Optimal. Leaf size=64 \[ -\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \log (\tan (c+d x))}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 64, 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} \[ \frac {3 a^2 b \log (\tan (c+d x))}{d}-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d} \]
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))^3 \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {(a+x)^3}{x^2} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac {b \operatorname {Subst}\left (\int \left (3 a+\frac {a^3}{x^2}+\frac {3 a^2}{x}+x\right ) \, dx,x,b \tan (c+d x)\right )}{d}\\ &=-\frac {a^3 \cot (c+d x)}{d}+\frac {3 a^2 b \log (\tan (c+d x))}{d}+\frac {3 a b^2 \tan (c+d x)}{d}+\frac {b^3 \tan ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 126, normalized size = 1.97 \[ -\frac {\csc (c+d x) \sec ^2(c+d x) \left (\left (a^3+3 a b^2\right ) \cos (3 (c+d x))+3 a \left (a^2-b^2\right ) \cos (c+d x)-2 b \sin (c+d x) \left (3 a^2 \log (\sin (c+d x))-3 a^2 \log (\cos (c+d x))-3 a^2 \cos (2 (c+d x)) (\log (\cos (c+d x))-\log (\sin (c+d x)))+b^2\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 127, normalized size = 1.98 \[ -\frac {3 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 3 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{4}\right ) \sin \left (d x + c\right ) - 6 \, a b^{2} \cos \left (d x + c\right ) + 2 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - b^{3} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.80, size = 70, normalized size = 1.09 \[ \frac {b^{3} \tan \left (d x + c\right )^{2} + 6 \, a^{2} b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 6 \, a b^{2} \tan \left (d x + c\right ) - \frac {2 \, {\left (3 \, a^{2} b \tan \left (d x + c\right ) + a^{3}\right )}}{\tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 63, normalized size = 0.98 \[ -\frac {a^{3} \cot \left (d x +c \right )}{d}+\frac {3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \tan \left (d x +c \right )}{d}+\frac {b^{3}}{2 d \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 56, normalized size = 0.88 \[ \frac {b^{3} \tan \left (d x + c\right )^{2} + 6 \, a^{2} b \log \left (\tan \left (d x + c\right )\right ) + 6 \, a b^{2} \tan \left (d x + c\right ) - \frac {2 \, a^{3}}{\tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.66, size = 62, normalized size = 0.97 \[ \frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,d}-\frac {a^3\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {3\,a^2\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {3\,a\,b^2\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{3} \csc ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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