Optimal. Leaf size=155 \[ -\frac {F_1\left (-\frac {1}{2};1,-n;\frac {1}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}-\frac {F_1\left (-\frac {1}{2};1,-n;\frac {1}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d} \]
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Rubi [A]
time = 0.20, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4326, 3656,
926, 129, 525, 524} \begin {gather*} -\frac {\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac {1}{2};1,-n;\frac {1}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d}-\frac {\sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (-\frac {1}{2};1,-n;\frac {1}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 524
Rule 525
Rule 926
Rule 3656
Rule 4326
Rubi steps
\begin {align*} \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+b \tan (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{x^{3/2} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {i (a+b x)^n}{2 (i-x) x^{3/2}}+\frac {i (a+b x)^n}{2 x^{3/2} (i+x)}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{(i-x) x^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+b x)^n}{x^{3/2} (i+x)} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^n}{x^2 \left (i-x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^n}{x^2 \left (i+x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^n}{x^2 \left (i-x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (i \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^n}{x^2 \left (i+x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {F_1\left (-\frac {1}{2};1,-n;\frac {1}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}-\frac {F_1\left (-\frac {1}{2};1,-n;\frac {1}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\cot (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}\\ \end {align*}
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Mathematica [F]
time = 3.92, size = 0, normalized size = 0.00 \begin {gather*} \int \cot ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))^n \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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