Optimal. Leaf size=77 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a^{3/2} f}-\frac {b \tan (e+f x)}{a (a+b) f \sqrt {a+b+b \tan ^2(e+f x)}} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4213, 390, 385,
209} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{a^{3/2} f}-\frac {b \tan (e+f x)}{a f (a+b) \sqrt {a+b \tan ^2(e+f x)+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 385
Rule 390
Rule 4213
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sec ^2(e+f x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (a+b+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {b \tan (e+f x)}{a (a+b) f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=-\frac {b \tan (e+f x)}{a (a+b) f \sqrt {a+b+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a f}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{a^{3/2} f}-\frac {b \tan (e+f x)}{a (a+b) f \sqrt {a+b+b \tan ^2(e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(168\) vs. \(2(77)=154\).
time = 1.46, size = 168, normalized size = 2.18 \begin {gather*} \frac {(a+2 b+a \cos (2 (e+f x))) \sec ^3(e+f x) \left (\sqrt {a+b} \text {ArcSin}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right ) (a+2 b+a \cos (2 (e+f x)))-\sqrt {2} \sqrt {a} b \sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}} \sin (e+f x)\right )}{4 a^{3/2} (a+b) f \left (a+b \sec ^2(e+f x)\right )^{3/2} \sqrt {\frac {a+b-a \sin ^2(e+f x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.15, size = 1007, normalized size = 13.08
method | result | size |
default | \(-\frac {\left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right ) \left (\sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {-\frac {4 i a^{\frac {3}{2}} \sqrt {b}-4 i \sqrt {a}\, b^{\frac {3}{2}}-a^{2}+6 a b -b^{2}}{\left (a +b \right )^{2}}}\right ) a \sin \left (f x +e \right )+\sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticF \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, \sqrt {-\frac {4 i a^{\frac {3}{2}} \sqrt {b}-4 i \sqrt {a}\, b^{\frac {3}{2}}-a^{2}+6 a b -b^{2}}{\left (a +b \right )^{2}}}\right ) b \sin \left (f x +e \right )-2 \sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticPi \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, -\frac {a +b}{2 i \sqrt {a}\, \sqrt {b}+a -b}, \frac {\sqrt {-\frac {2 i \sqrt {a}\, \sqrt {b}-a +b}{a +b}}}{\sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}\right ) a \sin \left (f x +e \right )-2 \sqrt {2}\, \sqrt {\frac {i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}+\cos \left (f x +e \right ) a +b}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \sqrt {-\frac {2 \left (i \cos \left (f x +e \right ) \sqrt {a}\, \sqrt {b}-i \sqrt {a}\, \sqrt {b}-\cos \left (f x +e \right ) a -b \right )}{\left (1+\cos \left (f x +e \right )\right ) \left (a +b \right )}}\, \EllipticPi \left (\frac {\left (\cos \left (f x +e \right )-1\right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}{\sin \left (f x +e \right )}, -\frac {a +b}{2 i \sqrt {a}\, \sqrt {b}+a -b}, \frac {\sqrt {-\frac {2 i \sqrt {a}\, \sqrt {b}-a +b}{a +b}}}{\sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}}\right ) b \sin \left (f x +e \right )+\cos \left (f x +e \right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}\, b -\sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}\, b \right ) \sin \left (f x +e \right )}{f \left (\cos \left (f x +e \right )-1\right ) \cos \left (f x +e \right )^{3} \left (\frac {b +a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right )^{2}}\right )^{\frac {3}{2}} \left (a +b \right ) \sqrt {\frac {2 i \sqrt {a}\, \sqrt {b}+a -b}{a +b}}\, a}\) | \(1007\) |
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. 2169 vs.
\(2 (73) = 146\).
time = 0.70, size = 2169, normalized size = 28.17 \begin {gather*} \text {Too large to display} \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. 254 vs.
\(2 (73) = 146\).
time = 3.02, size = 632, normalized size = 8.21 \begin {gather*} \left [-\frac {8 \, a b \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {-a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a^{4} - a^{3} b\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{4} - 14 \, a^{3} b + 5 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{4} - 28 \, a^{3} b + 70 \, a^{2} b^{2} - 28 \, a b^{3} + b^{4} - 32 \, {\left (a^{4} - 7 \, a^{3} b + 7 \, a^{2} b^{2} - a b^{3}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a^{3} - a^{2} b\right )} \cos \left (f x + e\right )^{5} + 2 \, {\left (5 \, a^{3} - 14 \, a^{2} b + 5 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} - 7 \, a^{2} b + 7 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )\right )}{8 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}}, -\frac {4 \, a b \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left ({\left (a^{2} + a b\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {a} \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{5} - 8 \, {\left (a^{2} - a b\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )\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} - a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right )}{4 \, {\left ({\left (a^{4} + a^{3} b\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{3} b + a^{2} b^{2}\right )} f\right )}}\right ] \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 {1}{\left (a + b \sec ^{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 [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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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