Optimal. Leaf size=177 \[ -\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{9/4} d}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{9/4} d}+\frac {(a+b) \cos (c+d x)}{b^2 d}-\frac {2 \cos ^3(c+d x)}{3 b d}+\frac {\cos ^5(c+d x)}{5 b d} \]
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Rubi [A]
time = 0.18, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3294, 1184,
1107, 211, 214} \begin {gather*} -\frac {a^{3/2} \text {ArcTan}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 b^{9/4} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 b^{9/4} d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {(a+b) \cos (c+d x)}{b^2 d}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {2 \cos ^3(c+d x)}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 1107
Rule 1184
Rule 3294
Rubi steps
\begin {align*} \int \frac {\sin ^9(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^4}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (-\frac {a+b}{b^2}+\frac {2 x^2}{b}-\frac {x^4}{b}+\frac {a^2}{b^2 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {(a+b) \cos (c+d x)}{b^2 d}-\frac {2 \cos ^3(c+d x)}{3 b d}+\frac {\cos ^5(c+d x)}{5 b d}-\frac {a^2 \text {Subst}\left (\int \frac {1}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{b^2 d}\\ &=\frac {(a+b) \cos (c+d x)}{b^2 d}-\frac {2 \cos ^3(c+d x)}{3 b d}+\frac {\cos ^5(c+d x)}{5 b d}+\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 b^{3/2} d}-\frac {a^{3/2} \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 b^{3/2} d}\\ &=-\frac {a^{3/2} \tan ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^{9/4} d}-\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^{9/4} d}+\frac {(a+b) \cos (c+d x)}{b^2 d}-\frac {2 \cos ^3(c+d x)}{3 b d}+\frac {\cos ^5(c+d x)}{5 b d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.34, size = 228, normalized size = 1.29 \begin {gather*} \frac {\cos (c+d x) (120 a+89 b-28 b \cos (2 (c+d x))+3 b \cos (4 (c+d x)))+60 i a^2 \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^3-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^3}{-b-8 a \text {$\#$1}^2+3 b \text {$\#$1}^2-3 b \text {$\#$1}^4+b \text {$\#$1}^6}\&\right ]}{120 b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 137, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) b}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) b}{3}+a \cos \left (d x +c \right )+\cos \left (d x +c \right ) b}{b^{2}}+\frac {a^{2} \left (-\frac {\arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{b}}{d}\) | \(137\) |
default | \(\frac {\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) b}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) b}{3}+a \cos \left (d x +c \right )+\cos \left (d x +c \right ) b}{b^{2}}+\frac {a^{2} \left (-\frac {\arctanh \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}+b \right ) b}}-\frac {\arctan \left (\frac {b \cos \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 \sqrt {a b}\, \sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{b}}{d}\) | \(137\) |
risch | \(\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}+\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{16 b d}+\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}+\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 b d}-\frac {i \left (\munderset {\textit {\_R} =\RootOf \left (\left (a \,b^{9} d^{4}-b^{10} d^{4}\right ) \textit {\_Z}^{4}-32768 a^{3} b^{5} d^{2} \textit {\_Z}^{2}-268435456 a^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {i b^{7} d^{3}}{1048576 a^{4}}+\frac {i b^{8} d^{3}}{1048576 a^{5}}\right ) \textit {\_R}^{3}+\left (\frac {i b^{2} d}{64 a}+\frac {i b^{3} d}{64 a^{2}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )}{512}+\frac {\cos \left (5 d x +5 c \right )}{80 b d}-\frac {5 \cos \left (3 d x +3 c \right )}{48 b d}\) | \(231\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 872 vs.
