Optimal. Leaf size=221 \[ \frac {\sqrt {i a-b} \text {ArcTan}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\sqrt {i a+b} \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}} \]
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Rubi [A]
time = 0.54, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3649, 3730,
3697, 3696, 95, 209, 212} \begin {gather*} \frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}}+\frac {\sqrt {-b+i a} \text {ArcTan}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {\sqrt {b+i a} \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 95
Rule 209
Rule 212
Rule 3649
Rule 3696
Rule 3697
Rule 3730
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \tan (c+d x)}}{\tan ^{\frac {7}{2}}(c+d x)} \, dx &=-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int \frac {-\frac {b}{2}+\frac {5}{2} a \tan (c+d x)+2 b \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} \left (-15 a^2-2 b^2\right )-\frac {15}{4} a b \tan (c+d x)-\frac {1}{2} b^2 \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{15 a}\\ &=-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}}-\frac {8 \int \frac {\frac {15 a^2 b}{8}-\frac {15}{8} a^3 \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{15 a^2}\\ &=-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}}-\frac {1}{2} (-i a+b) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx-\frac {1}{2} (i a+b) \int \frac {1+i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}}+\frac {(i a-b) \text {Subst}\left (\int \frac {1}{(1+i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}-\frac {(i a+b) \text {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}}+\frac {(i a-b) \text {Subst}\left (\int \frac {1}{1-(-i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {(i a+b) \text {Subst}\left (\int \frac {1}{1-(i a+b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\\ &=\frac {\sqrt {i a-b} \tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {\sqrt {i a+b} \tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 \sqrt {a+b \tan (c+d x)}}{5 d \tan ^{\frac {5}{2}}(c+d x)}-\frac {2 b \sqrt {a+b \tan (c+d x)}}{15 a d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (15 a^2+2 b^2\right ) \sqrt {a+b \tan (c+d x)}}{15 a^2 d \sqrt {\tan (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.50, size = 197, normalized size = 0.89 \begin {gather*} \frac {15 \sqrt [4]{-1} \sqrt {-a+i b} \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+15 \sqrt [4]{-1} \sqrt {a+i b} \text {ArcTan}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )+\frac {2 \sqrt {a+b \tan (c+d x)} \left (-3 a^2-a b \tan (c+d x)+\left (15 a^2+2 b^2\right ) \tan ^2(c+d x)\right )}{a^2 \tan ^{\frac {5}{2}}(c+d x)}}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 0.40, size = 1092009, normalized size = 4941.22 \[\text {output too large to display}\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {\sqrt {a + b \tan {\left (c + d x \right )}}}{\tan ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{{\mathrm {tan}\left (c+d\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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