Optimal. Leaf size=83 \[ \frac {(8 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {1}{16} x (8 a+5 b)+\frac {b \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac {13 b \sinh (c+d x) \cosh ^3(c+d x)}{24 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3217, 1257, 1157, 385, 206} \[ \frac {(8 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {1}{16} x (8 a+5 b)+\frac {b \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac {13 b \sinh (c+d x) \cosh ^3(c+d x)}{24 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 385
Rule 1157
Rule 1257
Rule 3217
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (a-2 a x^2+(a+b) x^4\right )}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {\operatorname {Subst}\left (\int \frac {-b+6 (a-b) x^2-6 (a+b) x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 d}\\ &=-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\operatorname {Subst}\left (\int \frac {-9 b-24 (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{24 d}\\ &=\frac {(8 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {(8 a+5 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=-\frac {1}{16} (8 a+5 b) x+\frac {(8 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 63, normalized size = 0.76 \[ \frac {(48 a+45 b) \sinh (2 (c+d x))-96 a c-96 a d x-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))-60 b c-60 b d x}{192 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 3.30, size = 109, normalized size = 1.31 \[ \frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 6 \, {\left (8 \, a + 5 \, b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} + {\left (16 \, a + 15 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 113, normalized size = 1.36 \[ -\frac {1}{16} \, {\left (8 \, a + 5 \, b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {3 \, b e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a + 15 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a + 15 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {3 \, b e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 76, normalized size = 0.92 \[ \frac {b \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )+a \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 122, normalized size = 1.47 \[ -\frac {1}{8} \, a {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{384} \, b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.16, size = 64, normalized size = 0.77 \[ \frac {12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {45\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-24\,a\,d\,x-15\,b\,d\,x}{48\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.95, size = 206, normalized size = 2.48 \[ \begin {cases} \frac {a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {5 b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\relax (c )}\right ) \sinh ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________