Optimal. Leaf size=94 \[ -\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}-\frac {\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac {(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac {\cot (x) \csc ^3(x)}{4 (a+b)} \]
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Rubi [A]
time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3269, 425, 541,
536, 212, 211} \begin {gather*} -\frac {\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac {b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}-\frac {\cot (x) \csc ^3(x)}{4 (a+b)}-\frac {(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 425
Rule 536
Rule 541
Rule 3269
Rubi steps
\begin {align*} \int \frac {\csc ^5(x)}{a+b \cos ^2(x)} \, dx &=-\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\frac {\cot (x) \csc ^3(x)}{4 (a+b)}-\frac {\text {Subst}\left (\int \frac {3 a+4 b+3 b x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\cos (x)\right )}{4 (a+b)}\\ &=-\frac {(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac {\cot (x) \csc ^3(x)}{4 (a+b)}-\frac {\text {Subst}\left (\int \frac {3 a^2+7 a b+8 b^2+b (3 a+7 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\cos (x)\right )}{8 (a+b)^2}\\ &=-\frac {(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac {\cot (x) \csc ^3(x)}{4 (a+b)}-\frac {b^3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\cos (x)\right )}{(a+b)^3}-\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (x)\right )}{8 (a+b)^3}\\ &=-\frac {b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \cos (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}-\frac {\left (3 a^2+10 a b+15 b^2\right ) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac {(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}-\frac {\cot (x) \csc ^3(x)}{4 (a+b)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(204\) vs. \(2(94)=188\).
time = 1.50, size = 204, normalized size = 2.17 \begin {gather*} \frac {-64 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b}-\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a}}\right )-64 b^{5/2} \text {ArcTan}\left (\frac {\sqrt {b}+\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a}}\right )+\sqrt {a} \left (-2 \left (3 a^2+10 a b+7 b^2\right ) \csc ^2\left (\frac {x}{2}\right )-(a+b)^2 \csc ^4\left (\frac {x}{2}\right )-8 \left (3 a^2+10 a b+15 b^2\right ) \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+2 \left (3 a^2+10 a b+7 b^2\right ) \sec ^2\left (\frac {x}{2}\right )+(a+b)^2 \sec ^4\left (\frac {x}{2}\right )\right )}{64 \sqrt {a} (a+b)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 155, normalized size = 1.65
method | result | size |
default | \(\frac {1}{2 \left (8 a +8 b \right ) \left (\cos \left (x \right )+1\right )^{2}}-\frac {-3 a -7 b}{16 \left (a +b \right )^{2} \left (\cos \left (x \right )+1\right )}+\frac {\left (-3 a^{2}-10 a b -15 b^{2}\right ) \ln \left (\cos \left (x \right )+1\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a +8 b \right ) \left (-1+\cos \left (x \right )\right )^{2}}-\frac {-3 a -7 b}{16 \left (a +b \right )^{2} \left (-1+\cos \left (x \right )\right )}+\frac {\left (3 a^{2}+10 a b +15 b^{2}\right ) \ln \left (-1+\cos \left (x \right )\right )}{16 \left (a +b \right )^{3}}-\frac {b^{3} \arctan \left (\frac {b \cos \left (x \right )}{\sqrt {a b}}\right )}{\left (a +b \right )^{3} \sqrt {a b}}\) | \(155\) |
risch | \(\frac {3 a \,{\mathrm e}^{7 i x}+7 b \,{\mathrm e}^{7 i x}-11 a \,{\mathrm e}^{5 i x}-15 b \,{\mathrm e}^{5 i x}-11 a \,{\mathrm e}^{3 i x}-15 b \,{\mathrm e}^{3 i x}+3 a \,{\mathrm e}^{i x}+7 b \,{\mathrm e}^{i x}}{4 \left (a +b \right )^{2} \left ({\mathrm e}^{2 i x}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i x}-1\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {5 \ln \left ({\mathrm e}^{i x}-1\right ) a b}{4 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{i x}-1\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {3 \ln \left ({\mathrm e}^{i x}+1\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {5 \ln \left ({\mathrm e}^{i x}+1\right ) a b}{4 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {15 \ln \left ({\mathrm e}^{i x}+1\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {i \sqrt {a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {a b}\, {\mathrm e}^{i x}}{b}+1\right )}{2 a \left (a +b \right )^{3}}-\frac {i \sqrt {a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {a b}\, {\mathrm e}^{i x}}{b}+1\right )}{2 a \left (a +b \right )^{3}}\) | \(371\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 200 vs.
\(2 (80) = 160\).
time = 0.47, size = 200, normalized size = 2.13 \begin {gather*} -\frac {b^{3} \arctan \left (\frac {b \cos \left (x\right )}{\sqrt {a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b}} - \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (x\right ) + 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (x\right ) - 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (3 \, a + 7 \, b\right )} \cos \left (x\right )^{3} - {\left (5 \, a + 9 \, b\right )} \cos \left (x\right )}{8 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (x\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )}} \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. 283 vs.
