Optimal. Leaf size=254 \[ \frac {a (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {a (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {b (c+d x) \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}+\frac {a d \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac {a d \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac {d \log (a+b \sinh (e+f x))}{f^2 \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.44, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3324, 3322, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac {a d \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac {a d \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac {a (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {a (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {b (c+d x) \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}+\frac {d \log (a+b \sinh (e+f x))}{f^2 \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2668
Rule 3322
Rule 3324
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b \sinh (e+f x))^2} \, dx &=-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}+\frac {a \int \frac {c+d x}{a+b \sinh (e+f x)} \, dx}{a^2+b^2}+\frac {(b d) \int \frac {\cosh (e+f x)}{a+b \sinh (e+f x)} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}+\frac {(2 a) \int \frac {e^{e+f x} (c+d x)}{-b+2 a e^{e+f x}+b e^{2 (e+f x)}} \, dx}{a^2+b^2}+\frac {d \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (e+f x)\right )}{\left (a^2+b^2\right ) f^2}\\ &=\frac {d \log (a+b \sinh (e+f x))}{\left (a^2+b^2\right ) f^2}-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}+\frac {(2 a b) \int \frac {e^{e+f x} (c+d x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {(2 a b) \int \frac {e^{e+f x} (c+d x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\left (a^2+b^2\right )^{3/2}}\\ &=\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}-\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {d \log (a+b \sinh (e+f x))}{\left (a^2+b^2\right ) f^2}-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}-\frac {(a d) \int \log \left (1+\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} f}+\frac {(a d) \int \log \left (1+\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} f}\\ &=\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}-\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {d \log (a+b \sinh (e+f x))}{\left (a^2+b^2\right ) f^2}-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}-\frac {(a d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2+b^2\right )^{3/2} f^2}+\frac {(a d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2+b^2\right )^{3/2} f^2}\\ &=\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}-\frac {a (c+d x) \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f}+\frac {d \log (a+b \sinh (e+f x))}{\left (a^2+b^2\right ) f^2}+\frac {a d \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {a d \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^2}-\frac {b (c+d x) \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}\\ \end {align*}
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Mathematica [A] time = 1.03, size = 194, normalized size = 0.76 \[ \frac {\frac {a \left (f (c+d x) \left (\log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )-\log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )\right )+d \text {Li}_2\left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}-a}\right )-d \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}-\frac {b f (c+d x) \cosh (e+f x)}{a+b \sinh (e+f x)}+d \log (a+b \sinh (e+f x))}{f^2 \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 1717, normalized size = 6.76 \[ \text {result too large to display} \]
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 \[ \int \frac {d x + c}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 519, normalized size = 2.04 \[ \frac {2 \left (d x +c \right ) \left (a \,{\mathrm e}^{f x +e}-b \right )}{f \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 f x +2 e}+2 a \,{\mathrm e}^{f x +e}-b \right )}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right )}{f^{2} \left (a^{2}+b^{2}\right )}+\frac {d \ln \left (b \,{\mathrm e}^{2 f x +2 e}+2 a \,{\mathrm e}^{f x +e}-b \right )}{f^{2} \left (a^{2}+b^{2}\right )}-\frac {2 a c \arctanh \left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {d a \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {d a \ln \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {d a \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{f \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {d a \ln \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) e}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {d a \dilog \left (\frac {-b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {d a \dilog \left (\frac {b \,{\mathrm e}^{f x +e}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {2 a d e \arctanh \left (\frac {2 b \,{\mathrm e}^{f x +e}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{f^{2} \left (a^{2}+b^{2}\right )^{\frac {3}{2}}} \]
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 \[ {\left (2 \, a f \int \frac {x e^{\left (f x + e\right )}}{a^{2} b f e^{\left (2 \, f x + 2 \, e\right )} + b^{3} f e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a^{3} f e^{\left (f x + e\right )} + 2 \, a b^{2} f e^{\left (f x + e\right )} - a^{2} b f - b^{3} f}\,{d x} + b {\left (\frac {a \log \left (\frac {b e^{\left (f x + e\right )} + a - \sqrt {a^{2} + b^{2}}}{b e^{\left (f x + e\right )} + a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a^{2} + b^{2}} f^{2}} - \frac {2 \, {\left (f x + e\right )}}{{\left (a^{2} b + b^{3}\right )} f^{2}} + \frac {\log \left (b e^{\left (2 \, f x + 2 \, e\right )} + 2 \, a e^{\left (f x + e\right )} - b\right )}{{\left (a^{2} b + b^{3}\right )} f^{2}}\right )} - \frac {2 \, {\left (a x e^{\left (f x + e\right )} - b x\right )}}{a^{2} b f + b^{3} f - {\left (a^{2} b f e^{\left (2 \, e\right )} + b^{3} f e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 2 \, {\left (a^{3} f e^{e} + a b^{2} f e^{e}\right )} e^{\left (f x\right )}} - \frac {a \log \left (\frac {b e^{\left (f x + e\right )} + a - \sqrt {a^{2} + b^{2}}}{b e^{\left (f x + e\right )} + a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} f^{2}}\right )} d + c {\left (\frac {a \log \left (\frac {b e^{\left (-f x - e\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-f x - e\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} + b^{2}\right )}^{\frac {3}{2}} f} - \frac {2 \, {\left (a e^{\left (-f x - e\right )} + b\right )}}{{\left (a^{2} b + b^{3} + 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-f x - e\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, f x - 2 \, e\right )}\right )} f}\right )} \]
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 \[ \int \frac {c+d\,x}{{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2} \,d x \]
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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