Optimal. Leaf size=167 \[ \frac {12 \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \cosh \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {12 \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 c \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {6 (c+d x) \sinh \left (a+b \sqrt {c+d x}\right )}{b^2 d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, 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 = {5472, 5394,
3377, 2717} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2717
Rule 3377
Rule 5394
Rule 5472
Rubi steps
\begin {align*} \int x \sinh \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {\text {Subst}\left (\int (-c+x) \sinh \left (a+b \sqrt {x}\right ) \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 \text {Subst}\left (\int x \left (-c+x^2\right ) \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=\frac {2 \text {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 \text {Subst}\left (\int x^3 \sinh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}-\frac {(2 c) \text {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 \text {Subst}\left (\int x^2 \cosh (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2}+\frac {(2 c) \text {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 \text {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 \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 72, normalized size = 0.43 \begin {gather*} \frac {2 b \sqrt {c+d x} \left (6+b^2 d x\right ) \cosh \left (a+b \sqrt {c+d x}\right )-2 \left (6+b^2 (2 c+3 d x)\right ) \sinh \left (a+b \sqrt {c+d x}\right )}{b^4 d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(302\) vs.
\(2(149)=298\).
time = 0.83, size = 303, normalized size = 1.81
method | result | size |
derivativedivides | \(\frac {\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-6 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2} d^{2}}\) | \(303\) |
default | \(\frac {\frac {6 a^{2} \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \cosh \left (a +b \sqrt {d x +c}\right )-2 \left (a +b \sqrt {d x +c}\right ) \sinh \left (a +b \sqrt {d x +c}\right )+2 \cosh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \cosh \left (a +b \sqrt {d x +c}\right )-3 \left (a +b \sqrt {d x +c}\right )^{2} \sinh \left (a +b \sqrt {d x +c}\right )+6 \left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-6 \sinh \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-2 c \left (\left (a +b \sqrt {d x +c}\right ) \cosh \left (a +b \sqrt {d x +c}\right )-\sinh \left (a +b \sqrt {d x +c}\right )\right )+2 c a \cosh \left (a +b \sqrt {d x +c}\right )}{b^{2} d^{2}}\) | \(303\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 293, normalized size = 1.75 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.48, size = 68, normalized size = 0.41 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.17, size = 151, normalized size = 0.90 \begin {gather*} \begin {cases} \frac {x^{2} \sinh {\left (a \right )}}{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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (149) = 298\).
time = 0.42, size = 299, normalized size = 1.79 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\mathrm {sinh}\left (a+b\,\sqrt {c+d\,x}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________