Optimal. Leaf size=248 \[ -\frac {\left (5 a^2+10 a b+b^2\right ) \cos (e+f x)}{5 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a-b) \cos ^3(e+f x)}{15 (a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 (a-b)^5 f \sqrt {a-b+b \sec ^2(e+f x)}} \]
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Rubi [A]
time = 0.16, antiderivative size = 248, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3745, 473, 464,
277, 198, 197} \begin {gather*} -\frac {8 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 f (a-b)^5 \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {4 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 f (a-b)^4 \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {\left (5 a^2+10 a b+b^2\right ) \cos (e+f x)}{5 f (a-b)^3 \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 f (a-b) \left (a+b \sec ^2(e+f x)-b\right )^{3/2}}+\frac {2 (5 a-b) \cos ^3(e+f x)}{15 f (a-b)^2 \left (a+b \sec ^2(e+f x)-b\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 277
Rule 464
Rule 473
Rule 3745
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x^6 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {-2 (5 a-b)+5 (a-b) x^2}{x^4 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 (a-b) f}\\ &=\frac {2 (5 a-b) \cos ^3(e+f x)}{15 (a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {\left (5 a^2+10 a b+b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 (a-b)^2 f}\\ &=-\frac {\left (5 a^2+10 a b+b^2\right ) \cos (e+f x)}{5 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a-b) \cos ^3(e+f x)}{15 (a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\left (4 b \left (5 a^2+10 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{5 (a-b)^3 f}\\ &=-\frac {\left (5 a^2+10 a b+b^2\right ) \cos (e+f x)}{5 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a-b) \cos ^3(e+f x)}{15 (a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\left (8 b \left (5 a^2+10 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{15 (a-b)^4 f}\\ &=-\frac {\left (5 a^2+10 a b+b^2\right ) \cos (e+f x)}{5 (a-b)^3 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}+\frac {2 (5 a-b) \cos ^3(e+f x)}{15 (a-b)^2 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {\cos ^5(e+f x)}{5 (a-b) f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {4 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 (a-b)^4 f \left (a-b+b \sec ^2(e+f x)\right )^{3/2}}-\frac {8 b \left (5 a^2+10 a b+b^2\right ) \sec (e+f x)}{15 (a-b)^5 f \sqrt {a-b+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 2.42, size = 294, normalized size = 1.19 \begin {gather*} -\frac {\cos (e+f x) \left (425 a^4+4700 a^3 b+6134 a^2 b^2+4700 a b^3+425 b^4+48 \left (11 a^4+106 a^3 b-106 a b^3-11 b^4\right ) \cos (2 (e+f x))+12 (a-b)^2 \left (7 a^2+50 a b+7 b^2\right ) \cos (4 (e+f x))-16 a^4 \cos (6 (e+f x))+32 a^3 b \cos (6 (e+f x))-32 a b^3 \cos (6 (e+f x))+16 b^4 \cos (6 (e+f x))+3 a^4 \cos (8 (e+f x))-12 a^3 b \cos (8 (e+f x))+18 a^2 b^2 \cos (8 (e+f x))-12 a b^3 \cos (8 (e+f x))+3 b^4 \cos (8 (e+f x))\right ) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{480 \sqrt {2} (a-b)^5 f (a+b+(a-b) \cos (2 (e+f x)))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.95, size = 391, normalized size = 1.58
method | result | size |
default | \(\frac {\left (a -b \right )^{2} \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right ) \left (3 \left (\cos ^{8}\left (f x +e \right )\right ) a^{4}-12 \left (\cos ^{8}\left (f x +e \right )\right ) a^{3} b +18 \left (\cos ^{8}\left (f x +e \right )\right ) a^{2} b^{2}-12 \left (\cos ^{8}\left (f x +e \right )\right ) a \,b^{3}+3 \left (\cos ^{8}\left (f x +e \right )\right ) b^{4}-10 \left (\cos ^{6}\left (f x +e \right )\right ) a^{4}+32 \left (\cos ^{6}\left (f x +e \right )\right ) a^{3} b -36 \left (\cos ^{6}\left (f x +e \right )\right ) a^{2} b^{2}+16 \left (\cos ^{6}\left (f x +e \right )\right ) a \,b^{3}-2 \left (\cos ^{6}\left (f x +e \right )\right ) b^{4}+15 \left (\cos ^{4}\left (f x +e \right )\right ) a^{4}-42 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2} b^{2}+24 \left (\cos ^{4}\left (f x +e \right )\right ) a \,b^{3}+3 \left (\cos ^{4}\left (f x +e \right )\right ) b^{4}+60 \left (\cos ^{2}\left (f x +e \right )\right ) a^{3} b +60 \left (\cos ^{2}\left (f x +e \right )\right ) a^{2} b^{2}-108 \left (\cos ^{2}\left (f x +e \right )\right ) a \,b^{3}-12 \left (\cos ^{2}\left (f x +e \right )\right ) b^{4}+40 a^{2} b^{2}+80 a \,b^{3}+8 b^{4}\right ) \sqrt {4}\, a^{7}}{30 f \left (\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}\right )^{\frac {5}{2}} \cos \left (f x +e \right )^{5} \left (\sqrt {-b \left (a -b \right )}+a -b \right )^{7} \left (\sqrt {-b \left (a -b \right )}-a +b \right )^{7}}\) | \(391\) |
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. 558 vs.
\(2 (238) = 476\).
time = 0.30, size = 558, normalized size = 2.25 \begin {gather*} -\frac {\frac {15 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} + \frac {3 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {5}{2}} \cos \left (f x + e\right )^{5} - 20 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} b \cos \left (f x + e\right )^{3} + 90 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b^{2} \cos \left (f x + e\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {10 \, {\left ({\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3} - 9 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {5 \, {\left (12 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b^{3} \cos \left (f x + e\right )^{2} - b^{4}\right )}}{{\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3}} + \frac {10 \, {\left (9 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b^{2} \cos \left (f x + e\right )^{2} - b^{3}\right )}}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3}} + \frac {5 \, {\left (6 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )} b \cos \left (f x + e\right )^{2} - b^{2}\right )}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{3}}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.59, size = 379, normalized size = 1.53 \begin {gather*} -\frac {{\left (3 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{9} - 2 \, {\left (5 \, a^{4} - 16 \, a^{3} b + 18 \, a^{2} b^{2} - 8 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{7} + 3 \, {\left (5 \, a^{4} - 14 \, a^{2} b^{2} + 8 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{5} + 12 \, {\left (5 \, a^{3} b + 5 \, a^{2} b^{2} - 9 \, a b^{3} - b^{4}\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (5 \, a^{2} b^{2} + 10 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{15 \, {\left ({\left (a^{7} - 7 \, a^{6} b + 21 \, a^{5} b^{2} - 35 \, a^{4} b^{3} + 35 \, a^{3} b^{4} - 21 \, a^{2} b^{5} + 7 \, a b^{6} - b^{7}\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b - 6 \, a^{5} b^{2} + 15 \, a^{4} b^{3} - 20 \, a^{3} b^{4} + 15 \, a^{2} b^{5} - 6 \, a b^{6} + b^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} - 5 \, a^{4} b^{3} + 10 \, a^{3} b^{4} - 10 \, a^{2} b^{5} + 5 \, a b^{6} - b^{7}\right )} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 5506 vs.
\(2 (238) = 476\).
time = 4.38, size = 5506, normalized size = 22.20 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^5}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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