Optimal. Leaf size=549 \[ \frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 a d (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac {2 d (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac {2 d (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac {a (c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {a (c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {b (c+d x)^2 \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}-\frac {(c+d x)^2}{f \left (a^2+b^2\right )}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )}+\frac {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )}-\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )^{3/2}} \]
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Rubi [A] time = 1.04, antiderivative size = 549, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3324, 3322, 2264, 2190, 2531, 2282, 6589, 5561, 2279, 2391} \[ \frac {2 a d (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 a d (c+d x) \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}\right )}{f^2 \left (a^2+b^2\right )^{3/2}}+\frac {2 d^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )}+\frac {2 d^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}\right )}{f^3 \left (a^2+b^2\right )}-\frac {2 a d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a d^2 \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}\right )}{f^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 d (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac {2 d (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f^2 \left (a^2+b^2\right )}+\frac {a (c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {a (c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \left (a^2+b^2\right )^{3/2}}-\frac {b (c+d x)^2 \cosh (e+f x)}{f \left (a^2+b^2\right ) (a+b \sinh (e+f x))}-\frac {(c+d x)^2}{f \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3322
Rule 3324
Rule 5561
Rule 6589
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+b \sinh (e+f x))^2} \, dx &=-\frac {b (c+d x)^2 \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}+\frac {a \int \frac {(c+d x)^2}{a+b \sinh (e+f x)} \, dx}{a^2+b^2}+\frac {(2 b d) \int \frac {(c+d x) \cosh (e+f x)}{a+b \sinh (e+f x)} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac {(c+d x)^2}{\left (a^2+b^2\right ) f}-\frac {b (c+d x)^2 \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)^2}{-b+2 a e^{e+f x}+b e^{2 (e+f x)}} \, dx}{a^2+b^2}+\frac {(2 b d) \int \frac {e^{e+f x} (c+d x)}{a-\sqrt {a^2+b^2}+b e^{e+f x}} \, dx}{\left (a^2+b^2\right ) f}+\frac {(2 b d) \int \frac {e^{e+f x} (c+d x)}{a+\sqrt {a^2+b^2}+b e^{e+f x}} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac {(c+d x)^2}{\left (a^2+b^2\right ) f}+\frac {2 d (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 ) f^2}+\frac {2 d (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 ) f^2}-\frac {b (c+d x)^2 \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}{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}{2 a+2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {\left (2 d^2\right ) \int \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) f^2}-\frac {\left (2 d^2\right ) \int \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) f^2}\\ &=-\frac {(c+d x)^2}{\left (a^2+b^2\right ) f}+\frac {2 d (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 ) f^2}+\frac {a (c+d x)^2 \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 {2 d (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 ) f^2}-\frac {a (c+d x)^2 \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 {b (c+d x)^2 \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2+b^2\right ) f^3}-\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2+b^2\right ) f^3}-\frac {(2 a d) \int (c+d x) \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 {(2 a d) \int (c+d x) \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 {(c+d x)^2}{\left (a^2+b^2\right ) f}+\frac {2 d (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 ) f^2}+\frac {a (c+d x)^2 \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 {2 d (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 ) f^2}-\frac {a (c+d x)^2 \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 {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}+\frac {2 a d (c+d x) \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 {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}-\frac {2 a d (c+d x) \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)^2 \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}-\frac {\left (2 a d^2\right ) \int \text {Li}_2\left (-\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^2}+\frac {\left (2 a d^2\right ) \int \text {Li}_2\left (-\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^2}\\ &=-\frac {(c+d x)^2}{\left (a^2+b^2\right ) f}+\frac {2 d (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 ) f^2}+\frac {a (c+d x)^2 \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 {2 d (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 ) f^2}-\frac {a (c+d x)^2 \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 {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}+\frac {2 a d (c+d x) \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 {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}-\frac {2 a d (c+d x) \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)^2 \cosh (e+f x)}{\left (a^2+b^2\right ) f (a+b \sinh (e+f x))}-\frac {\left (2 a d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2+b^2\right )^{3/2} f^3}+\frac {\left (2 a d^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\left (a^2+b^2\right )^{3/2} f^3}\\ &=-\frac {(c+d x)^2}{\left (a^2+b^2\right ) f}+\frac {2 d (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 ) f^2}+\frac {a (c+d x)^2 \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 {2 d (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 ) f^2}-\frac {a (c+d x)^2 \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 {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}+\frac {2 a d (c+d x) \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 {2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) f^3}-\frac {2 a d (c+d x) \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 {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^3}+\frac {2 a d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} f^3}-\frac {b (c+d x)^2 \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 = 1.74, size = 428, normalized size = 0.78 \[ \frac {-\frac {a \left (-f^2 (c+d x)^2 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )+f^2 (c+d x)^2 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )-2 d f (c+d x) \text {Li}_2\left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}-a}\right )+2 d f (c+d x) \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \text {Li}_3\left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}-a}\right )-2 d^2 \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+2 d f (c+d x) \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )+2 d f (c+d x) \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )+2 d^2 \text {Li}_2\left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}-a}\right )+2 d^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-\frac {b f^2 (c+d x)^2 \cosh (e+f x)}{a+b \sinh (e+f x)}-f^2 (c+d x)^2}{f^3 \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.79, size = 3957, normalized size = 7.21 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (d x + c\right )}^{2}}{{\left (b \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{2}}{\left (a +b \sinh \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ 2 \, a d^{2} f \int \frac {x^{2} 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} + 4 \, a c d 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} + 2 \, b c d {\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 )} - 4 \, a d^{2} \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} + 4 \, b d^{2} \int \frac {x}{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} + c^{2} {\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 )} - \frac {2 \, a c d \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}} + \frac {2 \, {\left (b d^{2} x^{2} + 2 \, b c d x - {\left (a d^{2} x^{2} e^{e} + 2 \, a c d x e^{e}\right )} e^{\left (f x\right )}\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 )}} \]
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 \[ \int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^2} \,d x \]
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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