Optimal. Leaf size=291 \[ \frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{35} d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{35} d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}-\frac {12 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac {32 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}+\frac {16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{343} b^2 c^6 d^3 x^7+\frac {234 b^2 c^4 d^3 x^5}{6125}+\frac {1514 b^2 c^2 d^3 x^3}{11025}+\frac {4322 b^2 d^3 x}{3675} \]
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Rubi [A] time = 0.40, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5684, 5653, 5717, 8, 194} \[ \frac {1}{7} d^3 x \left (c^2 x^2+1\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{35} d^3 x \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{35} d^3 x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}-\frac {12 b d^3 \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac {32 b d^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}+\frac {16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{343} b^2 c^6 d^3 x^7+\frac {234 b^2 c^4 d^3 x^5}{6125}+\frac {1514 b^2 c^2 d^3 x^3}{11025}+\frac {4322 b^2 d^3 x}{3675} \]
Antiderivative was successfully verified.
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Rule 8
Rule 194
Rule 5653
Rule 5684
Rule 5717
Rubi steps
\begin {align*} \int \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} (6 d) \int \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{7} \left (2 b c d^3\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac {6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{35} \left (24 d^2\right ) \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {1}{49} \left (2 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^3 \, dx-\frac {1}{35} \left (12 b c d^3\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac {8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{35} \left (16 d^3\right ) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {1}{49} \left (2 b^2 d^3\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx+\frac {1}{175} \left (12 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac {1}{35} \left (16 b c d^3\right ) \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=\frac {2}{49} b^2 d^3 x+\frac {2}{49} b^2 c^2 d^3 x^3+\frac {6}{245} b^2 c^4 d^3 x^5+\frac {2}{343} b^2 c^6 d^3 x^7-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac {12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac {16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{175} \left (12 b^2 d^3\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{105} \left (16 b^2 d^3\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{35} \left (32 b c d^3\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {962 b^2 d^3 x}{3675}+\frac {1514 b^2 c^2 d^3 x^3}{11025}+\frac {234 b^2 c^4 d^3 x^5}{6125}+\frac {2}{343} b^2 c^6 d^3 x^7-\frac {32 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac {12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac {16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{35} \left (32 b^2 d^3\right ) \int 1 \, dx\\ &=\frac {4322 b^2 d^3 x}{3675}+\frac {1514 b^2 c^2 d^3 x^3}{11025}+\frac {234 b^2 c^4 d^3 x^5}{6125}+\frac {2}{343} b^2 c^6 d^3 x^7-\frac {32 b d^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{35 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{105 c}-\frac {12 b d^3 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{175 c}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{49 c}+\frac {16}{35} d^3 x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {8}{35} d^3 x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {6}{35} d^3 x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{7} d^3 x \left (1+c^2 x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A] time = 0.56, size = 239, normalized size = 0.82 \[ \frac {d^3 \left (11025 a^2 c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )-210 a b \sqrt {c^2 x^2+1} \left (75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right )-210 b \sinh ^{-1}(c x) \left (b \sqrt {c^2 x^2+1} \left (75 c^6 x^6+351 c^4 x^4+757 c^2 x^2+2161\right )-105 a c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right )\right )+2 b^2 c x \left (1125 c^6 x^6+7371 c^4 x^4+26495 c^2 x^2+226905\right )+11025 b^2 c x \left (5 c^6 x^6+21 c^4 x^4+35 c^2 x^2+35\right ) \sinh ^{-1}(c x)^2\right )}{385875 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 354, normalized size = 1.22 \[ \frac {1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (1225 \, a^{2} + 78 \, b^{2}\right )} c^{5} d^{3} x^{5} + 35 \, {\left (11025 \, a^{2} + 1514 \, b^{2}\right )} c^{3} d^{3} x^{3} + 105 \, {\left (3675 \, a^{2} + 4322 \, b^{2}\right )} c d^{3} x + 11025 \, {\left (5 \, b^{2} c^{7} d^{3} x^{7} + 21 \, b^{2} c^{5} d^{3} x^{5} + 35 \, b^{2} c^{3} d^{3} x^{3} + 35 \, b^{2} c d^{3} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (525 \, a b c^{7} d^{3} x^{7} + 2205 \, a b c^{5} d^{3} x^{5} + 3675 \, a b c^{3} d^{3} x^{3} + 3675 \, a b c d^{3} x - {\left (75 \, b^{2} c^{6} d^{3} x^{6} + 351 \, b^{2} c^{4} d^{3} x^{4} + 757 \, b^{2} c^{2} d^{3} x^{2} + 2161 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 210 \, {\left (75 \, a b c^{6} d^{3} x^{6} + 351 \, a b c^{4} d^{3} x^{4} + 757 \, a b c^{2} d^{3} x^{2} + 2161 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{385875 \, c} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 354, normalized size = 1.22 \[ \frac {d^{3} a^{2} \left (\frac {1}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+c^{3} x^{3}+c x \right )+d^{3} b^{2} \left (\frac {16 \arcsinh \left (c x \right )^{2} c x}{35}+\frac {\arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{7}+\frac {6 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{35}+\frac {8 \arcsinh \left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{35}-\frac {32 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{35}+\frac {413312 c x}{385875}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{49}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{3}}{343}+\frac {888 c x \left (c^{2} x^{2}+1\right )^{2}}{42875}+\frac {30256 c x \left (c^{2} x^{2}+1\right )}{385875}-\frac {12 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{175}-\frac {16 \arcsinh \left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{105}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \arcsinh \left (c x \right ) c^{5} x^{5}}{5}+\arcsinh \left (c x \right ) c^{3} x^{3}+\arcsinh \left (c x \right ) c x -\frac {c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{49}-\frac {117 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1225}-\frac {757 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3675}-\frac {2161 \sqrt {c^{2} x^{2}+1}}{3675}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 712, normalized size = 2.45 \[ \frac {1}{7} \, b^{2} c^{6} d^{3} x^{7} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{7} \, a^{2} c^{6} d^{3} x^{7} + \frac {3}{5} \, b^{2} c^{4} d^{3} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {3}{5} \, a^{2} c^{4} d^{3} x^{5} + \frac {2}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{6} d^{3} - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {75 \, c^{6} x^{7} - 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} - 1680 \, x}{c^{6}}\right )} b^{2} c^{6} d^{3} + b^{2} c^{2} d^{3} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{25} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{3} - \frac {2}{375} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{3} + a^{2} c^{2} d^{3} x^{3} + \frac {2}{3} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{3} - \frac {2}{9} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{3} + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.36, size = 524, normalized size = 1.80 \[ \begin {cases} \frac {a^{2} c^{6} d^{3} x^{7}}{7} + \frac {3 a^{2} c^{4} d^{3} x^{5}}{5} + a^{2} c^{2} d^{3} x^{3} + a^{2} d^{3} x + \frac {2 a b c^{6} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {2 a b c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{49} + \frac {6 a b c^{4} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {234 a b c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1225} + 2 a b c^{2} d^{3} x^{3} \operatorname {asinh}{\left (c x \right )} - \frac {1514 a b c d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{3675} + 2 a b d^{3} x \operatorname {asinh}{\left (c x \right )} - \frac {4322 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{3675 c} + \frac {b^{2} c^{6} d^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{6} d^{3} x^{7}}{343} - \frac {2 b^{2} c^{5} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{49} + \frac {3 b^{2} c^{4} d^{3} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {234 b^{2} c^{4} d^{3} x^{5}}{6125} - \frac {234 b^{2} c^{3} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1225} + b^{2} c^{2} d^{3} x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {1514 b^{2} c^{2} d^{3} x^{3}}{11025} - \frac {1514 b^{2} c d^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {4322 b^{2} d^{3} x}{3675} - \frac {4322 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3675 c} & \text {for}\: c \neq 0 \\a^{2} d^{3} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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