Optimal. Leaf size=120 \[ \frac{2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac{2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 \cosh ^9(c+d x)}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.129451, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3215, 1153} \[ \frac{2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac{2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac{(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^{11}(c+d x)}{11 d}-\frac{5 b^2 \cosh ^9(c+d x)}{9 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3215
Rule 1153
Rubi steps
\begin{align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^4(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b-2 b x^2+b x^4\right )^2 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left ((a+b)^2+(-a-5 b) (a+b) x^2+2 b (3 a+5 b) x^4-2 b (a+5 b) x^6+5 b^2 x^8-b^2 x^{10}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{(a+b) (a+5 b) \cosh ^3(c+d x)}{3 d}-\frac{2 b (3 a+5 b) \cosh ^5(c+d x)}{5 d}+\frac{2 b (a+5 b) \cosh ^7(c+d x)}{7 d}-\frac{5 b^2 \cosh ^9(c+d x)}{9 d}+\frac{b^2 \cosh ^{11}(c+d x)}{11 d}\\ \end{align*}
Mathematica [A] time = 0.0635944, size = 207, normalized size = 1.72 \[ -\frac{3 a^2 \cosh (c+d x)}{4 d}+\frac{a^2 \cosh (3 (c+d x))}{12 d}-\frac{35 a b \cosh (c+d x)}{32 d}+\frac{7 a b \cosh (3 (c+d x))}{32 d}-\frac{7 a b \cosh (5 (c+d x))}{160 d}+\frac{a b \cosh (7 (c+d x))}{224 d}-\frac{231 b^2 \cosh (c+d x)}{512 d}+\frac{55 b^2 \cosh (3 (c+d x))}{512 d}-\frac{33 b^2 \cosh (5 (c+d x))}{1024 d}+\frac{55 b^2 \cosh (7 (c+d x))}{7168 d}-\frac{11 b^2 \cosh (9 (c+d x))}{9216 d}+\frac{b^2 \cosh (11 (c+d x))}{11264 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 132, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \left ( -{\frac{256}{693}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{10}}{11}}-{\frac{10\, \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{99}}+{\frac{80\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{693}}-{\frac{32\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{231}}+{\frac{128\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{693}} \right ) \cosh \left ( dx+c \right ) +2\,ab \left ( -{\frac{16}{35}}+1/7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) +{a}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.04615, size = 414, normalized size = 3.45 \begin{align*} -\frac{1}{1419264} \, b^{2}{\left (\frac{{\left (847 \, e^{\left (-2 \, d x - 2 \, c\right )} - 5445 \, e^{\left (-4 \, d x - 4 \, c\right )} + 22869 \, e^{\left (-6 \, d x - 6 \, c\right )} - 76230 \, e^{\left (-8 \, d x - 8 \, c\right )} + 320166 \, e^{\left (-10 \, d x - 10 \, c\right )} - 63\right )} e^{\left (11 \, d x + 11 \, c\right )}}{d} + \frac{320166 \, e^{\left (-d x - c\right )} - 76230 \, e^{\left (-3 \, d x - 3 \, c\right )} + 22869 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5445 \, e^{\left (-7 \, d x - 7 \, c\right )} + 847 \, e^{\left (-9 \, d x - 9 \, c\right )} - 63 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d}\right )} - \frac{1}{2240} \, a b{\left (\frac{{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac{1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac{1}{24} \, a^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.63243, size = 1111, normalized size = 9.26 \begin{align*} \frac{315 \, b^{2} \cosh \left (d x + c\right )^{11} + 3465 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - 4235 \, b^{2} \cosh \left (d x + c\right )^{9} + 3465 \,{\left (15 \, b^{2} \cosh \left (d x + c\right )^{3} - 11 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 495 \,{\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{7} + 1155 \,{\left (126 \, b^{2} \cosh \left (d x + c\right )^{5} - 308 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \,{\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 693 \,{\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 3465 \,{\left (30 \, b^{2} \cosh \left (d x + c\right )^{7} - 154 \, b^{2} \cosh \left (d x + c\right )^{5} + 5 \,{\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} -{\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 2310 \,{\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3465 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{9} - 44 \, b^{2} \cosh \left (d x + c\right )^{7} + 3 \,{\left (32 \, a b + 55 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} - 2 \,{\left (224 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 2 \,{\left (128 \, a^{2} + 336 \, a b + 165 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 6930 \,{\left (384 \, a^{2} + 560 \, a b + 231 \, b^{2}\right )} \cosh \left (d x + c\right )}{3548160 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 74.8908, size = 280, normalized size = 2.33 \begin{align*} \begin{cases} \frac{a^{2} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{2} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{2 a b \sinh ^{6}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{16 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{32 a b \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{b^{2} \sinh ^{10}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{10 b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{3 d} - \frac{32 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{7 d} + \frac{128 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{63 d} - \frac{256 b^{2} \cosh ^{11}{\left (c + d x \right )}}{693 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{2} \sinh ^{3}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.31993, size = 443, normalized size = 3.69 \begin{align*} \frac{315 \, b^{2} e^{\left (11 \, d x + 11 \, c\right )} - 4235 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 15840 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 27225 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 155232 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 114345 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 295680 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 776160 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 381150 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2661120 \, a^{2} e^{\left (d x + c\right )} - 3880800 \, a b e^{\left (d x + c\right )} - 1600830 \, b^{2} e^{\left (d x + c\right )} -{\left (2661120 \, a^{2} e^{\left (10 \, d x + 10 \, c\right )} + 3880800 \, a b e^{\left (10 \, d x + 10 \, c\right )} + 1600830 \, b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 295680 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} - 776160 \, a b e^{\left (8 \, d x + 8 \, c\right )} - 381150 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 155232 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 114345 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 15840 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 27225 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 4235 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, b^{2}\right )} e^{\left (-11 \, d x - 11 \, c\right )}}{7096320 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]