Optimal. Leaf size=137 \[ -\frac {a^4 \cot ^3(c+d x)}{3 d}-\frac {2 a^3 b \cot ^2(c+d x)}{d}+\frac {b^2 \left (6 a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \left (a^2+6 b^2\right ) \cot (c+d x)}{d}+\frac {4 a b \left (a^2+b^2\right ) \log (\tan (c+d x))}{d}+\frac {2 a b^3 \tan ^2(c+d x)}{d}+\frac {b^4 \tan ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.10, antiderivative size = 137, 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, 894} \[ \frac {b^2 \left (6 a^2+b^2\right ) \tan (c+d x)}{d}-\frac {a^2 \left (a^2+6 b^2\right ) \cot (c+d x)}{d}+\frac {4 a b \left (a^2+b^2\right ) \log (\tan (c+d x))}{d}-\frac {2 a^3 b \cot ^2(c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}+\frac {2 a b^3 \tan ^2(c+d x)}{d}+\frac {b^4 \tan ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 3516
Rubi steps
\begin {align*} \int \csc ^4(c+d x) (a+b \tan (c+d x))^4 \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^4 \left (b^2+x^2\right )}{x^4} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (6 a^2 \left (1+\frac {b^2}{6 a^2}\right )+\frac {a^4 b^2}{x^4}+\frac {4 a^3 b^2}{x^3}+\frac {a^4+6 a^2 b^2}{x^2}+\frac {4 a \left (a^2+b^2\right )}{x}+4 a x+x^2\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {a^2 \left (a^2+6 b^2\right ) \cot (c+d x)}{d}-\frac {2 a^3 b \cot ^2(c+d x)}{d}-\frac {a^4 \cot ^3(c+d x)}{3 d}+\frac {4 a b \left (a^2+b^2\right ) \log (\tan (c+d x))}{d}+\frac {b^2 \left (6 a^2+b^2\right ) \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 = 3.84, size = 188, normalized size = 1.37 \[ -\frac {\sin (c+d x) \tan ^3(c+d x) (a \cot (c+d x)+b)^4 \left (-2 b^2 \left (9 a^2+b^2\right ) \sin (c+d x) \cos ^2(c+d x)+2 a \cos ^3(c+d x) \left (a \left (a^2+9 b^2\right ) \cot (c+d x)+6 b \left (a^2+b^2\right ) (\log (\cos (c+d x))-\log (\sin (c+d x)))\right )+\cos (c+d x) \left (a^4 \cot ^3(c+d x)+6 a^3 b \cot ^2(c+d x)-6 a b^3\right )+b^4 (-\sin (c+d x))\right )}{3 d (a \cos (c+d x)+b \sin (c+d x))^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 267, normalized size = 1.95 \[ -\frac {2 \, {\left (a^{4} + 18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 18 \, a^{2} b^{2} \cos \left (d x + c\right )^{2} - 3 \, {\left (a^{4} + 18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 6 \, {\left ({\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 6 \, {\left ({\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{5} - {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) + 6 \, {\left (a b^{3} \cos \left (d x + c\right ) - {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.80, size = 161, normalized size = 1.18 \[ \frac {b^{4} \tan \left (d x + c\right )^{3} + 6 \, a b^{3} \tan \left (d x + c\right )^{2} + 18 \, a^{2} b^{2} \tan \left (d x + c\right ) + 3 \, b^{4} \tan \left (d x + c\right ) + 12 \, {\left (a^{3} b + a b^{3}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac {22 \, a^{3} b \tan \left (d x + c\right )^{3} + 22 \, a b^{3} \tan \left (d x + c\right )^{3} + 3 \, a^{4} \tan \left (d x + c\right )^{2} + 18 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} + 6 \, a^{3} b \tan \left (d x + c\right ) + a^{4}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 184, normalized size = 1.34 \[ -\frac {2 a^{4} \cot \left (d x +c \right )}{3 d}-\frac {a^{4} \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 a^{3} b}{d \sin \left (d x +c \right )^{2}}+\frac {4 a^{3} b \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {6 a^{2} b^{2}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {12 a^{2} b^{2} \cot \left (d x +c \right )}{d}+\frac {2 a \,b^{3}}{d \cos \left (d x +c \right )^{2}}+\frac {4 a \,b^{3} \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {2 b^{4} \tan \left (d x +c \right )}{3 d}+\frac {b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 120, normalized size = 0.88 \[ \frac {b^{4} \tan \left (d x + c\right )^{3} + 6 \, a b^{3} \tan \left (d x + c\right )^{2} + 12 \, {\left (a^{3} b + a b^{3}\right )} \log \left (\tan \left (d x + c\right )\right ) + 3 \, {\left (6 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right ) - \frac {6 \, a^{3} b \tan \left (d x + c\right ) + a^{4} + 3 \, {\left (a^{4} + 6 \, a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.76, size = 132, normalized size = 0.96 \[ \frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (4\,a^3\,b+4\,a\,b^3\right )}{d}-\frac {{\mathrm {cot}\left (c+d\,x\right )}^3\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^4+6\,a^2\,b^2\right )+\frac {a^4}{3}+2\,a^3\,b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (6\,a^2\,b^2+b^4\right )}{d}+\frac {b^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d}+\frac {2\,a\,b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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