Optimal. Leaf size=103 \[ \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{b^2 d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )}{b^2 d}-\frac {e \sqrt {(c+d x)^2+1} (c+d x)}{b d \left (a+b \sinh ^{-1}(c+d x)\right )} \]
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Rubi [A] time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5865, 12, 5665, 3303, 3298, 3301} \[ \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^2 d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^2 d}-\frac {e \sqrt {(c+d x)^2+1} (c+d x)}{b d \left (a+b \sinh ^{-1}(c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5665
Rule 5865
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e x}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {\left (e \cosh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}-\frac {\left (e \sinh \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {e (c+d x) \sqrt {1+(c+d x)^2}}{b d \left (a+b \sinh ^{-1}(c+d x)\right )}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^2 d}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \sinh ^{-1}(c+d x)\right )}{b^2 d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 97, normalized size = 0.94 \[ \frac {e \left (-\frac {b \sqrt {c^2+2 c d x+d^2 x^2+1} (c+d x)}{a+b \sinh ^{-1}(c+d x)}+\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\sinh ^{-1}(c+d x)\right )\right )\right )}{b^2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d e x + c e}{b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 2 \, a b \operatorname {arsinh}\left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {d e x + c e}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 160, normalized size = 1.55 \[ \frac {\frac {\left (2 \left (d x +c \right )^{2}-2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+1\right ) e}{4 \left (a +b \arcsinh \left (d x +c \right )\right ) b}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \Ei \left (1, 2 \arcsinh \left (d x +c \right )+\frac {2 a}{b}\right )}{2 b^{2}}-\frac {e \left (2 \left (d x +c \right )^{2}+1+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\right )}{4 b \left (a +b \arcsinh \left (d x +c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \Ei \left (1, -2 \arcsinh \left (d x +c \right )-\frac {2 a}{b}\right )}{2 b^{2}}}{d} \]
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 {d^{4} e x^{4} + 4 \, c d^{3} e x^{3} + c^{4} e + c^{2} e + {\left (6 \, c^{2} d^{2} e + d^{2} e\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e + c d e\right )} x + {\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + c^{3} e + c e + {\left (3 \, c^{2} d e + d e\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{a b d^{3} x^{2} + 2 \, a b c d^{2} x + {\left (c^{2} d + d\right )} a b + {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + {\left (c^{2} d + d\right )} b^{2} + {\left (b^{2} d^{2} x + b^{2} c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (a b d^{2} x + a b c d\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}} + \int \frac {2 \, d^{5} e x^{5} + 10 \, c d^{4} e x^{4} + 2 \, c^{5} e + 4 \, c^{3} e + 4 \, {\left (5 \, c^{2} d^{3} e + d^{3} e\right )} x^{3} + 4 \, {\left (5 \, c^{3} d^{2} e + 3 \, c d^{2} e\right )} x^{2} + 2 \, {\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + 2 \, c e + 2 \, {\left (5 \, c^{4} d e + 6 \, c^{2} d e + d e\right )} x + {\left (4 \, d^{4} e x^{4} + 16 \, c d^{3} e x^{3} + 4 \, c^{4} e + 4 \, c^{2} e + 4 \, {\left (6 \, c^{2} d^{2} e + d^{2} e\right )} x^{2} + 8 \, {\left (2 \, c^{3} d e + c d e\right )} x + e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{a b d^{4} x^{4} + 4 \, a b c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} d^{2} + d^{2}\right )} a b x^{2} + 4 \, {\left (c^{3} d + c d\right )} a b x + {\left (c^{4} + 2 \, c^{2} + 1\right )} a b + {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 2 \, {\left (3 \, c^{2} d^{2} + d^{2}\right )} b^{2} x^{2} + 4 \, {\left (c^{3} d + c d\right )} b^{2} x + {\left (c^{4} + 2 \, c^{2} + 1\right )} b^{2} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + {\left (3 \, c^{2} d + d\right )} b^{2} x + {\left (c^{3} + c\right )} b^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + {\left (3 \, c^{2} d + d\right )} a b x + {\left (c^{3} + c\right )} a b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2} \,d x \]
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 \[ e \left (\int \frac {c}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{2} + 2 a b \operatorname {asinh}{\left (c + d x \right )} + b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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