Optimal. Leaf size=121 \[ -\frac {2 b}{15 f (b \sec (e+f x))^{3/2} \sin ^{\frac {15}{2}}(e+f x)}-\frac {8 b}{55 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}-\frac {64 b}{385 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}-\frac {256 b}{1155 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)} \]
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Rubi [A]
time = 0.11, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2664, 2658}
\begin {gather*} -\frac {256 b}{1155 f \sin ^{\frac {3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac {64 b}{385 f \sin ^{\frac {7}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac {8 b}{55 f \sin ^{\frac {11}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac {2 b}{15 f \sin ^{\frac {15}{2}}(e+f x) (b \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2658
Rule 2664
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {17}{2}}(e+f x)} \, dx &=-\frac {2 b}{15 f (b \sec (e+f x))^{3/2} \sin ^{\frac {15}{2}}(e+f x)}+\frac {4}{5} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {13}{2}}(e+f x)} \, dx\\ &=-\frac {2 b}{15 f (b \sec (e+f x))^{3/2} \sin ^{\frac {15}{2}}(e+f x)}-\frac {8 b}{55 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}+\frac {32}{55} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {9}{2}}(e+f x)} \, dx\\ &=-\frac {2 b}{15 f (b \sec (e+f x))^{3/2} \sin ^{\frac {15}{2}}(e+f x)}-\frac {8 b}{55 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}-\frac {64 b}{385 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}+\frac {128}{385} \int \frac {1}{\sqrt {b \sec (e+f x)} \sin ^{\frac {5}{2}}(e+f x)} \, dx\\ &=-\frac {2 b}{15 f (b \sec (e+f x))^{3/2} \sin ^{\frac {15}{2}}(e+f x)}-\frac {8 b}{55 f (b \sec (e+f x))^{3/2} \sin ^{\frac {11}{2}}(e+f x)}-\frac {64 b}{385 f (b \sec (e+f x))^{3/2} \sin ^{\frac {7}{2}}(e+f x)}-\frac {256 b}{1155 f (b \sec (e+f x))^{3/2} \sin ^{\frac {3}{2}}(e+f x)}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 62, normalized size = 0.51 \begin {gather*} \frac {2 b (-195+150 \cos (2 (e+f x))-36 \cos (4 (e+f x))+4 \cos (6 (e+f x)))}{1155 f (b \sec (e+f x))^{3/2} \sin ^{\frac {15}{2}}(e+f x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 102, normalized size = 0.84
method | result | size |
default | \(\frac {512 \cos \left (f x +e \right ) \left (128 \left (\cos ^{6}\left (f x +e \right )\right )-480 \left (\cos ^{4}\left (f x +e \right )\right )+660 \left (\cos ^{2}\left (f x +e \right )\right )-385\right ) \left (-1+\cos \left (f x +e \right )\right )^{8}}{1155 f \sin \left (f x +e \right )^{\frac {15}{2}} \left (\sin ^{2}\left (f x +e \right )+\cos ^{2}\left (f x +e \right )-2 \cos \left (f x +e \right )+1\right )^{8} \sqrt {\frac {b}{\cos \left (f x +e \right )}}}\) | \(102\) |
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 [A]
time = 0.43, size = 126, normalized size = 1.04 \begin {gather*} \frac {2 \, {\left (128 \, \cos \left (f x + e\right )^{8} - 480 \, \cos \left (f x + e\right )^{6} + 660 \, \cos \left (f x + e\right )^{4} - 385 \, \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sqrt {\sin \left (f x + e\right )}}{1155 \, {\left (b f \cos \left (f x + e\right )^{8} - 4 \, b f \cos \left (f x + e\right )^{6} + 6 \, b f \cos \left (f x + e\right )^{4} - 4 \, b f \cos \left (f x + e\right )^{2} + b f\right )}} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.41, size = 192, normalized size = 1.59 \begin {gather*} \frac {{\mathrm {e}}^{-e\,8{}\mathrm {i}-f\,x\,8{}\mathrm {i}}\,\sqrt {\frac {b}{\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}}{2}}}\,\left (\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,1024{}\mathrm {i}}{77\,b\,f}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (2\,e+2\,f\,x\right )\,384{}\mathrm {i}}{55\,b\,f}-\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (4\,e+4\,f\,x\right )\,5248{}\mathrm {i}}{1155\,b\,f}+\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (6\,e+6\,f\,x\right )\,256{}\mathrm {i}}{165\,b\,f}-\frac {{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\cos \left (8\,e+8\,f\,x\right )\,256{}\mathrm {i}}{1155\,b\,f}\right )\,1{}\mathrm {i}}{128\,{\sin \left (e+f\,x\right )}^{15/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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