Optimal. Leaf size=242 \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^3}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}+\frac{b^2 f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{d^3}+\frac{b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^3}-\frac{b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.446764, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3317, 3297, 3303, 3298, 3301, 3314, 31, 3312} \[ -\frac{a^2}{2 d (c+d x)^2}+\frac{a b f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^3}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}+\frac{b^2 f^2 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{d^3}+\frac{b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{d^3}-\frac{b^2 f \sinh (e+f x) \cosh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 3314
Rule 31
Rule 3312
Rubi steps
\begin{align*} \int \frac{(a+b \sinh (e+f x))^2}{(c+d x)^3} \, dx &=\int \left (\frac{a^2}{(c+d x)^3}+\frac{2 a b \sinh (e+f x)}{(c+d x)^3}+\frac{b^2 \sinh ^2(e+f x)}{(c+d x)^3}\right ) \, dx\\ &=-\frac{a^2}{2 d (c+d x)^2}+(2 a b) \int \frac{\sinh (e+f x)}{(c+d x)^3} \, dx+b^2 \int \frac{\sinh ^2(e+f x)}{(c+d x)^3} \, dx\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{(a b f) \int \frac{\cosh (e+f x)}{(c+d x)^2} \, dx}{d}+\frac{\left (b^2 f^2\right ) \int \frac{1}{c+d x} \, dx}{d^2}+\frac{\left (2 b^2 f^2\right ) \int \frac{\sinh ^2(e+f x)}{c+d x} \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac{b^2 f^2 \log (c+d x)}{d^3}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{\left (a b f^2\right ) \int \frac{\sinh (e+f x)}{c+d x} \, dx}{d^2}-\frac{\left (2 b^2 f^2\right ) \int \left (\frac{1}{2 (c+d x)}-\frac{\cosh (2 e+2 f x)}{2 (c+d x)}\right ) \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{\left (b^2 f^2\right ) \int \frac{\cosh (2 e+2 f x)}{c+d x} \, dx}{d^2}+\frac{\left (a b f^2 \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d^2}+\frac{\left (a b f^2 \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac{a b f^2 \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^3}+\frac{\left (b^2 f^2 \cosh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d^2}+\frac{\left (b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{d^2}\\ &=-\frac{a^2}{2 d (c+d x)^2}-\frac{a b f \cosh (e+f x)}{d^2 (c+d x)}+\frac{b^2 f^2 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Chi}\left (\frac{2 c f}{d}+2 f x\right )}{d^3}+\frac{a b f^2 \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^3}-\frac{a b \sinh (e+f x)}{d (c+d x)^2}-\frac{b^2 f \cosh (e+f x) \sinh (e+f x)}{d^2 (c+d x)}-\frac{b^2 \sinh ^2(e+f x)}{2 d (c+d x)^2}+\frac{a b f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^3}+\frac{b^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 c f}{d}+2 f x\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.960048, size = 395, normalized size = 1.63 \[ \frac{-2 a^2 d^2+4 a b c^2 f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+4 a b f^2 (c+d x)^2 \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+4 a b d^2 f^2 x^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )+8 a b c d f^2 x \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )-4 a b c d f \cosh (e+f x)-4 a b d^2 \sinh (e+f x)-4 a b d^2 f x \cosh (e+f x)+4 b^2 c^2 f^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+4 b^2 f^2 (c+d x)^2 \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )+4 b^2 d^2 f^2 x^2 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+8 b^2 c d f^2 x \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-2 b^2 c d f \sinh (2 (e+f x))-2 b^2 d^2 f x \sinh (2 (e+f x))-b^2 d^2 \cosh (2 (e+f x))+b^2 d^2}{4 d^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 626, normalized size = 2.6 \begin{align*} -{\frac{ab{f}^{2}{{\rm e}^{fx+e}}}{2\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{ab{f}^{2}{{\rm e}^{fx+e}}}{2\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{ab{f}^{2}}{2\,{d}^{3}}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) }-{\frac{{a}^{2}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{2}}{4\,d \left ( dx+c \right ) ^{2}}}+{\frac{{f}^{3}{b}^{2}{{\rm e}^{-2\,fx-2\,e}}x}{4\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{3}{b}^{2}{{\rm e}^{-2\,fx-2\,e}}c}{4\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}{b}^{2}{{\rm e}^{-2\,fx-2\,e}}}{8\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}{b}^{2}}{2\,{d}^{3}}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) }-{\frac{{f}^{2}{b}^{2}{{\rm e}^{2\,fx+2\,e}}}{8\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{{f}^{2}{b}^{2}{{\rm e}^{2\,fx+2\,e}}}{4\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{f}^{2}{b}^{2}}{2\,{d}^{3}}{{\rm e}^{-2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-2\,fx-2\,e-2\,{\frac{cf-de}{d}} \right ) }-{\frac{ab{f}^{3}{{\rm e}^{-fx-e}}x}{2\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{ab{f}^{3}{{\rm e}^{-fx-e}}c}{2\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{ab{f}^{2}{{\rm e}^{-fx-e}}}{2\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,dc{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{ab{f}^{2}}{2\,{d}^{3}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58655, size = 274, normalized size = 1.13 \begin{align*} \frac{1}{4} \, b^{2}{\left (\frac{1}{d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d} - \frac{e^{\left (-2 \, e + \frac{2 \, c f}{d}\right )} E_{3}\left (\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{e^{\left (2 \, e - \frac{2 \, c f}{d}\right )} E_{3}\left (-\frac{2 \,{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} + a b{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{3}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{e^{\left (e - \frac{c f}{d}\right )} E_{3}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac{a^{2}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47013, size = 1237, normalized size = 5.11 \begin{align*} -\frac{b^{2} d^{2} \cosh \left (f x + e\right )^{2} + b^{2} d^{2} \sinh \left (f x + e\right )^{2} +{\left (2 \, a^{2} - b^{2}\right )} d^{2} + 4 \,{\left (a b d^{2} f x + a b c d f\right )} \cosh \left (f x + e\right ) - 2 \,{\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) - 2 \,{\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 4 \,{\left (a b d^{2} +{\left (b^{2} d^{2} f x + b^{2} c d f\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 2 \,{\left ({\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (a b d^{2} f^{2} x^{2} + 2 \, a b c d f^{2} x + a b c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\right ) + 2 \,{\left ({\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) -{\left (b^{2} d^{2} f^{2} x^{2} + 2 \, b^{2} c d f^{2} x + b^{2} c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right )}{4 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sinh{\left (e + f x \right )}\right )^{2}}{\left (c + d x\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.30969, size = 948, normalized size = 3.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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