Optimal. Leaf size=83 \[ -\frac {a^4 \cot (c+d x)}{d}+\frac {4 a^3 b \log (\tan (c+d x))}{d}+\frac {6 a^2 b^2 \tan (c+d x)}{d}+\frac {2 a b^3 \tan ^2(c+d x)}{d}+\frac {b^4 \tan ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3516, 43} \[ \frac {6 a^2 b^2 \tan (c+d x)}{d}+\frac {4 a^3 b \log (\tan (c+d x))}{d}-\frac {a^4 \cot (c+d x)}{d}+\frac {2 a b^3 \tan ^2(c+d x)}{d}+\frac {b^4 \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3516
Rubi steps
\begin {align*} \int \csc ^2(c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^4}{x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (6 a^2+\frac {a^4}{x^2}+\frac {4 a^3}{x}+4 a x+x^2\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {a^4 \cot (c+d x)}{d}+\frac {4 a^3 b \log (\tan (c+d x))}{d}+\frac {6 a^2 b^2 \tan (c+d x)}{d}+\frac {2 a b^3 \tan ^2(c+d x)}{d}+\frac {b^4 \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 1.21, size = 162, normalized size = 1.95 \[ -\frac {\csc (c+d x) \sec ^3(c+d x) \left (4 \left (3 a^4+b^4\right ) \cos (2 (c+d x))+\left (3 a^4+18 a^2 b^2-b^4\right ) \cos (4 (c+d x))+3 \left (3 a^4+4 a^3 b \sin (4 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))+8 a b \sin (2 (c+d x)) \left (-a^2 \log (\sin (c+d x))+a^2 \log (\cos (c+d x))-b^2\right )-6 a^2 b^2-b^4\right )\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 159, normalized size = 1.92 \[ -\frac {6 \, a^{3} b \cos \left (d x + c\right )^{3} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 6 \, a^{3} b \cos \left (d x + c\right )^{3} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) - 6 \, a b^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (3 \, a^{4} + 18 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} - b^{4} - 2 \, {\left (9 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}}{3 \, d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.04, size = 86, normalized size = 1.04 \[ \frac {b^{4} \tan \left (d x + c\right )^{3} + 6 \, a b^{3} \tan \left (d x + c\right )^{2} + 12 \, a^{3} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 18 \, a^{2} b^{2} \tan \left (d x + c\right ) - \frac {3 \, {\left (4 \, a^{3} b \tan \left (d x + c\right ) + a^{4}\right )}}{\tan \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 90, normalized size = 1.08 \[ -\frac {a^{4} \cot \left (d x +c \right )}{d}+\frac {4 a^{3} b \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{2} b^{2} \tan \left (d x +c \right )}{d}+\frac {2 a \,b^{3}}{d \cos \left (d x +c \right )^{2}}+\frac {b^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 72, normalized size = 0.87 \[ \frac {b^{4} \tan \left (d x + c\right )^{3} + 6 \, a b^{3} \tan \left (d x + c\right )^{2} + 12 \, a^{3} b \log \left (\tan \left (d x + c\right )\right ) + 18 \, a^{2} b^{2} \tan \left (d x + c\right ) - \frac {3 \, a^{4}}{\tan \left (d x + c\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 81, normalized size = 0.98 \[ \frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d}-\frac {a^4\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {6\,a^2\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d}+\frac {2\,a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{d}+\frac {4\,a^3\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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