Optimal. Leaf size=144 \[ -\frac {\left (15 a^2-10 a b+3 b^2\right ) \cos (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{15 (a-b)^3 f}+\frac {2 (5 a-3 b) \cos ^3(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{15 (a-b)^2 f}-\frac {\cos ^5(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{5 (a-b) f} \]
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Rubi [A]
time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3745, 473, 464,
270} \begin {gather*} -\frac {\left (15 a^2-10 a b+3 b^2\right ) \cos (e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{15 f (a-b)^3}-\frac {\cos ^5(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{5 f (a-b)}+\frac {2 (5 a-3 b) \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)-b}}{15 f (a-b)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 464
Rule 473
Rule 3745
Rubi steps
\begin {align*} \int \frac {\sin ^5(e+f x)}{\sqrt {a+b \tan ^2(e+f x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{x^6 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos ^5(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{5 (a-b) f}+\frac {\text {Subst}\left (\int \frac {-2 (5 a-3 b)+5 (a-b) x^2}{x^4 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{5 (a-b) f}\\ &=\frac {2 (5 a-3 b) \cos ^3(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{15 (a-b)^2 f}-\frac {\cos ^5(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{5 (a-b) f}+\frac {\left (15 a^2-10 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a-b+b x^2}} \, dx,x,\sec (e+f x)\right )}{15 (a-b)^2 f}\\ &=-\frac {\left (15 a^2-10 a b+3 b^2\right ) \cos (e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{15 (a-b)^3 f}+\frac {2 (5 a-3 b) \cos ^3(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{15 (a-b)^2 f}-\frac {\cos ^5(e+f x) \sqrt {a-b+b \sec ^2(e+f x)}}{5 (a-b) f}\\ \end {align*}
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Mathematica [A]
time = 42.12, size = 112, normalized size = 0.78 \begin {gather*} \frac {\cos (e+f x) \left (-89 a^2+34 a b-9 b^2+4 \left (7 a^2-10 a b+3 b^2\right ) \cos (2 (e+f x))-3 (a-b)^2 \cos (4 (e+f x))\right ) \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}}{120 \sqrt {2} (a-b)^3 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 169, normalized size = 1.17
method | result | size |
default | \(-\frac {\left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right ) \left (3 \left (\cos ^{4}\left (f x +e \right )\right ) a^{2}-6 \left (\cos ^{4}\left (f x +e \right )\right ) a b +3 \left (\cos ^{4}\left (f x +e \right )\right ) b^{2}-10 \left (\cos ^{2}\left (f x +e \right )\right ) a^{2}+16 \left (\cos ^{2}\left (f x +e \right )\right ) a b -6 \left (\cos ^{2}\left (f x +e \right )\right ) b^{2}+15 a^{2}-10 a b +3 b^{2}\right )}{15 f \sqrt {\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}}\, \cos \left (f x +e \right ) \left (a -b \right )^{3}}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 226, normalized size = 1.57 \begin {gather*} -\frac {\frac {15 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a - b} + \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} - 10 \, {\left (a - b + \frac {b}{\cos \left (f x + e\right )^{2}}\right )}^{\frac {3}{2}} b \cos \left (f x + e\right )^{3} + 15 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b^{2} \cos \left (f x + e\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \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} - 3 \, \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} b \cos \left (f x + e\right )\right )}}{a^{2} - 2 \, a b + b^{2}}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.25, size = 129, normalized size = 0.90 \begin {gather*} -\frac {{\left (3 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (5 \, a^{2} - 8 \, a b + 3 \, b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (15 \, a^{2} - 10 \, a b + 3 \, b^{2}\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 (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} f} \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. 1376 vs.
\(2 (138) = 276\).
time = 1.34, size = 1376, normalized size = 9.56 \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 = -0.01 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^5}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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