Integrand size = 24, antiderivative size = 84 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5899, 266} \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \]
[In]
[Out]
Rule 266
Rule 5899
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}+\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d \sqrt {d-c^2 d x^2}} \\ & = \frac {x (a+b \text {arccosh}(c x))}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {2 a c x+2 b c x \text {arccosh}(c x)-b \sqrt {-1+c x} \sqrt {1+c x} \log \left (1-c^2 x^2\right )}{2 c d \sqrt {d-c^2 d x^2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(74)=148\).
Time = 0.93 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.14
method | result | size |
default | \(\frac {a x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(180\) |
parts | \(\frac {a x}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right )}{d^{2} c \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) x}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}-1\right )}{d^{2} c \left (c^{2} x^{2}-1\right )}\) | \(180\) |
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b c \sqrt {-\frac {1}{c^{4} d}} \log \left (x^{2} - \frac {1}{c^{2}}\right )}{2 \, d} + \frac {b x \operatorname {arcosh}\left (c x\right )}{\sqrt {-c^{2} d x^{2} + d} d} + \frac {a x}{\sqrt {-c^{2} d x^{2} + d} d} \]
[In]
[Out]
\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]
[In]
[Out]