Optimal. Leaf size=163 \[ -\frac {3 \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {3 \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
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Rubi [A] time = 0.26, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3575, 912, 130, 511, 510} \[ -\frac {3 \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {3 \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 912
Rule 3575
Rubi steps
\begin {align*} \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^{4/3} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {i}{2 (i-x) x^{4/3} \sqrt {a+b x}}+\frac {i}{2 x^{4/3} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {1}{(i-x) x^{4/3} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \operatorname {Subst}\left (\int \frac {1}{x^{4/3} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (i-x^3\right ) \sqrt {a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (i+x^3\right ) \sqrt {a+b x^3}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d}\\ &=\frac {\left (3 i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (i-x^3\right ) \sqrt {1+\frac {b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt {a+b \tan (c+d x)}}+\frac {\left (3 i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (i+x^3\right ) \sqrt {1+\frac {b x^3}{a}}} \, dx,x,\sqrt [3]{\tan (c+d x)}\right )}{2 d \sqrt {a+b \tan (c+d x)}}\\ &=-\frac {3 F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {3 F_1\left (-\frac {1}{3};1,\frac {1}{2};\frac {2}{3};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
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Mathematica [B] time = 52.93, size = 23355, normalized size = 143.28 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [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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maple [F] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +b \tan \left (d x +c \right )}\, \tan \left (d x +c \right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {1}{\sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{4/3}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{\frac {4}{3}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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