Optimal. Leaf size=79 \[ -\frac {2 \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left (-\frac {1}{2};1-n,1;\frac {1}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d} \]
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Rubi [A] time = 0.17, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4241, 3564, 130, 511, 510} \[ -\frac {2 \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n F_1\left (-\frac {1}{2};1-n,1;\frac {1}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 3564
Rule 4241
Rubi steps
\begin {align*} \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^n \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^n}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {\left (i a^2 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+n}}{\left (-\frac {i x}{a}\right )^{3/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (2 a^3 \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\left (a+i a x^2\right )^{-1+n}}{x^2 \left (-a^2+i a^2 x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {\left (2 a^2 \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^n\right ) \operatorname {Subst}\left (\int \frac {\left (1+i x^2\right )^{-1+n}}{x^2 \left (-a^2+i a^2 x^2\right )} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {2 F_1\left (-\frac {1}{2};1-n,1;\frac {1}{2};-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {\cot (c+d x)} (1+i \tan (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d}\\ \end {align*}
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Mathematica [F] time = 11.37, size = 0, normalized size = 0.00 \[ \int \cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.26, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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