Optimal. Leaf size=57 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4224, 272, 53,
65, 214} \begin {gather*} \frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 214
Rule 272
Rule 4224
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^2\right )^{3/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 a f}\\ &=\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sec ^2(e+f x)}\right )}{a b f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{a^{3/2} f}+\frac {1}{a f \sqrt {a+b \sec ^2(e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 6.96, size = 382, normalized size = 6.70 \begin {gather*} \frac {(a+2 b+a \cos (2 (e+f x)))^{3/2} \sec ^2(e+f x) \left (-\frac {2}{b \sqrt {a+2 b+a \cos (2 (e+f x))}}+\frac {\sqrt {2} e^{i (e+f x)} \sqrt {4 b+a e^{-2 i (e+f x)} \left (1+e^{2 i (e+f x)}\right )^2} \left (\frac {\sqrt {a} (a+4 b) \left (1+e^{2 i (e+f x)}\right )}{b \left (4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2\right )}+\frac {4 i f x-2 \log \left (a+2 b+a e^{2 i (e+f x)}+\sqrt {a} \sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2}\right )-2 \log \left (a+a e^{2 i (e+f x)}+2 b e^{2 i (e+f x)}+\sqrt {a} \sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2}\right )}{\sqrt {4 b e^{2 i (e+f x)}+a \left (1+e^{2 i (e+f x)}\right )^2}}\right ) \sec (e+f x)}{a^{3/2}}\right )}{16 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 62, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {1}{a \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sec \left (f x +e \right )}\right )}{a^{\frac {3}{2}}}}{f}\) | \(62\) |
default | \(\frac {\frac {1}{a \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sec ^{2}\left (f x +e \right )\right )}}{\sec \left (f x +e \right )}\right )}{a^{\frac {3}{2}}}}{f}\) | \(62\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 183 vs.
\(2 (51) = 102\).
time = 3.59, size = 418, normalized size = 7.33 \begin {gather*} \left [\frac {8 \, a \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )^{2} + {\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} + 256 \, a^{3} b \cos \left (f x + e\right )^{6} + 160 \, a^{2} b^{2} \cos \left (f x + e\right )^{4} + 32 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4} - 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{8} + 24 \, a^{2} b \cos \left (f x + e\right )^{6} + 10 \, a b^{2} \cos \left (f x + e\right )^{4} + b^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}\right )}{8 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f\right )}}, \frac {4 \, a \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right )^{2} + {\left (a \cos \left (f x + e\right )^{2} + b\right )} \sqrt {-a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{4} + 8 \, a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b \cos \left (f x + e\right )^{2} + a b^{2}\right )}}\right )}{4 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} + a^{2} b f\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.33, size = 53, normalized size = 0.93 \begin {gather*} \frac {1}{a f \sqrt {a + b \sec ^{2}{\left (e + f x \right )}}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {a + b \sec ^{2}{\left (e + f x \right )}}}{\sqrt {- a}} \right )}}{a f \sqrt {- a}} \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. 216 vs.
\(2 (49) = 98\).
time = 0.98, size = 216, normalized size = 3.79 \begin {gather*} -\frac {\frac {\frac {\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a \mathrm {sgn}\left (\cos \left (f x + e\right )\right )} - \frac {1}{a \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}}{\sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b}} - \frac {2 \, \arctan \left (-\frac {\sqrt {a + b} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a + b} + \sqrt {a + b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.56, size = 49, normalized size = 0.86 \begin {gather*} \frac {1}{a\,f\,\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}-\frac {\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}}{\sqrt {a}}\right )}{a^{3/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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