Optimal. Leaf size=314 \[ \frac {2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}-\frac {\sqrt {-b+i a} (-B+i A) \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (7 a B+A b) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}-\frac {\sqrt {b+i a} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 1.34, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3608, 3649, 3616, 3615, 93, 203, 206} \[ \frac {2 \left (35 a^2 A-7 a b B+4 A b^2\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b+105 a^3 B+14 a b^2 B-8 A b^3\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}-\frac {\sqrt {-b+i a} (-B+i A) \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 (7 a B+A b) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}-\frac {\sqrt {b+i a} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 93
Rule 203
Rule 206
Rule 3608
Rule 3615
Rule 3616
Rule 3649
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x))}{\tan ^{\frac {9}{2}}(c+d x)} \, dx &=-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int \frac {\frac {1}{2} (-A b-7 a B)+\frac {7}{2} (a A-b B) \tan (c+d x)+3 A b \tan ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {4 \int \frac {\frac {1}{4} \left (-35 a^2 A-4 A b^2+7 a b B\right )-\frac {35}{4} a (A b+a B) \tan (c+d x)-b (A b+7 a B) \tan ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{35 a}\\ &=-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {\frac {1}{8} \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right )-\frac {105}{8} a^2 (a A-b B) \tan (c+d x)-\frac {1}{4} b \left (35 a^2 A+4 A b^2-7 a b B\right ) \tan ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^2}\\ &=-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {16 \int \frac {\frac {105}{16} a^3 (a A-b B)+\frac {105}{16} a^3 (A b+a B) \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{105 a^3}\\ &=-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {1}{2} ((a-i b) (A-i 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} ((a+i b) (A+i B)) \int \frac {1-i \tan (c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {2 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {((a-i b) (A-i B)) \operatorname {Subst}\left (\int \frac {1}{(1-i x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {((a+i b) (A+i B)) \operatorname {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 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}+\frac {((a-i b) (A-i B)) \operatorname {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 {((a+i b) (A+i B)) \operatorname {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} (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} (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 A \sqrt {a+b \tan (c+d x)}}{7 d \tan ^{\frac {7}{2}}(c+d x)}-\frac {2 (A b+7 a B) \sqrt {a+b \tan (c+d x)}}{35 a d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (35 a^2 A+4 A b^2-7 a b B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^2 d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (35 a^2 A b-8 A b^3+105 a^3 B+14 a b^2 B\right ) \sqrt {a+b \tan (c+d x)}}{105 a^3 d \sqrt {\tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 3.98, size = 265, normalized size = 0.84 \[ \frac {\frac {2 \sqrt {a+b \tan (c+d x)} \left (-15 a^3 A+a \left (35 a^2 A-7 a b B+4 A b^2\right ) \tan ^2(c+d x)-3 a^2 (7 a B+A b) \tan (c+d x)+\left (105 a^3 B+35 a^2 A b+14 a b^2 B-8 A b^3\right ) \tan ^3(c+d x)\right )}{a^3 \tan ^{\frac {7}{2}}(c+d x)}+105 \sqrt [4]{-1} \sqrt {-a+i b} (B+i A) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-105 (-1)^{3/4} \sqrt {a+i b} (A+i B) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{105 d} \]
Antiderivative was successfully verified.
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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 [B] time = 1.18, size = 2184224, normalized size = 6956.13 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{{\mathrm {tan}\left (c+d\,x\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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