Optimal. Leaf size=103 \[ \frac {b^2 e (c+d x)^2}{4 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d} \]
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Rubi [A]
time = 0.10, 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 = {5859, 12, 5776,
5812, 5783, 30} \begin {gather*} \frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {b e \sqrt {(c+d x)^2+1} (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {b^2 e (c+d x)^2}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 5776
Rule 5783
Rule 5812
Rule 5859
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}+\frac {(b e) \text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}+\frac {\left (b^2 e\right ) \text {Subst}(\int x \, dx,x,c+d x)}{2 d}\\ &=\frac {b^2 e (c+d x)^2}{4 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{2 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^2}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 120, normalized size = 1.17 \begin {gather*} \frac {e \left (\left (2 a^2+b^2\right ) (c+d x)^2-2 a b (c+d x) \sqrt {1+(c+d x)^2}+2 a b \sinh ^{-1}(c+d x)+2 b (c+d x) \left (2 a (c+d x)-b \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+b^2 \left (1+2 (c+d x)^2\right ) \sinh ^{-1}(c+d x)^2\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.36, size = 135, normalized size = 1.31
method | result | size |
derivativedivides | \(\frac {\frac {e \left (d x +c \right )^{2} a^{2}}{2}+e \,b^{2} \left (\frac {\arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arcsinh \left (d x +c \right ) \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{2}-\frac {\arcsinh \left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\arcsinh \left (d x +c \right )}{4}\right )}{d}\) | \(135\) |
default | \(\frac {\frac {e \left (d x +c \right )^{2} a^{2}}{2}+e \,b^{2} \left (\frac {\arcsinh \left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\arcsinh \left (d x +c \right ) \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{2}-\frac {\arcsinh \left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \arcsinh \left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\arcsinh \left (d x +c \right )}{4}\right )}{d}\) | \(135\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 361 vs.
\(2 (97) = 194\).
time = 0.43, size = 361, normalized size = 3.50 \begin {gather*} \frac {{\left ({\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + b^{2}\right )} \cosh \left (1\right ) + {\left (2 \, b^{2} d^{2} x^{2} + 4 \, b^{2} c d x + 2 \, b^{2} c^{2} + b^{2}\right )} \sinh \left (1\right )\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + {\left ({\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d x\right )} \cosh \left (1\right ) + 2 \, {\left ({\left (2 \, a b d^{2} x^{2} + 4 \, a b c d x + 2 \, a b c^{2} + a b\right )} \cosh \left (1\right ) + {\left (2 \, a b d^{2} x^{2} + 4 \, a b c d x + 2 \, a b c^{2} + a b\right )} \sinh \left (1\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (b^{2} d x + b^{2} c\right )} \cosh \left (1\right ) + {\left (b^{2} d x + b^{2} c\right )} \sinh \left (1\right )\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left ({\left (2 \, a^{2} + b^{2}\right )} d^{2} x^{2} + 2 \, {\left (2 \, a^{2} + b^{2}\right )} c d x\right )} \sinh \left (1\right ) - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left ({\left (a b d x + a b c\right )} \cosh \left (1\right ) + {\left (a b d x + a b c\right )} \sinh \left (1\right )\right )}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 335 vs.
\(2 (88) = 176\).
time = 0.20, size = 335, normalized size = 3.25 \begin {gather*} \begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {asinh}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {asinh}{\left (c + d x \right )} - \frac {a b c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{2 d} + a b d e x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {a b e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{2} + \frac {a b e \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} c^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {b^{2} c e x}{2} - \frac {b^{2} c e \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2 d} + \frac {b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{2} + \frac {b^{2} d e x^{2}}{4} - \frac {b^{2} e x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{2} + \frac {b^{2} e \operatorname {asinh}^{2}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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