Optimal. Leaf size=90 \[ -\frac {5 \tanh ^{-1}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b} \]
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Rubi [A]
time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4290, 3853,
3855} \begin {gather*} -\frac {5 \tanh ^{-1}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 3853
Rule 3855
Rule 4290
Rubi steps
\begin {align*} \int x \csc ^7\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \csc ^7(a+b x) \, dx,x,x^2\right )\\ &=-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}+\frac {5}{12} \text {Subst}\left (\int \csc ^5(a+b x) \, dx,x,x^2\right )\\ &=-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}+\frac {5}{16} \text {Subst}\left (\int \csc ^3(a+b x) \, dx,x,x^2\right )\\ &=-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}+\frac {5}{32} \text {Subst}\left (\int \csc (a+b x) \, dx,x,x^2\right )\\ &=-\frac {5 \tanh ^{-1}\left (\cos \left (a+b x^2\right )\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc \left (a+b x^2\right )}{32 b}-\frac {5 \cot \left (a+b x^2\right ) \csc ^3\left (a+b x^2\right )}{48 b}-\frac {\cot \left (a+b x^2\right ) \csc ^5\left (a+b x^2\right )}{12 b}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 167, normalized size = 1.86 \begin {gather*} -\frac {5 \csc ^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\csc ^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}-\frac {\csc ^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b}-\frac {5 \log \left (\cos \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}+\frac {5 \log \left (\sin \left (\frac {1}{2} \left (a+b x^2\right )\right )\right )}{32 b}+\frac {5 \sec ^2\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\sec ^4\left (\frac {1}{2} \left (a+b x^2\right )\right )}{128 b}+\frac {\sec ^6\left (\frac {1}{2} \left (a+b x^2\right )\right )}{768 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 73, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {\left (-\frac {\left (\csc ^{5}\left (b \,x^{2}+a \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (b \,x^{2}+a \right )\right )}{24}-\frac {5 \csc \left (b \,x^{2}+a \right )}{16}\right ) \cot \left (b \,x^{2}+a \right )+\frac {5 \ln \left (\csc \left (b \,x^{2}+a \right )-\cot \left (b \,x^{2}+a \right )\right )}{16}}{2 b}\) | \(73\) |
default | \(\frac {\left (-\frac {\left (\csc ^{5}\left (b \,x^{2}+a \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (b \,x^{2}+a \right )\right )}{24}-\frac {5 \csc \left (b \,x^{2}+a \right )}{16}\right ) \cot \left (b \,x^{2}+a \right )+\frac {5 \ln \left (\csc \left (b \,x^{2}+a \right )-\cot \left (b \,x^{2}+a \right )\right )}{16}}{2 b}\) | \(73\) |
risch | \(\frac {15 \,{\mathrm e}^{11 i \left (b \,x^{2}+a \right )}-85 \,{\mathrm e}^{9 i \left (b \,x^{2}+a \right )}+198 \,{\mathrm e}^{7 i \left (b \,x^{2}+a \right )}+198 \,{\mathrm e}^{5 i \left (b \,x^{2}+a \right )}-85 \,{\mathrm e}^{3 i \left (b \,x^{2}+a \right )}+15 \,{\mathrm e}^{i \left (b \,x^{2}+a \right )}}{48 b \left ({\mathrm e}^{2 i \left (b \,x^{2}+a \right )}-1\right )^{6}}+\frac {5 \ln \left ({\mathrm e}^{i \left (b \,x^{2}+a \right )}-1\right )}{32 b}-\frac {5 \ln \left ({\mathrm e}^{i \left (b \,x^{2}+a \right )}+1\right )}{32 b}\) | \(139\) |
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. 3543 vs.
\(2 (82) = 164\).
time = 0.38, size = 3543, normalized size = 39.37 \begin {gather*} \text {Too large to display} \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. 183 vs.
\(2 (82) = 164\).
time = 2.86, size = 183, normalized size = 2.03 \begin {gather*} \frac {30 \, \cos \left (b x^{2} + a\right )^{5} - 80 \, \cos \left (b x^{2} + a\right )^{3} - 15 \, {\left (\cos \left (b x^{2} + a\right )^{6} - 3 \, \cos \left (b x^{2} + a\right )^{4} + 3 \, \cos \left (b x^{2} + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x^{2} + a\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (b x^{2} + a\right )^{6} - 3 \, \cos \left (b x^{2} + a\right )^{4} + 3 \, \cos \left (b x^{2} + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x^{2} + a\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (b x^{2} + a\right )}{192 \, {\left (b \cos \left (b x^{2} + a\right )^{6} - 3 \, b \cos \left (b x^{2} + a\right )^{4} + 3 \, b \cos \left (b x^{2} + a\right )^{2} - b\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 x \csc ^{7}{\left (a + b x^{2} \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. 211 vs.
\(2 (82) = 164\).
time = 0.41, size = 211, normalized size = 2.34 \begin {gather*} -\frac {\frac {{\left (\frac {9 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}}{\cos \left (b x^{2} + a\right ) + 1} - \frac {45 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}}{\cos \left (b x^{2} + a\right ) + 1} - \frac {9 \, {\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (b x^{2} + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x^{2} + a\right ) + 1\right )}^{3}} - 60 \, \log \left (-\frac {\cos \left (b x^{2} + a\right ) - 1}{\cos \left (b x^{2} + a\right ) + 1}\right )}{768 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.32, size = 491, normalized size = 5.46 \begin {gather*} -\frac {5\,\ln \left (-\frac {x\,5{}\mathrm {i}}{8}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}\,5{}\mathrm {i}}{8}\right )}{32\,b}+\frac {5\,\ln \left (\frac {x\,5{}\mathrm {i}}{8}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\mathrm {e}}^{b\,x^2\,1{}\mathrm {i}}\,5{}\mathrm {i}}{8}\right )}{32\,b}+\frac {8\,{\mathrm {e}}^{3{}\mathrm {i}\,b\,x^2+a\,3{}\mathrm {i}}}{3\,b\,\left (5\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-10\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+10\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}-5\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}+{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}-1\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{6\,b\,\left (3\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}+{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}-1\right )}+\frac {5\,{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{16\,b\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}-1\right )}+\frac {16\,{\mathrm {e}}^{5{}\mathrm {i}\,b\,x^2+a\,5{}\mathrm {i}}}{3\,b\,\left (1+15\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-20\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+15\,{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}-6\,{\mathrm {e}}^{10{}\mathrm {i}\,b\,x^2+a\,10{}\mathrm {i}}+{\mathrm {e}}^{12{}\mathrm {i}\,b\,x^2+a\,12{}\mathrm {i}}-6\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{b\,\left (1+6\,{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-4\,{\mathrm {e}}^{6{}\mathrm {i}\,b\,x^2+a\,6{}\mathrm {i}}+{\mathrm {e}}^{8{}\mathrm {i}\,b\,x^2+a\,8{}\mathrm {i}}-4\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )}-\frac {5\,{\mathrm {e}}^{1{}\mathrm {i}\,b\,x^2+a\,1{}\mathrm {i}}}{24\,b\,\left (1+{\mathrm {e}}^{4{}\mathrm {i}\,b\,x^2+a\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{2{}\mathrm {i}\,b\,x^2+a\,2{}\mathrm {i}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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