Optimal. Leaf size=167 \[ -\frac {12 \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {12 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2} \]
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Rubi [A] time = 0.19, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5364, 5286, 3296, 2637} \[ -\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {12 \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {12 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 5286
Rule 5364
Rubi steps
\begin {align*} \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int (-c+x) \sinh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 \operatorname {Subst}\left (\int x \left (-c+x^2\right ) \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-c x \sinh (a+b x)+x^3 \sinh (a+b x)\right ) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {2 \operatorname {Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}-\frac {(2 c) \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {6 \operatorname {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2}+\frac {(2 c) \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2}\\ &=-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {12 \operatorname {Subst}\left (\int x \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^2}\\ &=\frac {12 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {12 \operatorname {Subst}\left (\int \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^2}\\ &=\frac {12 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {12 \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 72, normalized size = 0.43 \[ \frac {2 b \left (b^2 d x+6\right ) \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )-2 \left (b^2 (2 c+3 d x)+6\right ) \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 68, normalized size = 0.41 \[ \frac {2 \, {\left ({\left (b^{3} d x + 6 \, b\right )} \sqrt {d x + c} \cosh \left (\sqrt {d x + c} b + a\right ) - {\left (3 \, b^{2} d x + 2 \, b^{2} c + 6\right )} \sinh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 299, normalized size = 1.79 \[ -\frac {\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} - b^{2} c + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} - 6 \, {\left (\sqrt {d x + c} b + a\right )} a + 3 \, a^{2} - 6 \, \sqrt {d x + c} b + 6\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{3} d} + \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} + b^{2} c - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 6 \, {\left (\sqrt {d x + c} b + a\right )} a - 3 \, a^{2} - 6 \, \sqrt {d x + c} b - 6\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3} d}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 303, normalized size = 1.81 \[ \frac {\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-3 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )^{2}+6 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-6 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \sinh \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \cosh \left (a +b \sqrt {d x +c}\right )}{d^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 293, normalized size = 1.75 \[ \frac {2 \, d^{2} x^{2} \sinh \left (\sqrt {d x + c} b + a\right ) - {\left (\frac {c^{2} e^{\left (\sqrt {d x + c} b + a\right )}}{b} - \frac {c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {2 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} + \frac {2 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {{\left ({\left (d x + c\right )}^{2} b^{4} e^{a} - 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 12 \, {\left (d x + c\right )} b^{2} e^{a} - 24 \, \sqrt {d x + c} b e^{a} + 24 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} - \frac {{\left ({\left (d x + c\right )}^{2} b^{4} + 4 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 12 \, {\left (d x + c\right )} b^{2} + 24 \, \sqrt {d x + c} b + 24\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}}\right )} b}{4 \, d^{2}} \]
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 \[ \int x\,\mathrm {sinh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 151, normalized size = 0.90 \[ \begin {cases} \frac {x^{2} \sinh {\relax (a )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{2} \sinh {\left (a + b \sqrt {c} \right )}}{2} & \text {for}\: d = 0 \\\frac {2 x \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {4 c \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {6 x \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {12 \sqrt {c + d x} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {12 \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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