Optimal. Leaf size=76 \[ -\frac {\cos (e+f x)}{(a-b) f \sqrt {a-b+b \sec ^2(e+f x)}}-\frac {2 b \sec (e+f x)}{(a-b)^2 f \sqrt {a-b+b \sec ^2(e+f x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3745, 277, 197}
\begin {gather*} -\frac {2 b \sec (e+f x)}{f (a-b)^2 \sqrt {a+b \sec ^2(e+f x)-b}}-\frac {\cos (e+f x)}{f (a-b) \sqrt {a+b \sec ^2(e+f x)-b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 277
Rule 3745
Rubi steps
\begin {align*} \int \frac {\sin (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x)}{(a-b) f \sqrt {a-b+b \sec ^2(e+f x)}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{\left (a-b+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{(a-b) f}\\ &=-\frac {\cos (e+f x)}{(a-b) f \sqrt {a-b+b \sec ^2(e+f x)}}-\frac {2 b \sec (e+f x)}{(a-b)^2 f \sqrt {a-b+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.69, size = 72, normalized size = 0.95 \begin {gather*} -\frac {(a+3 b+(a-b) \cos (2 (e+f x))) \sec (e+f x)}{\sqrt {2} (a-b)^2 f \sqrt {(a+b+(a-b) \cos (2 (e+f x))) \sec ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 103, normalized size = 1.36
method | result | size |
default | \(-\frac {\left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right ) \left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +2 b \right )}{f \left (\frac {a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b}{\cos \left (f x +e \right )^{2}}\right )^{\frac {3}{2}} \cos \left (f x +e \right )^{3} \left (a -b \right )^{2}}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 87, normalized size = 1.14 \begin {gather*} -\frac {\frac {\sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {b}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b + \frac {b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.51, size = 109, normalized size = 1.43 \begin {gather*} -\frac {{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \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}}}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} b - 2 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin {\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, 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.
\(2 (76) = 152\).
time = 1.60, size = 178, normalized size = 2.34 \begin {gather*} -\frac {f^{2} {\left (\frac {\sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}{a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} + \frac {b}{{\left (a {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right ) - b {\left | f \right |} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )\right )} \sqrt {a \cos \left (f x + e\right )^{2} - b \cos \left (f x + e\right )^{2} + b}}\right )}}{a f^{2} - b f^{2}} + \frac {2 \, \sqrt {b} \mathrm {sgn}\left (f\right ) \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{a^{2} {\left | f \right |} - 2 \, a b {\left | f \right |} + b^{2} {\left | f \right |}} \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 )}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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