Optimal. Leaf size=79 \[ \frac {2 F_1\left (\frac {1}{2};1-m,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-m} \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^m}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3645, 129, 441,
440} \begin {gather*} \frac {2 \sqrt {\tan (c+d x)} (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left (\frac {1}{2};1-m,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 440
Rule 441
Rule 3645
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^m}{\sqrt {\tan (c+d x)}} \, dx &=\frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {(a+x)^{-1+m}}{\sqrt {-\frac {i x}{a}} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {\left (a+i a x^2\right )^{-1+m}}{-a^2+i a^2 x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=-\frac {\left (2 a^2 (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m\right ) \text {Subst}\left (\int \frac {\left (1+i x^2\right )^{-1+m}}{-a^2+i a^2 x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {2 F_1\left (\frac {1}{2};1-m,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right ) (1+i \tan (c+d x))^{-m} \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^m}{d}\\ \end {align*}
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Mathematica [F]
time = 5.26, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+i a \tan (c+d x))^m}{\sqrt {\tan (c+d x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 1.70, size = 0, normalized size = 0.00 \[\int \frac {\left (a +i a \tan \left (d x +c \right )\right )^{m}}{\sqrt {\tan \left (d x +c \right )}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{m}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \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 \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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