Optimal. Leaf size=86 \[ \frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {b^2 (3 a-b) \sinh ^3(c+d x)}{3 d}+\frac {(a-b)^3 \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b^3 \sinh ^5(c+d x)}{5 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3190, 390, 203} \[ \frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {b^2 (3 a-b) \sinh ^3(c+d x)}{3 d}+\frac {(a-b)^3 \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b^3 \sinh ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 390
Rule 3190
Rubi steps
\begin {align*} \int \text {sech}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b \left (3 a^2-3 a b+b^2\right )+(3 a-b) b^2 x^2+b^3 x^4+\frac {(a-b)^3}{1+x^2}\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {(3 a-b) b^2 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \sinh ^5(c+d x)}{5 d}+\frac {(a-b)^3 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {(a-b)^3 \tan ^{-1}(\sinh (c+d x))}{d}+\frac {b \left (3 a^2-3 a b+b^2\right ) \sinh (c+d x)}{d}+\frac {(3 a-b) b^2 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \sinh ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.55, size = 100, normalized size = 1.16 \[ \frac {\sinh (c+d x) \left (b \left (45 a^2+15 a b \left (\sinh ^2(c+d x)-3\right )+b^2 \left (3 \sinh ^4(c+d x)-5 \sinh ^2(c+d x)+15\right )\right )+\frac {15 (a-b)^3 \tanh ^{-1}\left (\sqrt {-\sinh ^2(c+d x)}\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.47, size = 1114, normalized size = 12.95 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.21, size = 204, normalized size = 2.37 \[ \frac {3 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 60 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 35 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 720 \, a^{2} b e^{\left (d x + c\right )} - 900 \, a b^{2} e^{\left (d x + c\right )} + 330 \, b^{3} e^{\left (d x + c\right )} + 960 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - {\left (720 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 900 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 330 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 60 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 155, normalized size = 1.80 \[ \frac {2 a^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {3 a^{2} b \sinh \left (d x +c \right )}{d}-\frac {6 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {a \,b^{2} \left (\sinh ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a \,b^{2} \sinh \left (d x +c \right )}{d}+\frac {6 a \,b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {b^{3} \left (\sinh ^{5}\left (d x +c \right )\right )}{5 d}-\frac {b^{3} \left (\sinh ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b^{3} \sinh \left (d x +c \right )}{d}-\frac {2 b^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.44, size = 233, normalized size = 2.71 \[ -\frac {1}{480} \, b^{3} {\left (\frac {{\left (35 \, e^{\left (-2 \, d x - 2 \, c\right )} - 330 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {330 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d} - \frac {960 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} - \frac {1}{8} \, a b^{2} {\left (\frac {{\left (15 \, e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )} e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {15 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac {48 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \arctan \left (\sinh \left (d x + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.04, size = 294, normalized size = 3.42 \[ \frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b-30\,a\,b^2+11\,b^3\right )}{16\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (24\,a^2\,b-30\,a\,b^2+11\,b^3\right )}{16\,d}+\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3\,\sqrt {d^2}-b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}-3\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}}\right )\,\sqrt {a^6-6\,a^5\,b+15\,a^4\,b^2-20\,a^3\,b^3+15\,a^2\,b^4-6\,a\,b^5+b^6}}{\sqrt {d^2}}-\frac {b^3\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {b^3\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}-\frac {b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}\,\left (12\,a-7\,b\right )}{96\,d}+\frac {b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (12\,a-7\,b\right )}{96\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sinh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________