Integrand size = 27, antiderivative size = 298 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2} \]
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Time = 0.13 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {272, 45, 5922, 12, 380} \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rule 12
Rule 45
Rule 272
Rule 380
Rule 5922
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {d^2 \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 c^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-7 c^2 x^2\right ) \left (1-c^2 x^2\right )^3 \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+15 c^4 x^4-19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {2 b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^3 \sqrt {d-c^2 d x^2}}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^5 \sqrt {d-c^2 d x^2}}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {19 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{7/2} (a+b \text {arccosh}(c x))}{7 c^4 d}+\frac {\left (d-c^2 d x^2\right )^{9/2} (a+b \text {arccosh}(c x))}{9 c^4 d^2} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.49 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (b c \left (126 x+21 c^2 x^3-189 c^4 x^5+171 c^6 x^7-49 c^8 x^9\right )+126 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))+441 c^2 x^2 (-1+c x)^{7/2} (1+c x)^{7/2} (a+b \text {arccosh}(c x))\right )}{3969 c^4 \sqrt {-1+c x} \sqrt {1+c x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1101\) vs. \(2(250)=500\).
Time = 0.60 (sec) , antiderivative size = 1102, normalized size of antiderivative = 3.70
method | result | size |
default | \(\text {Expression too large to display}\) | \(1102\) |
parts | \(\text {Expression too large to display}\) | \(1102\) |
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Time = 0.25 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.94 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\frac {63 \, {\left (7 \, b c^{10} d^{2} x^{10} - 26 \, b c^{8} d^{2} x^{8} + 34 \, b c^{6} d^{2} x^{6} - 16 \, b c^{4} d^{2} x^{4} - b c^{2} d^{2} x^{2} + 2 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (49 \, b c^{9} d^{2} x^{9} - 171 \, b c^{7} d^{2} x^{7} + 189 \, b c^{5} d^{2} x^{5} - 21 \, b c^{3} d^{2} x^{3} - 126 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 63 \, {\left (7 \, a c^{10} d^{2} x^{10} - 26 \, a c^{8} d^{2} x^{8} + 34 \, a c^{6} d^{2} x^{6} - 16 \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + 2 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{3969 \, {\left (c^{6} x^{2} - c^{4}\right )}} \]
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Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.62 \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=-\frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, {\left (\frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{2}}{c^{2} d} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{c^{4} d}\right )} a - \frac {{\left (49 \, c^{8} \sqrt {-d} d^{2} x^{9} - 171 \, c^{6} \sqrt {-d} d^{2} x^{7} + 189 \, c^{4} \sqrt {-d} d^{2} x^{5} - 21 \, c^{2} \sqrt {-d} d^{2} x^{3} - 126 \, \sqrt {-d} d^{2} x\right )} b}{3969 \, c^{3}} \]
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Exception generated. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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