Optimal. Leaf size=298 \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}+\frac{2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}+\frac{d \sinh (c) \text{Chi}(d x)}{a^3}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{\cosh (c+d x)}{a^3 x} \]
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Rubi [A] time = 0.689996, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}+\frac{2 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{2 a^2 b}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^4}+\frac{2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{a^3}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}+\frac{d \sinh (c) \text{Chi}(d x)}{a^3}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{\cosh (c+d x)}{a^3 x} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{x^2 (a+b x)^3} \, dx &=\int \left (\frac{\cosh (c+d x)}{a^3 x^2}-\frac{3 b \cosh (c+d x)}{a^4 x}+\frac{b^2 \cosh (c+d x)}{a^2 (a+b x)^3}+\frac{2 b^2 \cosh (c+d x)}{a^3 (a+b x)^2}+\frac{3 b^2 \cosh (c+d x)}{a^4 (a+b x)}\right ) \, dx\\ &=\frac{\int \frac{\cosh (c+d x)}{x^2} \, dx}{a^3}-\frac{(3 b) \int \frac{\cosh (c+d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2\right ) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{a^4}+\frac{\left (2 b^2\right ) \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{a^3}+\frac{b^2 \int \frac{\cosh (c+d x)}{(a+b x)^3} \, dx}{a^2}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}+\frac{d \int \frac{\sinh (c+d x)}{x} \, dx}{a^3}+\frac{(2 b d) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{a^3}+\frac{(b d) \int \frac{\sinh (c+d x)}{(a+b x)^2} \, dx}{2 a^2}-\frac{(3 b \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2 \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a^4}-\frac{(3 b \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^4}+\frac{\left (3 b^2 \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a^4}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^4}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{d^2 \int \frac{\cosh (c+d x)}{a+b x} \, dx}{2 a^2}+\frac{(d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx}{a^3}+\frac{\left (2 b d \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a^3}+\frac{(d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx}{a^3}+\frac{\left (2 b d \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{a^3}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{d \text{Chi}(d x) \sinh (c)}{a^3}+\frac{2 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{a^3}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{\left (d^2 \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{2 a^2}+\frac{\left (d^2 \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{2 a^2}\\ &=-\frac{\cosh (c+d x)}{a^3 x}-\frac{b \cosh (c+d x)}{2 a^2 (a+b x)^2}-\frac{2 b \cosh (c+d x)}{a^3 (a+b x)}-\frac{3 b \cosh (c) \text{Chi}(d x)}{a^4}+\frac{3 b \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{d^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{2 a^2 b}+\frac{d \text{Chi}(d x) \sinh (c)}{a^3}+\frac{2 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{a^3}-\frac{d \sinh (c+d x)}{2 a^2 (a+b x)}+\frac{d \cosh (c) \text{Shi}(d x)}{a^3}-\frac{3 b \sinh (c) \text{Shi}(d x)}{a^4}+\frac{2 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^3}+\frac{3 b \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{a^4}+\frac{d^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{2 a^2 b}\\ \end{align*}
Mathematica [B] time = 1.43962, size = 710, normalized size = 2.38 \[ \frac{a^2 b^2 d^2 x^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+8 a^2 b^2 d x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^2 b^2 d^2 x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+4 a^2 b^2 d x^2 \cosh (c) \text{Shi}(d x)+8 a^2 b^2 d x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-6 a^2 b^2 x \sinh (c) \text{Shi}(d x)+6 a^2 b^2 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-a^2 b^2 d x^2 \sinh (c+d x)-9 a^2 b^2 x \cosh (c+d x)+2 a^3 b d^2 x^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+a^4 d^2 x \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+4 a^3 b d x \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d^2 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+a^4 d^2 x \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )+2 a^3 b d x \cosh (c) \text{Shi}(d x)+4 a^3 b d x \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-a^3 b d x \sinh (c+d x)-2 a^3 b \cosh (c+d x)+4 a b^3 d x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{d (a+b x)}{b}\right )+6 b^2 x (a+b x)^2 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-12 a b^3 x^2 \sinh (c) \text{Shi}(d x)+12 a b^3 x^2 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+6 b^4 x^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )+2 a b^3 d x^3 \cosh (c) \text{Shi}(d x)+4 a b^3 d x^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{d (a+b x)}{b}\right )-6 a b^3 x^2 \cosh (c+d x)+2 b x (a+b x)^2 \text{Chi}(d x) (a d \sinh (c)-3 b \cosh (c))-6 b^4 x^3 \sinh (c) \text{Shi}(d x)}{2 a^4 b x (a+b x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.073, size = 643, normalized size = 2.2 \begin{align*}{\frac{{{\rm e}^{-dx-c}}x{d}^{3}b}{4\,{a}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{{{\rm e}^{-dx-c}}{d}^{3}}{4\,a \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{3\,{{\rm e}^{-dx-c}}x{d}^{2}{b}^{2}}{2\,{a}^{3} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{9\,{{\rm e}^{-dx-c}}{d}^{2}b}{4\,{a}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}-{\frac{{{\rm e}^{-dx-c}}{d}^{2}}{2\,ax \left ({b}^{2}{d}^{2}{x}^{2}+2\,ab{d}^{2}x+{a}^{2}{d}^{2} \right ) }}+{\frac{d{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{3}}}+{\frac{3\,b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2\,{a}^{4}}}-{\frac{{d}^{2}}{4\,{a}^{2}b}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{d}{{a}^{3}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{3\,b}{2\,{a}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+{\frac{3\,b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{4}}}-{\frac{{{\rm e}^{dx+c}}}{2\,{a}^{3}x}}-{\frac{d{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2\,{a}^{3}}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,{a}^{2}b} \left ({\frac{da}{b}}+dx \right ) ^{-2}}-{\frac{{d}^{2}{{\rm e}^{dx+c}}}{4\,{a}^{2}b} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{{d}^{2}}{4\,{a}^{2}b}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{d{{\rm e}^{dx+c}}}{{a}^{3}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d}{{a}^{3}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{3\,b}{2\,{a}^{4}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh \left (d x + c\right )}{{\left (b x + a\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.2, size = 1563, normalized size = 5.24 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25263, size = 1358, normalized size = 4.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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