Optimal. Leaf size=583 \[ \frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {3 a d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 b^2 f \left (a^2+b^2\right )}+\frac {3 a d^2 (d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{4 b^2 f \left (a^2+b^2\right ) \sec ^2(e+f x)^{3/4}}+\frac {3 d^2 \left (a^2+2 b^2\right ) (d \sec (e+f x))^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 b^{5/2} f \left (a^2+b^2\right )^{5/4} \sec ^2(e+f x)^{3/4}}-\frac {3 d^2 \left (a^2+2 b^2\right ) (d \sec (e+f x))^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 b^{5/2} f \left (a^2+b^2\right )^{5/4} \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 \left (a^2+2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b^3 f \left (a^2+b^2\right )^{3/2} \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 \left (a^2+2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b^3 f \left (a^2+b^2\right )^{3/2} \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2} \]
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Rubi [A] time = 0.52, antiderivative size = 583, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3512, 733, 835, 844, 227, 196, 746, 399, 490, 1213, 537, 444, 63, 298, 205, 208} \[ \frac {3 d^2 \left (a^2+2 b^2\right ) (d \sec (e+f x))^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 b^{5/2} f \left (a^2+b^2\right )^{5/4} \sec ^2(e+f x)^{3/4}}-\frac {3 d^2 \left (a^2+2 b^2\right ) (d \sec (e+f x))^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right )}{8 b^{5/2} f \left (a^2+b^2\right )^{5/4} \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {3 a d^2 \sin (e+f x) \cos (e+f x) (d \sec (e+f x))^{3/2}}{4 b^2 f \left (a^2+b^2\right )}+\frac {3 a d^2 (d \sec (e+f x))^{3/2} E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right )}{4 b^2 f \left (a^2+b^2\right ) \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 \left (a^2+2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b^3 f \left (a^2+b^2\right )^{3/2} \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 \left (a^2+2 b^2\right ) \sqrt {-\tan ^2(e+f x)} \cot (e+f x) (d \sec (e+f x))^{3/2} \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right )}{8 b^3 f \left (a^2+b^2\right )^{3/2} \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 196
Rule 205
Rule 208
Rule 227
Rule 298
Rule 399
Rule 444
Rule 490
Rule 537
Rule 733
Rule 746
Rule 835
Rule 844
Rule 1213
Rule 3512
Rubi steps
\begin {align*} \int \frac {(d \sec (e+f x))^{7/2}}{(a+b \tan (e+f x))^3} \, dx &=\frac {\left (d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^{3/4}}{(a+x)^3} \, dx,x,b \tan (e+f x)\right )}{b f \sec ^2(e+f x)^{3/4}}\\ &=-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {\left (3 d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x}{(a+x)^2 \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{4 b^3 f \sec ^2(e+f x)^{3/4}}\\ &=-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\left (3 d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {-1+\frac {a x}{2 b^2}}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{4 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\left (3 a d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b^3 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 \left (-2-\frac {a^2}{b^2}\right ) d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=-\frac {3 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 b^2 \left (a^2+b^2\right ) f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\left (3 a d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{5/4}} \, dx,x,b \tan (e+f x)\right )}{8 b^3 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (3 \left (-2-\frac {a^2}{b^2}\right ) d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a \left (-2-\frac {a^2}{b^2}\right ) d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-x^2\right ) \sqrt [4]{1+\frac {x^2}{b^2}}} \, dx,x,b \tan (e+f x)\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=\frac {3 a d^2 E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 b^2 \left (a^2+b^2\right ) f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\left (3 \left (-2-\frac {a^2}{b^2}\right ) d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a^2-x\right ) \sqrt [4]{1+\frac {x}{b^2}}} \, dx,x,b^2 \tan ^2(e+f x)\right )}{16 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a \left (-2-\frac {a^2}{b^2}\right ) d^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4} \left (1+\frac {a^2}{b^2}-x^4\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{4 b^2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=\frac {3 a d^2 E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 b^2 \left (a^2+b^2\right ) f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\left (3 \left (-2-\frac {a^2}{b^2}\right ) b d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{a^2+b^2-b^2 x^4} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{4 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a \left (-2-\frac {a^2}{b^2}\right ) d^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}-b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (3 a \left (-2-\frac {a^2}{b^2}\right ) d^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\sqrt {a^2+b^2}+b x^2\right ) \sqrt {1-x^4}} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=\frac {3 a d^2 E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 b^2 \left (a^2+b^2\right ) f}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\left (3 \left (-2-\frac {a^2}{b^2}\right ) d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 \left (-2-\frac {a^2}{b^2}\right ) d^2 (d \sec (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+b x^2} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {\left (3 a \left (-2-\frac {a^2}{b^2}\right ) d^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}-b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}+\frac {\left (3 a \left (-2-\frac {a^2}{b^2}\right ) d^2 \cot (e+f x) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (\sqrt {a^2+b^2}+b x^2\right )} \, dx,x,\sqrt [4]{\sec ^2(e+f x)}\right )}{8 b \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}\\ &=\frac {3 \left (a^2+2 b^2\right ) d^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 b^{5/2} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}-\frac {3 \left (a^2+2 b^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [4]{\sec ^2(e+f x)}}{\sqrt [4]{a^2+b^2}}\right ) (d \sec (e+f x))^{3/2}}{8 b^{5/2} \left (a^2+b^2\right )^{5/4} f \sec ^2(e+f x)^{3/4}}+\frac {3 a d^2 E\left (\left .\frac {1}{2} \tan ^{-1}(\tan (e+f x))\right |2\right ) (d \sec (e+f x))^{3/2}}{4 b^2 \left (a^2+b^2\right ) f \sec ^2(e+f x)^{3/4}}-\frac {3 a d^2 \cos (e+f x) (d \sec (e+f x))^{3/2} \sin (e+f x)}{4 b^2 \left (a^2+b^2\right ) f}-\frac {3 a \left (a^2+2 b^2\right ) d^2 \cot (e+f x) \Pi \left (-\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 b^3 \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}+\frac {3 a \left (a^2+2 b^2\right ) d^2 \cot (e+f x) \Pi \left (\frac {b}{\sqrt {a^2+b^2}};\left .\sin ^{-1}\left (\sqrt [4]{\sec ^2(e+f x)}\right )\right |-1\right ) (d \sec (e+f x))^{3/2} \sqrt {-\tan ^2(e+f x)}}{8 b^3 \left (a^2+b^2\right )^{3/2} f \sec ^2(e+f x)^{3/4}}-\frac {d^2 (d \sec (e+f x))^{3/2}}{2 b f (a+b \tan (e+f x))^2}+\frac {3 a d^2 (d \sec (e+f x))^{3/2}}{4 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}
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Mathematica [C] time = 29.50, size = 14225, normalized size = 24.40 \[ \text {Result too large to show} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}}{{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 7.56, size = 101372, normalized size = 173.88 \[ \text {output too large to display} \]
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 \[ \int \frac {{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \]
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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