Optimal. Leaf size=346 \[ -\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3} \]
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Rubi [A]
time = 0.30, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5473, 5395,
3377, 2718} \begin {gather*} -\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3377
Rule 5395
Rule 5473
Rubi steps
\begin {align*} \int x^2 \cosh \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int (-c+x)^2 \cosh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 \text {Subst}\left (\int x \left (c-x^2\right )^2 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \text {Subst}\left (\int \left (c^2 x \cosh (a+b x)-2 c x^3 \cosh (a+b x)+x^5 \cosh (a+b x)\right ) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 \text {Subst}\left (\int x^5 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}-\frac {(4 c) \text {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}+\frac {\left (2 c^2\right ) \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^3}\\ &=\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {10 \text {Subst}\left (\int x^4 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^3}+\frac {(12 c) \text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^3}-\frac {\left (2 c^2\right ) \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^3}\\ &=-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 \text {Subst}\left (\int x^3 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^3}-\frac {(24 c) \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^3}\\ &=-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {120 \text {Subst}\left (\int x^2 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^3}+\frac {(24 c) \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^3}\\ &=\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {240 \text {Subst}\left (\int x \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^4 d^3}\\ &=\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}-\frac {240 \text {Subst}\left (\int \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^5 d^3}\\ &=-\frac {240 \cosh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3}+\frac {24 c \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}-\frac {2 c^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {120 (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^4 d^3}+\frac {12 c (c+d x) \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}-\frac {10 (c+d x)^2 \cosh \left (a+b \sqrt {c+d x}\right )}{b^2 d^3}+\frac {240 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^5 d^3}-\frac {24 c \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}+\frac {2 c^2 \sqrt {c+d x} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {40 (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b^3 d^3}-\frac {4 c (c+d x)^{3/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}+\frac {2 (c+d x)^{5/2} \sinh \left (a+b \sqrt {c+d x}\right )}{b d^3}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 104, normalized size = 0.30 \begin {gather*} \frac {-2 \left (120+12 b^2 (4 c+5 d x)+b^4 d x (4 c+5 d x)\right ) \cosh \left (a+b \sqrt {c+d x}\right )+2 b \sqrt {c+d x} \left (120+b^4 d^2 x^2+4 b^2 (2 c+5 d x)\right ) \sinh \left (a+b \sqrt {c+d x}\right )}{b^6 d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs.
\(2(310)=620\).
time = 1.66, size = 831, normalized size = 2.40 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 486, normalized size = 1.40 \begin {gather*} \frac {2 \, d^{3} x^{3} \cosh \left (\sqrt {d x + c} b + a\right ) + {\left (\frac {c^{3} e^{\left (\sqrt {d x + c} b + a\right )}}{b} + \frac {c^{3} e^{\left (-\sqrt {d x + c} b - a\right )}}{b} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} e^{a} - 2 \, \sqrt {d x + c} b e^{a} + 2 \, e^{a}\right )} c^{2} e^{\left (\sqrt {d x + c} b\right )}}{b^{3}} - \frac {3 \, {\left ({\left (d x + c\right )} b^{2} + 2 \, \sqrt {d x + c} b + 2\right )} c^{2} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{3}} + \frac {3 \, {\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 )} c e^{\left (\sqrt {d x + c} b\right )}}{b^{5}} + \frac {3 \, {\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 )} c e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} e^{a} - 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} e^{a} + 30 \, {\left (d x + c\right )}^{2} b^{4} e^{a} - 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} e^{a} + 360 \, {\left (d x + c\right )} b^{2} e^{a} - 720 \, \sqrt {d x + c} b e^{a} + 720 \, e^{a}\right )} e^{\left (\sqrt {d x + c} b\right )}}{b^{7}} - \frac {{\left ({\left (d x + c\right )}^{3} b^{6} + 6 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} + 30 \, {\left (d x + c\right )}^{2} b^{4} + 120 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} + 360 \, {\left (d x + c\right )} b^{2} + 720 \, \sqrt {d x + c} b + 720\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{7}}\right )} b}{6 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 104, normalized size = 0.30 \begin {gather*} \frac {2 \, {\left ({\left (b^{5} d^{2} x^{2} + 20 \, b^{3} d x + 8 \, b^{3} c + 120 \, b\right )} \sqrt {d x + c} \sinh \left (\sqrt {d x + c} b + a\right ) - {\left (5 \, b^{4} d^{2} x^{2} + 48 \, b^{2} c + 4 \, {\left (b^{4} c + 15 \, b^{2}\right )} d x + 120\right )} \cosh \left (\sqrt {d x + c} b + a\right )\right )}}{b^{6} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 269, normalized size = 0.78 \begin {gather*} \begin {cases} \frac {x^{3} \cosh {\left (a \right )}}{3} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{3} \cosh {\left (a + b \sqrt {c} \right )}}{3} & \text {for}\: d = 0 \\\frac {2 x^{2} \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b d} - \frac {8 c x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} - \frac {10 x^{2} \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} + \frac {16 c \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{3}} + \frac {40 x \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {96 c \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{3}} - \frac {120 x \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} + \frac {240 \sqrt {c + d x} \sinh {\left (a + b \sqrt {c + d x} \right )}}{b^{5} d^{3}} - \frac {240 \cosh {\left (a + b \sqrt {c + d x} \right )}}{b^{6} d^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 915 vs.
\(2 (310) = 620\).
time = 0.42, size = 915, normalized size = 2.64 \begin {gather*} \frac {\frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c - b^{4} c^{2} + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c - 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c + 6 \, a^{2} b^{2} c - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a - 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} + 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} - 5 \, a^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c + 12 \, a b^{2} c + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a + 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} - 20 \, a^{3} + 12 \, b^{2} c - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 120 \, {\left (\sqrt {d x + c} b + a\right )} a - 60 \, a^{2} + 120 \, \sqrt {d x + c} b - 120\right )} e^{\left (\sqrt {d x + c} b + a\right )}}{b^{5} d^{2}} - \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{4} c^{2} - a b^{4} c^{2} - 2 \, {\left (\sqrt {d x + c} b + a\right )}^{3} b^{2} c + 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a b^{2} c - 6 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} b^{2} c + 2 \, a^{3} b^{2} c + b^{4} c^{2} + {\left (\sqrt {d x + c} b + a\right )}^{5} - 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} a + 10 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a^{2} - 10 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{3} + 5 \, {\left (\sqrt {d x + c} b + a\right )} a^{4} - a^{5} - 6 \, {\left (\sqrt {d x + c} b + a\right )}^{2} b^{2} c + 12 \, {\left (\sqrt {d x + c} b + a\right )} a b^{2} c - 6 \, a^{2} b^{2} c + 5 \, {\left (\sqrt {d x + c} b + a\right )}^{4} - 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} a + 30 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a^{2} - 20 \, {\left (\sqrt {d x + c} b + a\right )} a^{3} + 5 \, a^{4} - 12 \, {\left (\sqrt {d x + c} b + a\right )} b^{2} c + 12 \, a b^{2} c + 20 \, {\left (\sqrt {d x + c} b + a\right )}^{3} - 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a + 60 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} - 20 \, a^{3} - 12 \, b^{2} c + 60 \, {\left (\sqrt {d x + c} b + a\right )}^{2} - 120 \, {\left (\sqrt {d x + c} b + a\right )} a + 60 \, a^{2} + 120 \, \sqrt {d x + c} b + 120\right )} e^{\left (-\sqrt {d x + c} b - a\right )}}{b^{5} d^{2}}}{b d} \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 = -0.00 \begin {gather*} \int x^2\,\mathrm {cosh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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