Optimal. Leaf size=343 \[ -\frac {2 a^2 b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac {4 a^2 b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac {a^3 b^2 \sin (c+d x)}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}-\frac {(a-b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}+\frac {(a+b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2} \]
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Rubi [A] time = 0.55, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3872, 2897, 2650, 2648, 2664, 12, 2659, 208} \[ \frac {a^3 b^2 \sin (c+d x)}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}-\frac {4 a^2 b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac {2 a^2 b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}-\frac {(a-b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}+\frac {(a+b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac {\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 2648
Rule 2650
Rule 2659
Rule 2664
Rule 2897
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac {1}{4 (a-b)^2 (-1-\cos (c+d x))^2}+\frac {-a-b}{4 (a-b)^3 (-1-\cos (c+d x))}+\frac {1}{4 (a+b)^2 (1-\cos (c+d x))^2}+\frac {a-b}{4 (a+b)^3 (1-\cos (c+d x))}+\frac {a^2 b^2}{\left (a^2-b^2\right )^2 (-b-a \cos (c+d x))^2}+\frac {2 a^2 b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3 (-b-a \cos (c+d x))}\right ) \, dx\\ &=\frac {\int \frac {1}{(-1-\cos (c+d x))^2} \, dx}{4 (a-b)^2}+\frac {(a-b) \int \frac {1}{1-\cos (c+d x)} \, dx}{4 (a+b)^3}+\frac {\int \frac {1}{(1-\cos (c+d x))^2} \, dx}{4 (a+b)^2}-\frac {(a+b) \int \frac {1}{-1-\cos (c+d x)} \, dx}{4 (a-b)^3}+\frac {\left (a^2 b^2\right ) \int \frac {1}{(-b-a \cos (c+d x))^2} \, dx}{\left (a^2-b^2\right )^2}+\frac {\left (2 a^2 b \left (a^2+b^2\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {(a-b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}+\frac {(a+b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {a^3 b^2 \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\int \frac {1}{-1-\cos (c+d x)} \, dx}{12 (a-b)^2}+\frac {\int \frac {1}{1-\cos (c+d x)} \, dx}{12 (a+b)^2}+\frac {\left (a^2 b^2\right ) \int \frac {b}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^3}+\frac {\left (4 a^2 b \left (a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {4 a^2 b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {(a-b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))}+\frac {(a+b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {a^3 b^2 \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (a^2 b^3\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=-\frac {4 a^2 b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {(a-b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))}+\frac {(a+b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {a^3 b^2 \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (2 a^2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac {2 a^2 b^3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {4 a^2 b \left (a^2+b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} d}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))^2}-\frac {(a-b) \sin (c+d x)}{4 (a+b)^3 d (1-\cos (c+d x))}-\frac {\sin (c+d x)}{12 (a+b)^2 d (1-\cos (c+d x))}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))^2}+\frac {\sin (c+d x)}{12 (a-b)^2 d (1+\cos (c+d x))}+\frac {(a+b) \sin (c+d x)}{4 (a-b)^3 d (1+\cos (c+d x))}+\frac {a^3 b^2 \sin (c+d x)}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 1.13, size = 281, normalized size = 0.82 \[ \frac {\sec ^2(c+d x) (a \cos (c+d x)+b) \left (\frac {24 a^3 b^2 \sin (c+d x)}{(a-b)^3 (a+b)^3}+\frac {48 a^2 b \left (2 a^2+3 b^2\right ) (a \cos (c+d x)+b) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac {4 (2 a+b) \tan \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^3}-\frac {4 (2 a-b) \cot \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^3}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a+b)^2}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b)}{(a-b)^2}\right )}{24 d (a+b \sec (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 1040, normalized size = 3.03 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 457, normalized size = 1.33 \[ -\frac {\frac {48 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {48 \, {\left (2 \, a^{4} b + 3 \, a^{2} b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}} + \frac {9 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 242, normalized size = 0.71 \[ \frac {\frac {\frac {a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{3}+3 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{8 \left (a^{2}-2 a b +b^{2}\right ) \left (a -b \right )}+\frac {2 a^{2} b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a -b}-\frac {\left (2 a^{2}+3 b^{2}\right ) \arctanh \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {1}{24 \left (a +b \right )^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {3 a -b}{8 \left (a +b \right )^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.62, size = 403, normalized size = 1.17 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d\,{\left (a-b\right )}^2}+\frac {\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{3\,\left (a+b\right )}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (4\,a^4-13\,a^3\,b+15\,a^2\,b^2-7\,a\,b^3+b^4\right )}{3\,{\left (a+b\right )}^2}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (3\,a^5-13\,a^4\,b+38\,a^3\,b^2-18\,a^2\,b^3+7\,a\,b^4-b^5\right )}{{\left (a+b\right )}^3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (8\,a^4-32\,a^3\,b+48\,a^2\,b^2-32\,a\,b^3+8\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (8\,a^4-16\,a^3\,b+16\,a\,b^3-8\,b^4\right )\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {16\,a^2-16\,b^2}{64\,{\left (a-b\right )}^4}+\frac {1}{8\,{\left (a-b\right )}^2}\right )}{d}+\frac {a^2\,b\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^6-3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^4\,b^2+3{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2\,b^4-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^6}{{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{5/2}}\right )\,\left (2\,a^2+3\,b^2\right )\,2{}\mathrm {i}}{d\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}} \]
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 \[ \int \frac {\csc ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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