Optimal. Leaf size=269 \[ -\frac {8 \sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 a \left (1-a^2 x^2\right ) \left (3 a^2 c+2 d\right )}{15 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {a \left (1-a^2 x^2\right )}{15 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
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Rubi [A] time = 0.81, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {192, 191, 5705, 12, 519, 6715, 949, 78, 63, 217, 206} \[ \frac {2 a \left (1-a^2 x^2\right ) \left (3 a^2 c+2 d\right )}{15 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}-\frac {8 \sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {a \left (1-a^2 x^2\right )}{15 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 78
Rule 191
Rule 192
Rule 206
Rule 217
Rule 519
Rule 949
Rule 5705
Rule 6715
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx &=\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}} \, dx\\ &=\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3}\\ &=\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\sqrt {-1+a^2 x^2} \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {15 c^2+20 c d x+8 d^2 x^2}{\sqrt {-1+a^2 x} (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {3 c \left (7 a^2 c+6 d\right )+12 d \left (a^2 c+d\right ) x}{\sqrt {-1+a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{45 c^3 \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (4 a \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{15 c^3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (8 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d}{a^2}+\frac {d x^2}{a^2}}} \, dx,x,\sqrt {-1+a^2 x^2}\right )}{15 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (8 \sqrt {-1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{a^2}} \, dx,x,\frac {\sqrt {-1+a^2 x^2}}{\sqrt {c+d x^2}}\right )}{15 a c^3 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \sqrt {-1+a^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {-1+a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d} \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}
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Mathematica [C] time = 3.51, size = 655, normalized size = 2.43 \[ \frac {\frac {16 (a x-1)^{3/2} \left (c+d x^2\right )^2 \sqrt {\frac {(a x+1) \left (a \sqrt {c}-i \sqrt {d}\right )}{(a x-1) \left (a \sqrt {c}+i \sqrt {d}\right )}} \left (a \sqrt {c} \left (-a \sqrt {c}+i \sqrt {d}\right ) \sqrt {\frac {\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (a x-1)^2}} \sqrt {-\frac {a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )+\frac {i \sqrt {d} x}{\sqrt {c}}-1}{1-a x}} \Pi \left (\frac {2 a \sqrt {c}}{\sqrt {c} a+i \sqrt {d}};\sin ^{-1}\left (\sqrt {-\frac {\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )-1}{2-2 a x}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )+\frac {a \left (\sqrt {d}-i a \sqrt {c}\right ) \left (\sqrt {d} x+i \sqrt {c}\right ) \sqrt {\frac {\frac {i a \sqrt {c}}{\sqrt {d}}+a (-x)+\frac {i \sqrt {d} x}{\sqrt {c}}+1}{1-a x}} F\left (\sin ^{-1}\left (\sqrt {-\frac {\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )-1}{2-2 a x}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )}{a x-1}\right )}{a c^3 \sqrt {a x+1} \left (a^2 c+d\right ) \sqrt {-\frac {a \left (x+\frac {i \sqrt {c}}{\sqrt {d}}\right )+\frac {i \sqrt {d} x}{\sqrt {c}}-1}{1-a x}}}-\frac {a \sqrt {a x-1} \sqrt {a x+1} \left (c+d x^2\right ) \left (a^2 c \left (7 c+6 d x^2\right )+d \left (5 c+4 d x^2\right )\right )}{c^2 \left (a^2 c+d\right )^2}+\frac {x \cosh ^{-1}(a x) \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{c^3}}{15 \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 1098, normalized size = 4.08 \[ \left [\frac {2 \, {\left (a^{4} c^{5} + 2 \, a^{2} c^{4} d + {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{6} + c^{3} d^{2} + 3 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{4} + 3 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x^{2}\right )} \sqrt {d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \, {\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c} \sqrt {d} + d^{2}\right ) + {\left (8 \, {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{5} + 20 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{3} + 15 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (7 \, a^{3} c^{4} d + 5 \, a c^{3} d^{2} + 2 \, {\left (3 \, a^{3} c^{2} d^{3} + 2 \, a c d^{4}\right )} x^{4} + {\left (13 \, a^{3} c^{3} d^{2} + 9 \, a c^{2} d^{3}\right )} x^{2}\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c}}{15 \, {\left (a^{4} c^{8} d + 2 \, a^{2} c^{7} d^{2} + c^{6} d^{3} + {\left (a^{4} c^{5} d^{4} + 2 \, a^{2} c^{4} d^{5} + c^{3} d^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{6} d^{3} + 2 \, a^{2} c^{5} d^{4} + c^{4} d^{5}\right )} x^{4} + 3 \, {\left (a^{4} c^{7} d^{2} + 2 \, a^{2} c^{6} d^{3} + c^{5} d^{4}\right )} x^{2}\right )}}, \frac {4 \, {\left (a^{4} c^{5} + 2 \, a^{2} c^{4} d + {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{6} + c^{3} d^{2} + 3 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{4} + 3 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c} \sqrt {-d}}{2 \, {\left (a^{3} d^{2} x^{4} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right ) + {\left (8 \, {\left (a^{4} c^{2} d^{3} + 2 \, a^{2} c d^{4} + d^{5}\right )} x^{5} + 20 \, {\left (a^{4} c^{3} d^{2} + 2 \, a^{2} c^{2} d^{3} + c d^{4}\right )} x^{3} + 15 \, {\left (a^{4} c^{4} d + 2 \, a^{2} c^{3} d^{2} + c^{2} d^{3}\right )} x\right )} \sqrt {d x^{2} + c} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (7 \, a^{3} c^{4} d + 5 \, a c^{3} d^{2} + 2 \, {\left (3 \, a^{3} c^{2} d^{3} + 2 \, a c d^{4}\right )} x^{4} + {\left (13 \, a^{3} c^{3} d^{2} + 9 \, a c^{2} d^{3}\right )} x^{2}\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {d x^{2} + c}}{15 \, {\left (a^{4} c^{8} d + 2 \, a^{2} c^{7} d^{2} + c^{6} d^{3} + {\left (a^{4} c^{5} d^{4} + 2 \, a^{2} c^{4} d^{5} + c^{3} d^{6}\right )} x^{6} + 3 \, {\left (a^{4} c^{6} d^{3} + 2 \, a^{2} c^{5} d^{4} + c^{4} d^{5}\right )} x^{4} + 3 \, {\left (a^{4} c^{7} d^{2} + 2 \, a^{2} c^{6} d^{3} + c^{5} d^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.76, size = 411, normalized size = 1.53 \[ \frac {4}{15} \, a {\left (\frac {{\left | d \right |} \log \left ({\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2}\right )}{c^{3} d^{\frac {3}{2}} {\left | a \right |}} - \frac {3 \, a^{6} c^{2} d^{\frac {5}{2}} {\left | d \right |} + 7 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{4} c d^{\frac {3}{2}} {\left | d \right |} + 5 \, a^{4} c d^{\frac {7}{2}} {\left | d \right |} + 2 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{2} \sqrt {d} {\left | d \right |} + 4 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{2} d^{\frac {5}{2}} {\left | d \right |} + 2 \, a^{2} d^{\frac {9}{2}} {\left | d \right |}}{{\left (a^{2} c d + {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} + d^{2}\right )}^{3} c^{2} d {\left | a \right |}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{15 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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