Optimal. Leaf size=79 \[ \frac {2 \sqrt {\tan (c+d x)} (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left (\frac {1}{2};1-m,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d} \]
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Rubi [A] time = 0.10, 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 = {3564, 130, 430, 429} \[ \frac {2 \sqrt {\tan (c+d x)} (1+i \tan (c+d x))^{-m} (a+i a \tan (c+d x))^m F_1\left (\frac {1}{2};1-m,1;\frac {3}{2};-i \tan (c+d x),i \tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 430
Rule 3564
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 5.45, size = 0, normalized size = 0.00 \[ \int \frac {(a+i a \tan (c+d x))^m}{\sqrt {\tan (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}, x\right ) \]
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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m}}{\sqrt {\tan \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.10, 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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{m}}{\sqrt {\tan \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^m}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{m}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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