Optimal. Leaf size=88 \[ \frac {\sinh (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac {\sinh (2 (a+c)+2 x (b+d))}{16 (b+d)}-\frac {\sinh (2 a+2 b x)}{8 b}-\frac {\sinh (2 c+2 d x)}{8 d}+\frac {x}{4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5613, 2637} \[ \frac {\sinh (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac {\sinh (2 (a+c)+2 x (b+d))}{16 (b+d)}-\frac {\sinh (2 a+2 b x)}{8 b}-\frac {\sinh (2 c+2 d x)}{8 d}+\frac {x}{4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2637
Rule 5613
Rubi steps
\begin {align*} \int \sinh ^2(a+b x) \sinh ^2(c+d x) \, dx &=\int \left (\frac {1}{4}-\frac {1}{4} \cosh (2 a+2 b x)+\frac {1}{8} \cosh (2 (a-c)+2 (b-d) x)-\frac {1}{4} \cosh (2 c+2 d x)+\frac {1}{8} \cosh (2 (a+c)+2 (b+d) x)\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{8} \int \cosh (2 (a-c)+2 (b-d) x) \, dx+\frac {1}{8} \int \cosh (2 (a+c)+2 (b+d) x) \, dx-\frac {1}{4} \int \cosh (2 a+2 b x) \, dx-\frac {1}{4} \int \cosh (2 c+2 d x) \, dx\\ &=\frac {x}{4}-\frac {\sinh (2 a+2 b x)}{8 b}+\frac {\sinh (2 (a-c)+2 (b-d) x)}{16 (b-d)}-\frac {\sinh (2 c+2 d x)}{8 d}+\frac {\sinh (2 (a+c)+2 (b+d) x)}{16 (b+d)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.74, size = 106, normalized size = 1.20 \[ \frac {\left (2 d^3-2 b^2 d\right ) \sinh (2 (a+b x))+b d (b+d) \sinh (2 (a+x (b-d)-c))+b (b-d) (d (\sinh (2 (a+x (b+d)+c))+4 x (b+d))-2 (b+d) \sinh (2 (c+d x)))}{16 b d (b-d) (b+d)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 192, normalized size = 2.18 \[ \frac {b^{2} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} + {\left (b^{3} d - b d^{3}\right )} x + {\left (b^{2} d \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - {\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - {\left (b d^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left (b d^{2} \cosh \left (b x + a\right )^{2} + b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{4 \, {\left ({\left (b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{3} d - b d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.14, size = 156, normalized size = 1.77 \[ \frac {1}{4} \, x + \frac {e^{\left (2 \, b x + 2 \, d x + 2 \, a + 2 \, c\right )}}{32 \, {\left (b + d\right )}} + \frac {e^{\left (2 \, b x - 2 \, d x + 2 \, a - 2 \, c\right )}}{32 \, {\left (b - d\right )}} - \frac {e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} - \frac {e^{\left (-2 \, b x + 2 \, d x - 2 \, a + 2 \, c\right )}}{32 \, {\left (b - d\right )}} - \frac {e^{\left (-2 \, b x - 2 \, d x - 2 \, a - 2 \, c\right )}}{32 \, {\left (b + d\right )}} + \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{16 \, d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.33, size = 83, normalized size = 0.94 \[ \frac {x}{4}-\frac {\sinh \left (2 b x +2 a \right )}{8 b}-\frac {\sinh \left (2 d x +2 c \right )}{8 d}+\frac {\sinh \left (\left (2 b -2 d \right ) x +2 a -2 c \right )}{16 b -16 d}+\frac {\sinh \left (\left (2 b +2 d \right ) x +2 a +2 c \right )}{16 b +16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.92, size = 152, normalized size = 1.73 \[ -\frac {d^3\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )-b^3\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )-b\,d^3\,x+b^3\,d\,x-2\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )+2\,b\,d^2\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )+2\,b^2\,d\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )-2\,b\,d^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{4\,b\,d^3-4\,b^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 21.62, size = 1027, normalized size = 11.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________