \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x) \sinh (c+d x)}{a^3 d}+\frac {b f \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {(e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x) \sinh ^2(c+d x)}{2 a^2 d}\\ \end {align*}
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Mathematica [C] time = 5.91, size = 743, normalized size = 1.86 \[ -\frac {\text {csch}(c+d x) (a \sinh (c+d x)+b) \left (\frac {9}{2} a^2 f \left (8 b \left (\text {Li}_2\left (\frac {\left (b-\sqrt {a^2+b^2}\right ) e^{c+d x}}{a}\right )+\text {Li}_2\left (\frac {\left (b+\sqrt {a^2+b^2}\right ) e^{c+d x}}{a}\right )\right )+4 b \log \left (\frac {\left (\sqrt {a^2+b^2}-b\right ) e^{c+d x}}{a}+1\right ) \left (4 i \sin ^{-1}\left (\frac {\sqrt {1+\frac {i b}{a}}}{\sqrt {2}}\right )+2 c+2 d x+i \pi \right )+4 b \log \left (1-\frac {\left (\sqrt {a^2+b^2}+b\right ) e^{c+d x}}{a}\right ) \left (-4 i \sin ^{-1}\left (\frac {\sqrt {1+\frac {i b}{a}}}{\sqrt {2}}\right )+2 c+2 d x+i \pi \right )-32 b \sin ^{-1}\left (\frac {\sqrt {1+\frac {i b}{a}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {(b+i a) \cot \left (\frac {1}{4} (2 i c+2 i d x+\pi )\right )}{\sqrt {a^2+b^2}}\right )-4 i \pi b \log (a \sinh (c+d x)+b)-8 b c \log \left (\frac {a \sinh (c+d x)}{b}+1\right )-8 a d x \sinh (c+d x)+8 a \cosh (c+d x)-b (2 c+2 d x+i \pi )^2\right )-36 a^2 d e (a \sinh (c+d x)-b \log (a \sinh (c+d x)+b))+12 d e \left (-2 a^3 \sinh ^3(c+d x)-3 a \left (a^2+2 b^2\right ) \sinh (c+d x)+3 b \left (a^2+2 b^2\right ) \log (a \sinh (c+d x)+b)+3 a^2 b \sinh ^2(c+d x)\right )+f \left (-6 a^3 d x \sinh (3 (c+d x))+2 a^3 \cosh (3 (c+d x))+36 b \left (a^2+2 b^2\right ) \left (\text {Li}_2\left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}-b}\right )+\text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )+(c+d x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )+(c+d x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )-c \log (a \sinh (c+d x)+b)-\frac {1}{2} (c+d x)^2\right )-18 a d x \left (a^2+4 b^2\right ) \sinh (c+d x)+18 a \left (a^2+4 b^2\right ) \cosh (c+d x)-9 a^2 b \sinh (2 (c+d x))+18 a^2 b d x \cosh (2 (c+d x))\right )\right )}{72 a^4 d^2 (a+b \text {csch}(c+d x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, 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 \[ \int \frac {{\left (f x + e\right )} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.14, size = 1102, normalized size = 2.76 \[ \frac {2 b f c x}{d \,a^{2}}+\frac {b^{3} f \,c^{2}}{d^{2} a^{4}}+\frac {2 b^{3} e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,a^{4}}-\frac {b^{3} e \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d \,a^{4}}-\frac {b^{3} f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{4}}-\frac {b^{3} f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{4}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{2}}-\frac {b f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{2}}-\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{2}}-\frac {b f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{2}}-\frac {2 b f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{2}}-\frac {\left (3 a^{2}+4 b^{2}\right ) \left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{8 a^{3} d^{2}}-\frac {b \left (2 d f x +2 d e +f \right ) {\mathrm e}^{-2 d x -2 c}}{16 a^{2} d^{2}}-\frac {b^{3} f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{4}}-\frac {2 b^{3} f