\(2 (133) = 266\).
time = 0.48, size = 872, normalized size = 4.93 \begin {gather*} \frac {12 \, b \cos \left (d x + c\right )^{5} - 15 \, b^{2} d \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (a^{5} \cos \left (d x + c\right ) + {\left (a^{4} b^{2} d - {\left (a b^{7} - b^{8}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{3}\right )} \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) + 15 \, b^{2} d \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (a^{5} \cos \left (d x + c\right ) - {\left (a^{4} b^{2} d + {\left (a b^{7} - b^{8}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{3}\right )} \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) + 15 \, b^{2} d \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (-a^{5} \cos \left (d x + c\right ) + {\left (a^{4} b^{2} d - {\left (a b^{7} - b^{8}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{3}\right )} \sqrt {-\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} + a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) - 15 \, b^{2} d \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}} \log \left (-a^{5} \cos \left (d x + c\right ) - {\left (a^{4} b^{2} d + {\left (a b^{7} - b^{8}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{3}\right )} \sqrt {\frac {{\left (a b^{4} - b^{5}\right )} \sqrt {\frac {a^{7}}{{\left (a^{2} b^{9} - 2 \, a b^{10} + b^{11}\right )} d^{4}}} d^{2} - a^{3}}{{\left (a b^{4} - b^{5}\right )} d^{2}}}\right ) - 40 \, b \cos \left (d x + c\right )^{3} + 60 \, {\left (a + b\right )} \cos \left (d x + c\right )}{60 \, b^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 14.44, size = 1067, normalized size = 6.03 \begin {gather*} \frac {{\cos \left (c+d\,x\right )}^5}{5\,b\,d}-\frac {2\,{\cos \left (c+d\,x\right )}^3}{3\,b\,d}+\frac {\cos \left (c+d\,x\right )\,\left (\frac {a-b}{b^2}+\frac {2}{b}\right )}{d}+\frac {\mathrm {atan}\left (-\frac {a^4\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^9}}{16\,\left (a\,b^9-b^{10}\right )}-\frac {a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,8{}\mathrm {i}}{\frac {2\,a^6\,b^7}{a\,b^9-b^{10}}+\frac {2\,a^3\,b^2\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}}+\frac {a^4\,b^9\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^9}}{16\,\left (a\,b^9-b^{10}\right )}-\frac {a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,8{}\mathrm {i}}{\frac {2\,a^6\,b^{16}}{a\,b^9-b^{10}}-\frac {2\,a^7\,b^{15}}{a\,b^9-b^{10}}+\frac {2\,a^3\,b^{11}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}-\frac {2\,a^4\,b^{10}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}}+\frac {a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {-\frac {\sqrt {a^7\,b^9}}{16\,\left (a\,b^9-b^{10}\right )}-\frac {a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,\sqrt {a^7\,b^9}\,8{}\mathrm {i}}{\frac {2\,a^6\,b^{16}}{a\,b^9-b^{10}}-\frac {2\,a^7\,b^{15}}{a\,b^9-b^{10}}+\frac {2\,a^3\,b^{11}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}-\frac {2\,a^4\,b^{10}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}}\right )\,\sqrt {-\frac {\sqrt {a^7\,b^9}+a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,2{}\mathrm {i}}{d}-\frac {\mathrm {atan}\left (\frac {a^4\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^9}}{16\,\left (a\,b^9-b^{10}\right )}-\frac {a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,8{}\mathrm {i}}{\frac {2\,a^6\,b^7}{a\,b^9-b^{10}}-\frac {2\,a^3\,b^2\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}}-\frac {a^4\,b^9\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^9}}{16\,\left (a\,b^9-b^{10}\right )}-\frac {a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,8{}\mathrm {i}}{\frac {2\,a^6\,b^{16}}{a\,b^9-b^{10}}-\frac {2\,a^7\,b^{15}}{a\,b^9-b^{10}}-\frac {2\,a^3\,b^{11}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}+\frac {2\,a^4\,b^{10}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}}+\frac {a\,b^4\,\cos \left (c+d\,x\right )\,\sqrt {\frac {\sqrt {a^7\,b^9}}{16\,\left (a\,b^9-b^{10}\right )}-\frac {a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,\sqrt {a^7\,b^9}\,8{}\mathrm {i}}{\frac {2\,a^6\,b^{16}}{a\,b^9-b^{10}}-\frac {2\,a^7\,b^{15}}{a\,b^9-b^{10}}-\frac {2\,a^3\,b^{11}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}+\frac {2\,a^4\,b^{10}\,\sqrt {a^7\,b^9}}{a\,b^9-b^{10}}}\right )\,\sqrt {\frac {\sqrt {a^7\,b^9}-a^3\,b^5}{16\,\left (a\,b^9-b^{10}\right )}}\,2{}\mathrm {i}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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