Optimal. Leaf size=211 \[ -\frac {8 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d}}-\frac {2 a \sqrt {1-a^2 x^2} \left (3 a^2 c+2 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
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Rubi [A] time = 0.85, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {192, 191, 4666, 12, 6715, 949, 78, 63, 217, 203} \[ -\frac {2 a \sqrt {1-a^2 x^2} \left (3 a^2 c+2 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}-\frac {8 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d}}-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 78
Rule 191
Rule 192
Rule 203
Rule 217
Rule 949
Rule 4666
Rule 6715
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx &=\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-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^2 x^2} \left (c+d x^2\right )^{5/2}} \, dx\\ &=\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-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^2 x^2} \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3}\\ &=\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {a \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}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \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 )}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {(4 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{15 c^3}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \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}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \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}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {2 a \left (3 a^2 c+2 d\right ) \sqrt {1-a^2 x^2}}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \cos ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \cos ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {8 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{15 c^3 \sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 0.35, size = 162, normalized size = 0.77 \[ \frac {4 a x^2 \sqrt {\frac {d x^2}{c}+1} \left (c+d x^2\right )^2 F_1\left (1;\frac {1}{2},\frac {1}{2};2;a^2 x^2,-\frac {d x^2}{c}\right )-\frac {a c \sqrt {1-a^2 x^2} \left (c+d x^2\right ) \left (a^2 c \left (7 c+6 d x^2\right )+d \left (5 c+4 d x^2\right )\right )}{\left (a^2 c+d\right )^2}+x \cos ^{-1}(a x) \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \left (c+d x^2\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.61, size = 1066, normalized size = 5.05 \[ \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 ) - \sqrt {d x^{2} + c} {\left ({\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 )} \arccos \left (a x\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}\right )}}{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 ) - \sqrt {d x^{2} + c} {\left ({\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 )} \arccos \left (a x\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}\right )}}{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 [B] time = 2.42, size = 398, normalized size = 1.89 \[ -\frac {4}{15} \, a {\left (\frac {{\left | d \right |} \log \left ({\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{2}\right )}{c^{3} \sqrt {-d} d {\left | a \right |}} - \frac {3 \, a^{6} c^{2} d^{2} {\left | d \right |} - 7 \, {\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{2} a^{4} c d {\left | d \right |} + 5 \, a^{4} c d^{3} {\left | d \right |} + 2 \, {\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{4} a^{2} {\left | d \right |} - 4 \, {\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{2} a^{2} d^{2} {\left | d \right |} + 2 \, a^{2} d^{4} {\left | d \right |}}{{\left (a^{2} c d - {\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{2} + d^{2}\right )}^{3} c^{2} \sqrt {-d} {\left | a \right |}}\right )} + \frac {{\left (4 \, x^{2} {\left (\frac {2 \, d^{2} x^{2}}{c^{3}} + \frac {5 \, d}{c^{2}}\right )} + \frac {15}{c}\right )} x \arccos \left (a x\right )}{15 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \frac {\arccos \left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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