c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{4}}+\frac {b^{3} f c \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d^{2} a^{4}}-\frac {b^{3} f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{4}}-\frac {b^{3} f \ln \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{4}}-\frac {b^{3} f \ln \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{4}}+\frac {b f \,x^{2}}{2 a^{2}}-\frac {b e x}{a^{2}}+\frac {2 b^{3} f c x}{d \,a^{4}}-\frac {b \left (2 d f x +2 d e -f \right ) {\mathrm e}^{2 d x +2 c}}{16 a^{2} d^{2}}+\frac {b f \,c^{2}}{d^{2} a^{2}}+\frac {2 b e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,a^{2}}-\frac {b e \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d \,a^{2}}-\frac {b f \dilog \left (\frac {a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{2}}-\frac {b f \dilog \left (\frac {-a \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{2}}+\frac {b f c \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d^{2} a^{2}}+\frac {\left (3 a^{2} d f x +4 b^{2} d f x +3 a^{2} d e +4 b^{2} d e -3 a^{2} f -4 f \,b^{2}\right ) {\mathrm e}^{d x +c}}{8 a^{3} d^{2}}+\frac {\left (3 d f x +3 d e -f \right ) {\mathrm e}^{3 d x +3 c}}{72 a \,d^{2}}-\frac {\left (3 d f x +3 d e +f \right ) {\mathrm e}^{-3 d x -3 c}}{72 a \,d^{2}}+\frac {b^{3} f \,x^{2}}{2 a^{4}}-\frac {b^{3} e x}{a^{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 )} - a^{2} - 3 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{a^{3} d} + \frac {24 \, {\left (a^{2} b + b^{3}\right )} {\left (d x + c\right )}}{a^{4} d} + \frac {3 \, a b e^{\left (-2 \, d x - 2 \, c\right )} + a^{2} e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, {\left (3 \, a^{2} + 4 \, b^{2}\right )} e^{\left (-d x - c\right )}}{a^{3} d} + \frac {24 \, {\left (a^{2} b + b^{3}\right )} \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{4} d}\right )} - \frac {1}{144} \, f {\left (\frac {{\left (72 \, {\left (a^{2} b d^{2} e^{\left (3 \, c\right )} + b^{3} d^{2} e^{\left (3 \, c\right )}\right )} x^{2} - 2 \, {\left (3 \, a^{3} d x e^{\left (6 \, c\right )} - a^{3} e^{\left (6 \, c\right )}\right )} e^{\left (3 \, d x\right )} + 9 \, {\left (2 \, a^{2} b d x e^{\left (5 \, c\right )} - a^{2} b e^{\left (5 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 18 \, {\left (3 \, a^{3} e^{\left (4 \, c\right )} + 4 \, a b^{2} e^{\left (4 \, c\right )} - {\left (3 \, a^{3} d e^{\left (4 \, c\right )} + 4 \, a b^{2} d e^{\left (4 \, c\right )}\right )} x\right )} e^{\left (d x\right )} + 18 \, {\left (3 \, a^{3} e^{\left (2 \, c\right )} + 4 \, a b^{2} e^{\left (2 \, c\right )} + {\left (3 \, a^{3} d e^{\left (2 \, c\right )} + 4 \, a b^{2} d e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (-d x\right )} + 9 \, {\left (2 \, a^{2} b d x e^{c} + a^{2} b e^{c}\right )} e^{\left (-2 \, d x\right )} + 2 \, {\left (3 \, a^{3} d x + a^{3}\right )} e^{\left (-3 \, d x\right )}\right )} e^{\left (-3 \, c\right )}}{a^{4} d^{2}} - 18 \, \int \frac {16 \, {\left ({\left (a^{2} b^{2} e^{c} + b^{4} e^{c}\right )} x e^{\left (d x\right )} - {\left (a^{3} b + a b^{3}\right )} x\right )}}{a^{5} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b e^{\left (d x + c\right )} - a^{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 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right ) \cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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