Optimal. Leaf size=880 \[ -\frac {37384 b^2 d^2 \sqrt {d-c^2 d x^2}}{694575 c^4}+\frac {3358 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{694575 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac {20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt {d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d x^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.23, antiderivative size = 880, normalized size of antiderivative = 1.00, number of steps
used = 35, number of rules used = 18, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {5930, 5926,
5939, 5915, 5879, 75, 5883, 102, 12, 5912, 14, 5921, 471, 276, 534, 1265, 911, 1167}
\begin {gather*} -\frac {2 b c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^9}{81 \sqrt {c x-1} \sqrt {c x+1}}+\frac {38 b c^3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^7}{441 \sqrt {c x-1} \sqrt {c x+1}}-\frac {10 b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^6}{3087}-\frac {2 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^5}{21 \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{9} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac {5}{63} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac {1}{21} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^4+\frac {484 b^2 d^2 \sqrt {d-c^2 d x^2} x^4}{77175}+\frac {2 b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) x^3}{189 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 x^2}{63 c^2}+\frac {3358 b^2 d^2 \sqrt {d-c^2 d x^2} x^2}{694575 c^2}+\frac {4 b^2 d^2 \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) x}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 a b d^2 \sqrt {d-c^2 d x^2} x}{63 c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt {d-c^2 d x^2}}{729 c^4 (1-c x) (c x+1)}-\frac {20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{3969 c^4 (1-c x) (c x+1)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{4725 c^4 (1-c x) (c x+1)}-\frac {37384 b^2 d^2 \sqrt {d-c^2 d x^2}}{694575 c^4}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{8505 c^4 (1-c x) (c x+1)}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2835 c^4 (1-c x) (c x+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 75
Rule 102
Rule 276
Rule 471
Rule 534
Rule 911
Rule 1167
Rule 1265
Rule 5879
Rule 5883
Rule 5912
Rule 5915
Rule 5921
Rule 5926
Rule 5930
Rule 5939
Rubi steps
\begin {align*} \int x^3 \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^3 (-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}{9} d^2 x^4 (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^3 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (10 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{63 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{21 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2835 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (10 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (-7+5 c^2 x^2\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{63 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{63 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{63 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (-7+5 c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{441 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{2835 (-1+c x) (1+c x)}\\ &=\frac {2}{525} b^2 d^2 x^4 \sqrt {d-c^2 d x^2}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{525 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{189 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (38 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3087 \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 ) \text {Subst}\left (\int \frac {x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{2835 (-1+c x) (1+c x)}\\ &=-\frac {2 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{567 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (38 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {4 x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15435 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (8 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{525 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{567 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}+\frac {x^2}{c^2}\right )^2 \left (8-20 x^2+35 x^4\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{2835 (-1+c x) (1+c x)}\\ &=\frac {22 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{14175 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (152 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{15435 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (8 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1575 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{567 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (4 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{63 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{c^4}-\frac {4 x^2}{c^4}+\frac {3 x^4}{c^4}+\frac {50 x^6}{c^4}+\frac {35 x^8}{c^4}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{2835 (-1+c x) (1+c x)}\\ &=-\frac {40 b^2 d^2 \sqrt {d-c^2 d x^2}}{567 c^4}+\frac {3358 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{694575 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac {20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt {d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (152 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{46305 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (16 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{1575 c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {856 b^2 d^2 \sqrt {d-c^2 d x^2}}{14175 c^4}+\frac {3358 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{694575 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac {20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt {d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (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 (304 b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{46305 c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {37384 b^2 d^2 \sqrt {d-c^2 d x^2}}{694575 c^4}+\frac {3358 b^2 d^2 x^2 \sqrt {d-c^2 d x^2}}{694575 c^2}+\frac {484 b^2 d^2 x^4 \sqrt {d-c^2 d x^2}}{77175}-\frac {10 b^2 c^2 d^2 x^6 \sqrt {d-c^2 d x^2}}{3087}+\frac {4 a b d^2 x \sqrt {d-c^2 d x^2}}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{2835 c^4 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{8505 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{4725 c^4 (1-c x) (1+c x)}-\frac {20 b^2 d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{3969 c^4 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^5 \sqrt {d-c^2 d x^2}}{729 c^4 (1-c x) (1+c x)}+\frac {4 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{63 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{189 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {38 b c^3 d^2 x^7 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{441 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^9 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^4}-\frac {d^2 x^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{63 c^2}+\frac {1}{21} d^2 x^4 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {5}{63} d^2 x^4 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{9} d^2 x^4 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 288, normalized size = 0.33 \begin {gather*} \frac {d^2 \sqrt {d-c^2 d x^2} \left (3969 a^2 \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right )-126 a b c x \sqrt {-1+c x} \sqrt {1+c x} \left (-126-21 c^2 x^2+189 c^4 x^4-171 c^6 x^6+49 c^8 x^8\right )+2 b^2 \left (6140-7039 c^2 x^2-106 c^4 x^4+2152 c^6 x^6-1490 c^8 x^8+343 c^{10} x^{10}\right )+126 b \left (63 a \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right )+b c x \sqrt {-1+c x} \sqrt {1+c x} \left (126+21 c^2 x^2-189 c^4 x^4+171 c^6 x^6-49 c^8 x^8\right )\right ) \cosh ^{-1}(c x)+3969 b^2 \left (-1+c^2 x^2\right )^4 \left (2+7 c^2 x^2\right ) \cosh ^{-1}(c x)^2\right )}{250047 c^4 \left (-1+c^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2223\) vs.