\(2 (80) = 160\).
time = 0.49, size = 592, normalized size = 6.30 \begin {gather*} \left [\frac {2 \, {\left (3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right )} \cos \left (x\right )^{3} + 8 \, {\left (b^{2} \cos \left (x\right )^{4} - 2 \, b^{2} \cos \left (x\right )^{2} + b^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b \cos \left (x\right )^{2} - 2 \, a \sqrt {-\frac {b}{a}} \cos \left (x\right ) - a}{b \cos \left (x\right )^{2} + a}\right ) - 2 \, {\left (5 \, a^{2} + 14 \, a b + 9 \, b^{2}\right )} \cos \left (x\right ) - {\left ({\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{2} + 3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{2} + 3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{16 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )}}, \frac {2 \, {\left (3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right )} \cos \left (x\right )^{3} - 16 \, {\left (b^{2} \cos \left (x\right )^{4} - 2 \, b^{2} \cos \left (x\right )^{2} + b^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \cos \left (x\right )\right ) - 2 \, {\left (5 \, a^{2} + 14 \, a b + 9 \, b^{2}\right )} \cos \left (x\right ) - {\left ({\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{2} + 3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} - 2 \, {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{2} + 3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{16 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3} - 2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2}\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\csc ^{5}{\left (x \right )}}{a + b \cos ^{2}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 178 vs.
\(2 (80) = 160\).
time = 0.40, size = 178, normalized size = 1.89 \begin {gather*} -\frac {b^{3} \arctan \left (\frac {b \cos \left (x\right )}{\sqrt {a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b}} - \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (x\right ) + 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (-\cos \left (x\right ) + 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {3 \, a \cos \left (x\right )^{3} + 7 \, b \cos \left (x\right )^{3} - 5 \, a \cos \left (x\right ) - 9 \, b \cos \left (x\right )}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (\cos \left (x\right )^{2} - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.28, size = 833, normalized size = 8.86 \begin {gather*} -\frac {3\,a^3\,\mathrm {atanh}\left (\cos \left (x\right )\right )-3\,a^3\,{\cos \left (x\right )}^3+5\,a^3\,\cos \left (x\right )+9\,a\,b^2\,\cos \left (x\right )+14\,a^2\,b\,\cos \left (x\right )-6\,a^3\,\mathrm {atanh}\left (\cos \left (x\right )\right )\,{\cos \left (x\right )}^2+3\,a^3\,\mathrm {atanh}\left (\cos \left (x\right )\right )\,{\cos \left (x\right )}^4-7\,a\,b^2\,{\cos \left (x\right )}^3-10\,a^2\,b\,{\cos \left (x\right )}^3+15\,a\,b^2\,\mathrm {atanh}\left (\cos \left (x\right )\right )+10\,a^2\,b\,\mathrm {atanh}\left (\cos \left (x\right )\right )-30\,a\,b^2\,\mathrm {atanh}\left (\cos \left (x\right )\right )\,{\cos \left (x\right )}^2-20\,a^2\,b\,\mathrm {atanh}\left (\cos \left (x\right )\right )\,{\cos \left (x\right )}^2+15\,a\,b^2\,\mathrm {atanh}\left (\cos \left (x\right )\right )\,{\cos \left (x\right )}^4+10\,a^2\,b\,\mathrm {atanh}\left (\cos \left (x\right )\right )\,{\cos \left (x\right )}^4+\mathrm {atan}\left (\frac {a\,\cos \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}-b\,\cos \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}+a^6\,b\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,9{}\mathrm {i}+a^2\,b^5\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,289{}\mathrm {i}+a^3\,b^4\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,300{}\mathrm {i}+a^4\,b^3\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,190{}\mathrm {i}+a^5\,b^2\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,60{}\mathrm {i}}{9\,a^7\,b^3+60\,a^6\,b^4+190\,a^5\,b^5+300\,a^4\,b^6+225\,a^3\,b^7+64\,a^2\,b^8}\right )\,\sqrt {-a\,b^5}\,8{}\mathrm {i}-\mathrm {atan}\left (\frac {a\,\cos \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}-b\,\cos \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}+a^6\,b\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,9{}\mathrm {i}+a^2\,b^5\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,289{}\mathrm {i}+a^3\,b^4\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,300{}\mathrm {i}+a^4\,b^3\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,190{}\mathrm {i}+a^5\,b^2\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,60{}\mathrm {i}}{9\,a^7\,b^3+60\,a^6\,b^4+190\,a^5\,b^5+300\,a^4\,b^6+225\,a^3\,b^7+64\,a^2\,b^8}\right )\,{\cos \left (x\right )}^2\,\sqrt {-a\,b^5}\,16{}\mathrm {i}+\mathrm {atan}\left (\frac {a\,\cos \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}-b\,\cos \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}+a^6\,b\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,9{}\mathrm {i}+a^2\,b^5\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,289{}\mathrm {i}+a^3\,b^4\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,300{}\mathrm {i}+a^4\,b^3\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,190{}\mathrm {i}+a^5\,b^2\,\cos \left (x\right )\,\sqrt {-a\,b^5}\,60{}\mathrm {i}}{9\,a^7\,b^3+60\,a^6\,b^4+190\,a^5\,b^5+300\,a^4\,b^6+225\,a^3\,b^7+64\,a^2\,b^8}\right )\,{\cos \left (x\right )}^4\,\sqrt {-a\,b^5}\,8{}\mathrm {i}}{8\,a^4\,{\cos \left (x\right )}^4-16\,a^4\,{\cos \left (x\right )}^2+8\,a^4+24\,a^3\,b\,{\cos \left (x\right )}^4-48\,a^3\,b\,{\cos \left (x\right )}^2+24\,a^3\,b+24\,a^2\,b^2\,{\cos \left (x\right )}^4-48\,a^2\,b^2\,{\cos \left (x\right )}^2+24\,a^2\,b^2+8\,a\,b^3\,{\cos \left (x\right )}^4-16\,a\,b^3\,{\cos \left (x\right )}^2+8\,a\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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