Optimal. Leaf size=841 \[ \frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d x^3 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} x^3 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.10, antiderivative size = 841, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 21, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {5930, 5926,
5939, 5893, 5883, 92, 54, 102, 12, 5912, 14, 5921, 471, 272, 45, 534, 1281, 470, 327, 223, 212}
\begin {gather*} -\frac {b c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^8}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^4 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x^7}{256 (1-c x) (c x+1)}+\frac {17 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^6}{144 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{864} b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^5-\frac {43 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x^5}{4608 (1-c x) (c x+1)}-\frac {59 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^4}{384 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{8} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac {5}{48} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac {5}{64} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^3+\frac {215 b^2 d^2 \sqrt {d-c^2 d x^2} x^3}{13824}+\frac {73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x^3}{18432 (1-c x) (c x+1)}+\frac {5 b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^2}{128 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x}{128 c^2}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} x}{9216 c^2}+\frac {73 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} x}{12288 c^2 (1-c x) (c x+1)}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {73 b^2 d^2 \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{12288 c^3 (1-c x) (c x+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 45
Rule 54
Rule 92
Rule 102
Rule 212
Rule 223
Rule 272
Rule 327
Rule 470
Rule 471
Rule 534
Rule 1281
Rule 5883
Rule 5893
Rule 5912
Rule 5921
Rule 5926
Rule 5930
Rule 5939
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{12 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {11 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{96 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-3+2 c^2 x^2\right )}{12 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{64 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (-3+2 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{288 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (6-8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{96 (-1+c x) (1+c x)}\\ &=\frac {5}{512} b^2 d^2 x^3 \sqrt {d-c^2 d x^2}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{512 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{216 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (48 c^2-43 c^4 x^2\right )}{\sqrt {-1+c^2 x^2}} \, dx}{768 (-1+c x) (1+c x)}\\ &=-\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2}}{256 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {3 x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{864 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{512 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{256 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {-1+c^2 x^2}} \, dx}{4608 (-1+c x) (1+c x)}\\ &=-\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2}}{1024 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}-\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{288 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1024 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {-1+c^2 x^2}} \, dx}{6144 (-1+c x) (1+c x)}\\ &=\frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}-\frac {5 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{1024 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{576 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2}} \, dx}{12288 c^2 (-1+c x) (1+c x)}\\ &=\frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^2 (-1+c x) (1+c x)}\\ &=\frac {35 b^2 d^2 x \sqrt {d-c^2 d x^2}}{9216 c^2}+\frac {215 b^2 d^2 x^3 \sqrt {d-c^2 d x^2}}{13824}-\frac {5}{864} b^2 c^2 d^2 x^5 \sqrt {d-c^2 d x^2}+\frac {73 b^2 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{12288 c^2 (1-c x) (1+c x)}+\frac {73 b^2 d^2 x^3 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{18432 (1-c x) (1+c x)}-\frac {43 b^2 c^2 d^2 x^5 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{4608 (1-c x) (1+c x)}+\frac {b^2 c^4 d^2 x^7 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{256 (1-c x) (1+c x)}+\frac {35 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{9216 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {59 b c d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{384 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {17 b c^3 d^2 x^6 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{144 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^8 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{48} d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^3 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{384 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {73 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{12288 c^3 (1-c x) (1+c x)}\\ \end {align*}
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Mathematica [A]
time = 3.95, size = 910, normalized size = 1.08 \begin {gather*} -\frac {d^2 \left (34560 a^2 c x \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+34560 a^2 c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^3 x^3 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-271872 a^2 c^4 x^4 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^5 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+313344 a^2 c^6 x^6 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^7 x^7 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}-110592 a^2 c^8 x^8 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}+11520 b^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)^3+34560 a^2 \sqrt {d} \sqrt {\frac {-1+c x}{1+c x}} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+34560 a^2 c \sqrt {d} x \sqrt {\frac {-1+c x}{1+c x}} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+13824 a b \sqrt {d-c^2 d x^2} \cosh \left (2 \cosh ^{-1}(c x)\right )+3456 a b \sqrt {d-c^2 d x^2} \cosh \left (4 \cosh ^{-1}(c x)\right )-1536 a b \sqrt {d-c^2 d x^2} \cosh \left (6 \cosh ^{-1}(c x)\right )+216 a b \sqrt {d-c^2 d x^2} \cosh \left (8 \cosh ^{-1}(c x)\right )-6912 b^2 \sqrt {d-c^2 d x^2} \sinh \left (2 \cosh ^{-1}(c x)\right )-864 b^2 \sqrt {d-c^2 d x^2} \sinh \left (4 \cosh ^{-1}(c x)\right )+256 b^2 \sqrt {d-c^2 d x^2} \sinh \left (6 \cosh ^{-1}(c x)\right )-27 b^2 \sqrt {d-c^2 d x^2} \sinh \left (8 \cosh ^{-1}(c x)\right )+24 b \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) \left (576 b \cosh \left (2 \cosh ^{-1}(c x)\right )+144 b \cosh \left (4 \cosh ^{-1}(c x)\right )-64 b \cosh \left (6 \cosh ^{-1}(c x)\right )+9 b \cosh \left (8 \cosh ^{-1}(c x)\right )-1152 a \sinh \left (2 \cosh ^{-1}(c x)\right )-576 a \sinh \left (4 \cosh ^{-1}(c x)\right )+384 a \sinh \left (6 \cosh ^{-1}(c x)\right )-72 a \sinh \left (8 \cosh ^{-1}(c x)\right )\right )-288 b \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)^2 \left (-120 a+48 b \sinh \left (2 \cosh ^{-1}(c x)\right )+24 b \sinh \left (4 \cosh ^{-1}(c x)\right )-16 b \sinh \left (6 \cosh ^{-1}(c x)\right )+3 b \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{884736 c^3 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2527\) vs.
\(2(741)=1482\).
time = 2.97, size = 2528, normalized size = 3.01
method | result | size |
default | \(\text {Expression too large to display}\) | \(2528\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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