Optimal. Leaf size=318 \[ \frac {4 a \cot (e+f x) E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) (a+b)^{3/2} f}-\frac {(3 a-b) \cot (e+f x) F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) (a+b)^{3/2} f}-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \]
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Rubi [A]
time = 0.36, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3960, 3918,
4088, 4090, 3917, 4089} \begin {gather*} \frac {4 a b^2 \tan (e+f x)}{f \left (a^2-b^2\right )^2 \sqrt {a+b \sec (e+f x)}}+\frac {b^2 \tan (e+f x)}{f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}-\frac {(3 a-b) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f (a-b) (a+b)^{3/2}}+\frac {4 a \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{f (a-b) (a+b)^{3/2}}-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3917
Rule 3918
Rule 3960
Rule 4088
Rule 4089
Rule 4090
Rubi steps
\begin {align*} \int \frac {\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx &=-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}-\frac {1}{2} (3 b) \int \frac {\sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx\\ &=-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {b \int \frac {\sec (e+f x) \left (-\frac {3 a}{2}+\frac {1}{2} b \sec (e+f x)\right )}{(a+b \sec (e+f x))^{3/2}} \, dx}{a^2-b^2}\\ &=-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}-\frac {(2 b) \int \frac {\sec (e+f x) \left (\frac {1}{4} \left (3 a^2+b^2\right )+a b \sec (e+f x)\right )}{\sqrt {a+b \sec (e+f x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}-\frac {((3 a-b) b) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{2 (a-b) (a+b)^2}-\frac {\left (2 a b^2\right ) \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {4 a \cot (e+f x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) (a+b)^{3/2} f}-\frac {(3 a-b) \cot (e+f x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{(a-b) (a+b)^{3/2} f}-\frac {\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac {b^2 \tan (e+f x)}{\left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {4 a b^2 \tan (e+f x)}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 5.07, size = 259, normalized size = 0.81 \begin {gather*} \frac {-((a-b) ((3 a-b) b+a (a-3 b) \cos (e+f x)) \csc (e+f x))+8 a b (a+b) \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} E\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sec (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}-2 b \left (3 a^2+4 a b+b^2\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {\frac {b+a \cos (e+f x)}{(a+b) (1+\cos (e+f x))}} F\left (\text {ArcSin}\left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sec (e+f x) \sqrt {\frac {1}{1+\sec (e+f x)}}}{\left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1064\) vs.
\(2(294)=588\).
time = 0.20, size = 1065, normalized size = 3.35
method | result | size |
default | \(\text {Expression too large to display}\) | \(1065\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc ^{2}{\left (e + f x \right )}}{\left (a + b \sec {\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 = -0.00 \begin {gather*} \int \frac {1}{{\sin \left (e+f\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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