Optimal. Leaf size=449 \[ \frac {2 b^2 \left (a^2+b^2\right ) \tan (c+d x)}{a d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}+\frac {b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}-\frac {\left (a^2-a b+2 b^2\right ) \cot (c+d x) \sqrt {-\frac {b (\sec (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a d (a-b) (a+b)^{3/2}}+\frac {2 \left (a^2+b^2\right ) \cot (c+d x) \sqrt {-\frac {b (\sec (c+d x)-1)}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a d (a-b) (a+b)^{3/2}}+\frac {2 \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a^2 d}-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}} \]
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Rubi [A] time = 0.95, antiderivative size = 664, normalized size of antiderivative = 1.48, number of steps used = 14, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3896, 3785, 4058, 3921, 3784, 3832, 4004, 3875, 3833, 4003, 4005} \[ -\frac {2 b^2 \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {4 a b^2 \tan (c+d x)}{d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}+\frac {b^2 \tan (c+d x)}{d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {2 \sqrt {a+b} \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a^2 d}-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac {2 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a d \sqrt {a+b}}-\frac {(3 a-b) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}}-\frac {2 \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{a d \sqrt {a+b}}+\frac {4 a \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{d (a-b) (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3784
Rule 3785
Rule 3832
Rule 3833
Rule 3875
Rule 3896
Rule 3921
Rule 4003
Rule 4004
Rule 4005
Rule 4058
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx &=\int \left (-\frac {1}{(a+b \sec (c+d x))^{3/2}}+\frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}}\right ) \, dx\\ &=-\int \frac {1}{(a+b \sec (c+d x))^{3/2}} \, dx+\int \frac {\csc ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx\\ &=-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}-\frac {2 b^2 \tan (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {1}{2} (3 b) \int \frac {\sec (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx+\frac {2 \int \frac {\frac {1}{2} \left (-a^2+b^2\right )+\frac {1}{2} a b \sec (c+d x)+\frac {1}{2} b^2 \sec ^2(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b^2 \tan (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \int \frac {\frac {1}{2} \left (-a^2+b^2\right )+\left (\frac {a b}{2}-\frac {b^2}{2}\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{a \left (a^2-b^2\right )}+\frac {b \int \frac {\sec (c+d x) \left (-\frac {3 a}{2}+\frac {1}{2} b \sec (c+d x)\right )}{(a+b \sec (c+d x))^{3/2}} \, dx}{a^2-b^2}+\frac {b^2 \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {2 \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a \sqrt {a+b} d}-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {4 a b^2 \tan (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 b^2 \tan (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {1}{\sqrt {a+b \sec (c+d x)}} \, dx}{a}+\frac {b \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{a (a+b)}-\frac {(2 b) \int \frac {\sec (c+d x) \left (\frac {1}{4} \left (3 a^2+b^2\right )+a b \sec (c+d x)\right )}{\sqrt {a+b \sec (c+d x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=-\frac {2 \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {2 \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {2 \sqrt {a+b} \cot (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^2 d}-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {4 a b^2 \tan (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 b^2 \tan (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {((3 a-b) b) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx}{2 (a-b) (a+b)^2}-\frac {\left (2 a b^2\right ) \int \frac {\sec (c+d x) (1+\sec (c+d x))}{\sqrt {a+b \sec (c+d x)}} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac {4 a \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{(a-b) (a+b)^{3/2} d}-\frac {2 \cot (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a \sqrt {a+b} d}-\frac {(3 a-b) \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{(a-b) (a+b)^{3/2} d}+\frac {2 \cot (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {2 \sqrt {a+b} \cot (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{a^2 d}-\frac {\cot (c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac {b^2 \tan (c+d x)}{\left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}+\frac {4 a b^2 \tan (c+d x)}{\left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}-\frac {2 b^2 \tan (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 13.64, size = 663, normalized size = 1.48 \[ \frac {\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left (-\frac {2 b \left (a^2+b^2\right ) \sin (c+d x)}{a \left (a^2-b^2\right )^2}+\frac {\csc (c+d x) \left (a^2 (-\cos (c+d x))+2 a b-b^2 \cos (c+d x)\right )}{\left (b^2-a^2\right )^2}+\frac {2 b^4 \sin (c+d x)}{a \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}\right )}{d (a+b \sec (c+d x))^{3/2}}-\frac {2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) (a \cos (c+d x)+b) \left (-b \sqrt {\frac {b-a}{a+b}} \left (a^2+b^2\right ) \cos (c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)-4 i \left (a^2-b^2\right )^2 \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left (-\frac {a+b}{a-b};i \sinh ^{-1}\left (\sqrt {\frac {b-a}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right )-2 i b \left (-a^3+a^2 b-a b^2+b^3\right ) \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {b-a}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right )+i \left (2 a^4-a^3 b-2 a^2 b^2-3 a b^3+4 b^4\right ) \sqrt {\frac {\cos (c+d x)}{\cos (c+d x)+1}} \sqrt {\frac {a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {b-a}{a+b}} \tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {a+b}{a-b}\right )\right )}{a d \sqrt {\frac {b-a}{a+b}} \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\cot \left (d x + c\right )^{2}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.32, size = 2238, normalized size = 4.98 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^2}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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