Optimal. Leaf size=251 \[ -\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}-\frac {\tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
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Rubi [A] time = 1.76, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3565, 3645, 3655, 6725, 63, 217, 206, 93, 205, 208} \[ -\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}-\frac {\tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (b+i a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 205
Rule 206
Rule 208
Rule 217
Rule 3565
Rule 3645
Rule 3655
Rule 6725
Rubi steps
\begin {align*} \int \frac {\tan ^{\frac {7}{2}}(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 \int \frac {\sqrt {\tan (c+d x)} \left (\frac {3 a^2}{2}-\frac {3}{2} a b \tan (c+d x)+\frac {3}{2} \left (a^2+b^2\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {4 \int \frac {\frac {3}{4} a^2 \left (a^2+3 b^2\right )-\frac {3}{2} a b^3 \tan (c+d x)+\frac {3}{4} \left (a^2+b^2\right )^2 \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \, dx}{3 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {4 \operatorname {Subst}\left (\int \frac {\frac {3}{4} a^2 \left (a^2+3 b^2\right )-\frac {3}{2} a b^3 x+\frac {3}{4} \left (a^2+b^2\right )^2 x^2}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{3 b^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {4 \operatorname {Subst}\left (\int \left (\frac {3 \left (a^2+b^2\right )^2}{4 \sqrt {x} \sqrt {a+b x}}+\frac {3 \left (b^2 \left (a^2-b^2\right )-2 a b^3 x\right )}{4 \sqrt {x} \sqrt {a+b x} \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{3 b^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{b^2 d}+\frac {\operatorname {Subst}\left (\int \frac {b^2 \left (a^2-b^2\right )-2 a b^3 x}{\sqrt {x} \sqrt {a+b x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {\tan (c+d x)}\right )}{b^2 d}+\frac {\operatorname {Subst}\left (\int \left (\frac {2 a b^3+i b^2 \left (a^2-b^2\right )}{2 (i-x) \sqrt {x} \sqrt {a+b x}}+\frac {-2 a b^3+i b^2 \left (a^2-b^2\right )}{2 \sqrt {x} (i+x) \sqrt {a+b x}}\right ) \, dx,x,\tan (c+d x)\right )}{b^2 \left (a^2+b^2\right )^2 d}\\ &=-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (i+x) \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a-i b)^2 d}+\frac {i \operatorname {Subst}\left (\int \frac {1}{(i-x) \sqrt {x} \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{2 (a+i b)^2 d}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^2 d}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {i \operatorname {Subst}\left (\int \frac {1}{i-(-a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a-i b)^2 d}+\frac {i \operatorname {Subst}\left (\int \frac {1}{i-(a+i b) x^2} \, dx,x,\frac {\sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(a+i b)^2 d}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a-b)^{5/2} d}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{b^{5/2} d}-\frac {\tanh ^{-1}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{(i a+b)^{5/2} d}-\frac {2 a^2 \tan ^{\frac {3}{2}}(c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {2 a^2 \left (a^2+3 b^2\right ) \sqrt {\tan (c+d x)}}{b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 6.23, size = 468, normalized size = 1.86 \[ \frac {2 \sqrt {a+b \tan (c+d x)} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2} d \sqrt {\frac {b \tan (c+d x)}{a}+1}}-\frac {2 \sqrt {\tan (c+d x)}}{b^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\sqrt [4]{-1} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-a+i b)^{3/2} (b+i a)}+\frac {\sqrt [4]{-1} \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d (-b+i a) (a+i b)^{3/2}}-\frac {i \tan ^{\frac {3}{2}}(c+d x)}{3 d (a-i b) (a+b \tan (c+d x))^{3/2}}+\frac {i \tan ^{\frac {3}{2}}(c+d x)}{3 d (a+i b) (a+b \tan (c+d x))^{3/2}}-\frac {2 \tan ^{\frac {3}{2}}(c+d x)}{3 b d (a+b \tan (c+d x))^{3/2}}-\frac {i \sqrt {\tan (c+d x)}}{d (-b+i a) (a+i b) \sqrt {a+b \tan (c+d x)}}-\frac {i \sqrt {\tan (c+d x)}}{d (a-i b) (b+i a) \sqrt {a+b \tan (c+d x)}} \]
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 = 1.29, size = 1490358, normalized size = 5937.68 \[ \text {output too large to display} \]
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 \[ \int \frac {\tan \left (d x + c\right )^{\frac {7}{2}}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^{7/2}}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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