\(2(776)=1552\).
time = 2.39, size = 2224, normalized size = 2.53
method | result | size |
default | \(\text {Expression too large to display}\) | \(2224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 471, normalized size = 0.54 \begin {gather*} -\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^{2} \operatorname {arcosh}\left (c x\right )^{2} - \frac {2}{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 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^{2} + \frac {2}{250047} \, b^{2} {\left (\frac {343 \, \sqrt {c^{2} x^{2} - 1} c^{6} \sqrt {-d} d^{2} x^{8} - 1147 \, \sqrt {c^{2} x^{2} - 1} c^{4} \sqrt {-d} d^{2} x^{6} + 1005 \, \sqrt {c^{2} x^{2} - 1} c^{2} \sqrt {-d} d^{2} x^{4} + 899 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2} x^{2} - \frac {6140 \, \sqrt {c^{2} x^{2} - 1} \sqrt {-d} d^{2}}{c^{2}}}{c^{2}} - \frac {63 \, {\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 )} \operatorname {arcosh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\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 )} a b}{3969 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 558, normalized size = 0.63 \begin {gather*} \frac {3969 \, {\left (7 \, b^{2} c^{10} d^{2} x^{10} - 26 \, b^{2} c^{8} d^{2} x^{8} + 34 \, b^{2} c^{6} d^{2} x^{6} - 16 \, b^{2} c^{4} d^{2} x^{4} - b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )^{2} - 126 \, {\left (49 \, a b c^{9} d^{2} x^{9} - 171 \, a b c^{7} d^{2} x^{7} + 189 \, a b c^{5} d^{2} x^{5} - 21 \, a b c^{3} d^{2} x^{3} - 126 \, a b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 126 \, {\left ({\left (49 \, b^{2} c^{9} d^{2} x^{9} - 171 \, b^{2} c^{7} d^{2} x^{7} + 189 \, b^{2} c^{5} d^{2} x^{5} - 21 \, b^{2} c^{3} d^{2} x^{3} - 126 \, b^{2} c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 63 \, {\left (7 \, a b c^{10} d^{2} x^{10} - 26 \, a b c^{8} d^{2} x^{8} + 34 \, a b c^{6} d^{2} x^{6} - 16 \, a b c^{4} d^{2} x^{4} - a b c^{2} d^{2} x^{2} + 2 \, a b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (343 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{10} d^{2} x^{10} - 2 \, {\left (51597 \, a^{2} + 1490 \, b^{2}\right )} c^{8} d^{2} x^{8} + 2 \, {\left (67473 \, a^{2} + 2152 \, b^{2}\right )} c^{6} d^{2} x^{6} - 4 \, {\left (15876 \, a^{2} + 53 \, b^{2}\right )} c^{4} d^{2} x^{4} - {\left (3969 \, a^{2} + 14078 \, b^{2}\right )} c^{2} d^{2} x^{2} + 2 \, {\left (3969 \, a^{2} + 6140 \, b^{2}\right )} d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{250047 \, {\left (c^{6} x^{2} - c^{4}\right )}} \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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^3\,{\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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