Optimal. Leaf size=91 \[ \frac {\sqrt {1+i \tan (e+f x)} F_1\left (n+1;\frac {5}{2},1;n+2;-i \tan (e+f x),i \tan (e+f x)\right ) (d \tan (e+f x))^{n+1}}{a d f (n+1) \sqrt {a+i a \tan (e+f x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3564, 135, 133} \[ \frac {\sqrt {1+i \tan (e+f x)} F_1\left (n+1;\frac {5}{2},1;n+2;-i \tan (e+f x),i \tan (e+f x)\right ) (d \tan (e+f x))^{n+1}}{a d f (n+1) \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 3564
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n}{(a+x)^{5/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {\left (i \sqrt {1+i \tan (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i d x}{a}\right )^n}{\left (1+\frac {x}{a}\right )^{5/2} \left (-a^2+a x\right )} \, dx,x,i a \tan (e+f x)\right )}{f \sqrt {a+i a \tan (e+f x)}}\\ &=\frac {F_1\left (1+n;\frac {5}{2},1;2+n;-i \tan (e+f x),i \tan (e+f x)\right ) \sqrt {1+i \tan (e+f x)} (d \tan (e+f x))^{1+n}}{a d f (1+n) \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [F] time = 34.45, size = 0, normalized size = 0.00 \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2} \left (\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{4 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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